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Lie algebra E16
Semisimple complex Lie subalgebras
Up to linear equivalence, there are total 119 semisimple subalgebras (including the full subalgebra). The subalgebras are ordered by rank, Dynkin indices of simple constituents and dimensions of simple constituents.
The upper index indicates the Dynkin index, the lower index indicates the rank of the subalgebra.
Generation comments.
Computation time in seconds: 14146.8.
1885412174 total arithmetic operations performed = 1569905347 additions and 315506827 multiplications.
The base field over which the subalgebras were realized is: Q[√−1,√3,√7]
Number of root subalgebras other than the Cartan and full subalgebra: 19
Number of sl(2)'s: 20
Subalgebra A11 ↪ E16
1 out of 119
Subalgebra type: A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Centralizer: A15 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A11
Basis of Cartan of centralizer: 5 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: 2A11 , A21+A11 , A31+A11 , A41+A11 , A51+A11 , A81+A11 , A101+A11 , A111+A11 , A201+A11 , A351+A11 , 3A11 , A21+2A11 , 2A21+A11 , A31+A21+A11 , A41+2A11 , 2A41+A11 , A81+A31+A11 , A101+2A11 , A12+A11 , B12+A11 , A22+A11 , A22+A11 , B22+A11 , A52+A11 , 4A11 , 2A21+2A11 , A12+2A11 , A12+A41+A11 , B12+2A11 , A22+A31+A11 , A13+A11 , C13+A11 , A23+A11 , 2A12+A11 , A13+2A11 , A14+A11 , A15+A11 .
Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2, 2, 3, 2, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−36
Positive simple generators: g36
Cartan symmetric matrix: (2)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2)
Decomposition of ambient Lie algebra: V2ω1⊕20Vω1⊕35V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ψ1+2ψ5⊕Vω1+2ψ1−2ψ3+2ψ5⊕Vω1+2ψ1−2ψ2+2ψ3−2ψ4+2ψ5⊕Vω1+2ψ2−2ψ4+2ψ5⊕Vω1+2ψ1−2ψ2+2ψ4⊕Vω1+2ψ2−2ψ3+2ψ4⊕Vω1+2ψ3⊕V−2ψ4+4ψ5⊕V−2ψ3+2ψ4+2ψ5⊕V−2ψ2+2ψ3+2ψ5⊕V−2ψ1+2ψ2+2ψ5⊕V2ω1⊕V4ψ1−2ψ2⊕V2ψ1+2ψ2−2ψ3⊕V2ψ1+2ψ3−2ψ4⊕V2ψ1+2ψ4−2ψ5⊕Vω1−2ψ2+2ψ5⊕Vω1−2ψ1+2ψ2−2ψ3+2ψ5⊕Vω1−2ψ1+2ψ3−2ψ4+2ψ5⊕Vω1−2ψ1+2ψ4⊕Vω1+2ψ1−2ψ4⊕Vω1+2ψ1−2ψ3+2ψ4−2ψ5⊕Vω1+2ψ1−2ψ2+2ψ3−2ψ5⊕Vω1+2ψ2−2ψ5⊕V2ψ1−2ψ2−2ψ4+2ψ5⊕V2ψ2−2ψ3−2ψ4+2ψ5⊕V2ψ3−4ψ4+2ψ5⊕V2ψ1−2ψ2−2ψ3+2ψ4⊕V2ψ2−4ψ3+2ψ4⊕V2ψ1−4ψ2+2ψ3⊕5V0⊕V−2ψ1+4ψ2−2ψ3⊕V−2ψ2+4ψ3−2ψ4⊕V−2ψ1+2ψ2+2ψ3−2ψ4⊕V−2ψ3+4ψ4−2ψ5⊕V−2ψ2+2ψ3+2ψ4−2ψ5⊕V−2ψ1+2ψ2+2ψ4−2ψ5⊕Vω1−2ψ3⊕Vω1−2ψ2+2ψ3−2ψ4⊕Vω1−2ψ1+2ψ2−2ψ4⊕Vω1−2ψ2+2ψ4−2ψ5⊕Vω1−2ψ1+2ψ2−2ψ3+2ψ4−2ψ5⊕Vω1−2ψ1+2ψ3−2ψ5⊕V−2ψ1−2ψ4+2ψ5⊕V−2ψ1−2ψ3+2ψ4⊕V−2ψ1−2ψ2+2ψ3⊕V−4ψ1+2ψ2⊕V2ψ1−2ψ2−2ψ5⊕V2ψ2−2ψ3−2ψ5⊕V2ψ3−2ψ4−2ψ5⊕V2ψ4−4ψ5⊕V−2ψ1−2ψ5
Made total 282 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A21 ↪ E16
2 out of 119
Subalgebra type: A21 (click on type for detailed printout).
Centralizer: B13 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A21
Basis of Cartan of centralizer: 4 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A21+A11 , 2A21 , A31+A21 , A41+A21 , A101+A21 , A281+A21 , A21+2A11 , 2A21+A11 , 3A21 , A31+A21+A11 , A12+A21 , B12+A21 , G12+A21 , 2A21+2A11 , A13+A21 , B13+A21 .
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 3, 4, 3, 2): 4
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−30+g−34
Positive simple generators: g34+g30
Cartan symmetric matrix: (1)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4)
Decomposition of ambient Lie algebra: 8V2ω1⊕16Vω1⊕22V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ψ2+2ψ3+2ψ4⊕Vω1+2ψ1+2ψ3+2ψ4⊕V2ψ1+2ψ2+4ψ3−2ψ4⊕V2ω1+2ψ2+4ψ3−2ψ4⊕Vω1−2ψ1+2ψ2+2ψ3+2ψ4⊕Vω1+2ψ3+2ψ4⊕V2ω1+2ψ1⊕V−2ψ1+4ψ2+4ψ3−2ψ4⊕V2ψ2+4ψ3−2ψ4⊕V2ψ1+4ψ3−2ψ4⊕Vω1−2ψ3+4ψ4⊕Vω1+2ψ1−2ψ2−2ψ3+4ψ4⊕V2ψ2⊕V2ω1−2ψ1+2ψ2⊕V2ψ1⊕2V2ω1⊕V4ψ1−2ψ2⊕V2ω1+2ψ1−2ψ2⊕V−2ψ1+2ψ2+4ψ3−2ψ4⊕Vω1−2ψ1−2ψ3+4ψ4⊕Vω1−2ψ2−2ψ3+4ψ4⊕Vω1+2ψ2+2ψ3−4ψ4⊕Vω1+2ψ1+2ψ3−4ψ4⊕V−2ψ1+2ψ2⊕4V0⊕V2ω1−2ψ1⊕V2ψ1−2ψ2⊕Vω1−2ψ1+2ψ2+2ψ3−4ψ4⊕Vω1+2ψ3−4ψ4⊕V2ψ1−2ψ2−4ψ3+2ψ4⊕V2ω1−2ψ2−4ψ3+2ψ4⊕V−4ψ1+2ψ2⊕V−2ψ1⊕V−2ψ2⊕Vω1−2ψ3−2ψ4⊕Vω1+2ψ1−2ψ2−2ψ3−2ψ4⊕V−2ψ1−4ψ3+2ψ4⊕V−2ψ2−4ψ3+2ψ4⊕V2ψ1−4ψ2−4ψ3+2ψ4⊕Vω1−2ψ1−2ψ3−2ψ4⊕Vω1−2ψ2−2ψ3−2ψ4⊕V−2ψ1−2ψ2−4ψ3+2ψ4
Made total 6531159 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A31 ↪ E16
3 out of 119
Subalgebra type: A31 (click on type for detailed printout).
Centralizer: A22+A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A31
Basis of Cartan of centralizer: 3 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A31+A11 , A31+A21 , 2A31 , A81+A31 , A91+A31 , A31+A21+A11 , A81+A31+A11 , A22+A31 , A22+A31+A11 .
Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4, 6, 4, 2): 6
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−23+g−30+g−34
Positive simple generators: g34+g30+g23
Cartan symmetric matrix: (2/3)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6)
Decomposition of ambient Lie algebra: 2V3ω1⊕9V2ω1⊕16Vω1⊕11V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ψ1+2ψ2+2ψ3⊕V2ω1+2ψ2+2ψ3⊕Vω1+2ψ1−2ψ2+4ψ3⊕V3ω1+2ψ1⊕Vω1+2ψ1+4ψ2−2ψ3⊕V2ω1−2ψ2+4ψ3⊕V2ψ2+2ψ3⊕V4ψ1⊕V2ω1+4ψ2−2ψ3⊕Vω1−2ψ1+2ψ2+2ψ3⊕2Vω1+2ψ1⊕V−2ψ2+4ψ3⊕3V2ω1⊕V4ψ2−2ψ3⊕Vω1−2ψ1−2ψ2+4ψ3⊕Vω1+2ψ1−4ψ2+2ψ3⊕V3ω1−2ψ1⊕Vω1−2ψ1+4ψ2−2ψ3⊕Vω1+2ψ1+2ψ2−4ψ3⊕V2ω1−4ψ2+2ψ3⊕3V0⊕V2ω1+2ψ2−4ψ3⊕2Vω1−2ψ1⊕Vω1+2ψ1−2ψ2−2ψ3⊕V−4ψ2+2ψ3⊕V2ω1−2ψ2−2ψ3⊕V2ψ2−4ψ3⊕Vω1−2ψ1−4ψ2+2ψ3⊕Vω1−2ψ1+2ψ2−4ψ3⊕V−4ψ1⊕V−2ψ2−2ψ3⊕Vω1−2ψ1−2ψ2−2ψ3
Made total 5853604 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A41 ↪ E16
4 out of 119
Subalgebra type: A41 (click on type for detailed printout).
Centralizer: 2A12 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A12
Basis of Cartan of centralizer: 4 vectors:
(0, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A41+A11 , A41+A21 , 2A41 , A51+A41 , A81+A41 , A41+2A11 , 2A41+A11 , 3A41 , A12+A41 , A22+A41 , A22+A41 , A12+A41+A11 , A12+2A41 , 2A12+A41 .
Elements Cartan subalgebra scaled to act by two by components: A41: (2, 4, 4, 6, 4, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−23+g−27
Positive simple generators: 2g27+2g23
Cartan symmetric matrix: (1/2)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (8)
Decomposition of ambient Lie algebra: V4ω1⊕19V2ω1⊕16V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ψ2+2ψ3+4ψ4⊕V2ω1+2ψ3+4ψ4⊕V2ω1−2ψ1+4ψ2+4ψ4⊕V2ω1−2ψ1+2ψ2+2ψ3+2ψ4⊕V2ω1+2ψ1+2ψ4⊕V−2ψ1+2ψ2+4ψ4⊕V−2ψ2+4ψ3+2ψ4⊕V2ω1−2ψ1+2ψ3+2ψ4⊕V4ψ2−2ψ3+2ψ4⊕V2ω1+2ψ2−2ψ3+2ψ4⊕V2ω1+2ψ1−2ψ2+2ψ3⊕V2ω1+2ψ2⊕V4ω1⊕V−4ψ1+4ψ2+2ψ4⊕V2ψ1−2ψ2+2ψ4⊕V2ω1+2ψ1−4ψ2+2ψ3⊕V2ω1⊕V2ω1−2ψ1+4ψ2−2ψ3⊕4V0⊕V2ω1−2ψ2⊕V2ω1−2ψ1+2ψ2−2ψ3⊕V2ω1−2ψ2+2ψ3−2ψ4⊕V2ω1+2ψ1−2ψ3−2ψ4⊕V−2ψ1+2ψ2−2ψ4⊕V2ω1−2ψ1−2ψ4⊕V4ψ1−4ψ2−2ψ4⊕V2ω1+2ψ1−2ψ2−2ψ3−2ψ4⊕V−4ψ2+2ψ3−2ψ4⊕V2ψ2−4ψ3−2ψ4⊕V2ψ1−2ψ2−4ψ4⊕V2ω1+2ψ1−4ψ2−4ψ4⊕V2ω1−2ψ3−4ψ4⊕V−2ψ2−2ψ3−4ψ4
Made total 45203957 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A51 ↪ E16
5 out of 119
Subalgebra type: A51 (click on type for detailed printout).
Centralizer: A12 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: 2A12
Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0), (1, 0, 0, 2, 0, -1)
Contained up to conjugation as a direct summand of: A51+A11 , A51+A41 , A12+A51 .
Elements Cartan subalgebra scaled to act by two by components: A51: (3, 4, 5, 7, 5, 3): 10
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−22+g−24+g−28
Positive simple generators: 2g28+g24+2g22
Cartan symmetric matrix: (2/5)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (10)
Decomposition of ambient Lie algebra: V4ω1⊕6V3ω1⊕8V2ω1⊕8Vω1⊕9V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω1−2ψ1+2ψ2+8ψ3⊕V3ω1+2ψ2+4ψ3⊕V−2ψ1+4ψ2+6ψ3⊕Vω1+6ψ3⊕Vω1+2ψ2+4ψ3⊕V2ω1+2ψ1+2ψ3⊕V3ω1−2ψ1+4ψ3⊕V3ω1−2ψ1+2ψ2+2ψ3⊕V−4ψ1+2ψ2+6ψ3⊕V2ψ1+2ψ2⊕V4ω1⊕Vω1−2ψ1+4ψ3⊕Vω1−2ψ1+2ψ2+2ψ3⊕V2ω1−2ψ2+2ψ3⊕2V2ω1⊕V2ω1+2ψ2−2ψ3⊕V3ω1+2ψ1−2ψ2−2ψ3⊕V3ω1+2ψ1−4ψ3⊕3V0⊕Vω1+2ψ1−2ψ2−2ψ3⊕Vω1+2ψ1−4ψ3⊕V2ω1−2ψ1−2ψ3⊕V3ω1−2ψ2−4ψ3⊕V−2ψ1−2ψ2⊕V4ψ1−2ψ2−6ψ3⊕Vω1−2ψ2−4ψ3⊕Vω1−6ψ3⊕V2ω1+2ψ1−2ψ2−8ψ3⊕V2ψ1−4ψ2−6ψ3
Made total 9538973 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A61 ↪ E16
6 out of 119
Subalgebra type: A61 (click on type for detailed printout).
Centralizer: A61 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A61
Basis of Cartan of centralizer: 2 vectors:
(1, -1, 1, -1, -1, 0), (1, -2, 0, -2, 0, 1)
Contained up to conjugation as a direct summand of: 2A61 .
Elements Cartan subalgebra scaled to act by two by components: A61: (3, 4, 6, 8, 6, 3): 12
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−15+g−22+g−25+g−30
Positive simple generators: 2g30+g25+g22+2g15
Cartan symmetric matrix: (1/3)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (12)
Decomposition of ambient Lie algebra: 3V4ω1⊕4V3ω1⊕9V2ω1⊕8Vω1⊕4V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V2ω1+4ψ1+8ψ2⊕Vω1+6ψ1+6ψ2⊕V4ω1+2ψ1+4ψ2⊕V3ω1+4ψ1+2ψ2⊕2V2ω1+2ψ1+4ψ2⊕Vω1+6ψ2⊕Vω1+4ψ1+2ψ2⊕V2ψ1+4ψ2⊕V4ω1⊕V3ω1−2ψ1+2ψ2⊕V3ω1+2ψ1−2ψ2⊕3V2ω1⊕Vω1−2ψ1+2ψ2⊕Vω1+2ψ1−2ψ2⊕2V0⊕V4ω1−2ψ1−4ψ2⊕V3ω1−4ψ1−2ψ2⊕2V2ω1−2ψ1−4ψ2⊕Vω1−4ψ1−2ψ2⊕Vω1−6ψ2⊕V−2ψ1−4ψ2⊕V2ω1−4ψ1−8ψ2⊕Vω1−6ψ1−6ψ2
Made total 220476270 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A81 ↪ E16
7 out of 119
Subalgebra type: A81 (click on type for detailed printout).
Centralizer: G12 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A22
Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 1, 1, 0)
Contained up to conjugation as a direct summand of: A81+A11 , A81+A31 , A81+A41 , A281+A81 , A81+A31+A11 , A12+A81 , G12+A81 .
Elements Cartan subalgebra scaled to act by two by components: A81: (4, 4, 6, 8, 6, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−7+g−11+g−26+g−28
Positive simple generators: 2g28+2g26+2g11+2g7
Cartan symmetric matrix: (1/4)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (16)
Decomposition of ambient Lie algebra: 8V4ω1⊕8V2ω1⊕14V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+2ψ2⊕V4ω1+2ψ1⊕V2ψ1+2ψ2⊕V2ω1+2ψ2⊕V4ω1−2ψ1+2ψ2⊕V2ω1+2ψ1⊕2V4ω1⊕V4ω1+2ψ1−2ψ2⊕V−2ψ1+4ψ2⊕V2ψ2⊕V2ω1−2ψ1+2ψ2⊕V2ψ1⊕2V2ω1⊕V4ω1−2ψ1⊕V4ψ1−2ψ2⊕V2ω1+2ψ1−2ψ2⊕V4ω1−2ψ2⊕V−2ψ1+2ψ2⊕2V0⊕V2ω1−2ψ1⊕V2ψ1−2ψ2⊕V2ω1−2ψ2⊕V−4ψ1+2ψ2⊕V−2ψ1⊕V−2ψ2⊕V2ψ1−4ψ2⊕V−2ψ1−2ψ2
Made total 22819045 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A91 ↪ E16
8 out of 119
Subalgebra type: A91 (click on type for detailed printout).
Centralizer: A31 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A22+A11
Basis of Cartan of centralizer: 1 vectors:
(0, 1, -1, 0, -1, 0)
Contained up to conjugation as a direct summand of: A91+A31 .
Elements Cartan subalgebra scaled to act by two by components: A91: (4, 5, 7, 10, 7, 4): 18
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−12+g−16+g−22+g−23+g−25
Positive simple generators: 2g25+g23+2g22+2g16+2g12
Cartan symmetric matrix: (2/9)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (18)
Decomposition of ambient Lie algebra: 2V5ω1⊕4V4ω1⊕4V3ω1⊕5V2ω1⊕6Vω1⊕3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+4ψ⊕Vω1+6ψ⊕V5ω1+2ψ⊕V2ω1+4ψ⊕2V3ω1+2ψ⊕V4ψ⊕2V4ω1⊕2Vω1+2ψ⊕V5ω1−2ψ⊕3V2ω1⊕2V3ω1−2ψ⊕V0⊕V4ω1−4ψ⊕2Vω1−2ψ⊕V2ω1−4ψ⊕V−4ψ⊕Vω1−6ψ
Made total 3644384 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A101 ↪ E16
9 out of 119
Subalgebra type: A101 (click on type for detailed printout).
Centralizer: B12 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B12
Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A101+A11 , A101+A21 , 2A101 , A101+2A11 , B12+A101 .
Elements Cartan subalgebra scaled to act by two by components: A101: (4, 6, 7, 10, 7, 4): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−8+g−19+g−24
Positive simple generators: 4g24+3g19+3g8
Cartan symmetric matrix: (1/5)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (20)
Decomposition of ambient Lie algebra: V6ω1⊕5V4ω1⊕8V3ω1⊕V2ω1⊕11V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V3ω1+2ψ1+2ψ2+2ψ3⊕V4ω1+2ψ1+4ψ2−2ψ3⊕V3ω1+2ψ2+2ψ3⊕V4ω1+2ψ1⊕V6ω1⊕V4ψ1+4ψ2−2ψ3⊕V3ω1−2ψ2+4ψ3⊕V4ω1⊕V2ψ1+4ψ2−2ψ3⊕V3ω1−2ψ1−2ψ2+4ψ3⊕V3ω1+2ψ1+2ψ2−4ψ3⊕V2ψ1⊕V2ω1⊕V4ω1−2ψ1⊕V4ψ2−2ψ3⊕V3ω1+2ψ2−4ψ3⊕V4ω1−2ψ1−4ψ2+2ψ3⊕3V0⊕V3ω1−2ψ2−2ψ3⊕V−4ψ2+2ψ3⊕V−2ψ1⊕V3ω1−2ψ1−2ψ2−2ψ3⊕V−2ψ1−4ψ2+2ψ3⊕V−4ψ1−4ψ2+2ψ3
Made total 4613803 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A111 ↪ E16
10 out of 119
Subalgebra type: A111 (click on type for detailed printout).
Centralizer: A11 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2)
Contained up to conjugation as a direct summand of: A111+A11 .
Elements Cartan subalgebra scaled to act by two by components: A111: (4, 6, 8, 11, 8, 4): 22
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−13+g−14+g−15+g−24
Positive simple generators: 4g24+g15+3g14+3g13
Cartan symmetric matrix: (2/11)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (22)
Decomposition of ambient Lie algebra: V6ω1⊕2V5ω1⊕3V4ω1⊕6V3ω1⊕4V2ω1⊕2Vω1⊕4V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V3ω1+2ψ1+6ψ2⊕V4ω1+6ψ2⊕V2ω1+6ψ2⊕V3ω1−2ψ1+6ψ2⊕V5ω1+2ψ1⊕V6ω1⊕V3ω1+2ψ1⊕V4ψ1⊕V4ω1⊕Vω1+2ψ1⊕V5ω1−2ψ1⊕2V2ω1⊕V3ω1−2ψ1⊕2V0⊕Vω1−2ψ1⊕V3ω1+2ψ1−6ψ2⊕V4ω1−6ψ2⊕V−4ψ1⊕V2ω1−6ψ2⊕V3ω1−2ψ1−6ψ2
Made total 2168003 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A121 ↪ E16
11 out of 119
Subalgebra type: A121 (click on type for detailed printout).
Centralizer: T2 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: A121: (4, 6, 8, 12, 8, 4): 24
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−4−2g−15+g−19+g−27
Positive simple generators: 4g27+g24+2g19−g15+4g8+4g4
Cartan symmetric matrix: (1/6)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (24)
Decomposition of ambient Lie algebra: 2V6ω1⊕7V4ω1⊕9V2ω1⊕2V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+2ψ1+2ψ2⊕V4ω1−2ψ1+4ψ2⊕V2ω1+2ψ1+2ψ2⊕2V6ω1⊕V4ω1+4ψ1−2ψ2⊕V2ω1−2ψ1+4ψ2⊕V4ω1⊕V2ω1+4ψ1−2ψ2⊕V4ω1−4ψ1+2ψ2⊕3V2ω1⊕V4ω1+2ψ1−4ψ2⊕V2ω1−4ψ1+2ψ2⊕2V0⊕V4ω1−2ψ1−2ψ2⊕V2ω1+2ψ1−4ψ2⊕V2ω1−2ψ1−2ψ2
Made total 583010629 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A201 ↪ E16
12 out of 119
Subalgebra type: A201 (click on type for detailed printout).
Centralizer: A11 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (1, 0, -1, 0, 1, -1)
Contained up to conjugation as a direct summand of: A201+A11 .
Elements Cartan subalgebra scaled to act by two by components: A201: (6, 8, 10, 14, 10, 6): 40
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−7+g−8+g−16+g−19
Positive simple generators: 4g19+6g16+4g8+6g7
Cartan symmetric matrix: (1/10)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (40)
Decomposition of ambient Lie algebra: V8ω1⊕5V6ω1⊕3V4ω1⊕5V2ω1⊕4V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+6ψ2⊕V6ω1−2ψ1+6ψ2⊕V6ω1+2ψ1⊕V8ω1⊕V2ω1−2ψ1+6ψ2⊕V6ω1⊕V2ω1+2ψ1⊕V4ω1⊕V6ω1−2ψ1⊕V−4ψ1+6ψ2⊕V2ω1⊕V6ω1+2ψ1−6ψ2⊕2V0⊕V2ω1−2ψ1⊕V4ψ1−6ψ2⊕V2ω1+2ψ1−6ψ2⊕V4ω1−6ψ2
Made total 17521203 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A211 ↪ E16
13 out of 119
Subalgebra type: A211 (click on type for detailed printout).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 3, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: A211: (6, 8, 11, 15, 11, 6): 42
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−7+g−13+g−14+g−15+g−16
Positive simple generators: 6g16+g15+4g14+4g13+6g7
Cartan symmetric matrix: (2/21)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (42)
Decomposition of ambient Lie algebra: V8ω1⊕2V7ω1⊕V6ω1⊕2V5ω1⊕3V4ω1⊕2V3ω1⊕2V2ω1⊕2Vω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+12ψ⊕V7ω1+6ψ⊕V5ω1+6ψ⊕V3ω1+6ψ⊕V8ω1⊕Vω1+6ψ⊕V6ω1⊕V4ω1⊕2V2ω1⊕V7ω1−6ψ⊕V0⊕V5ω1−6ψ⊕V3ω1−6ψ⊕Vω1−6ψ⊕V4ω1−12ψ
Made total 2116053 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A281 ↪ E16
14 out of 119
Subalgebra type: A281 (click on type for detailed printout).
Centralizer: A22 .
The semisimple part of the centralizer of the semisimple part of my centralizer: G12
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A281+A21 , A281+A81 , A22+A281 .
Elements Cartan subalgebra scaled to act by two by components: A281: (6, 10, 12, 18, 12, 6): 56
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−2+g−4+g−15+g−24
Positive simple generators: 6g24+6g15+6g4+10g2
Cartan symmetric matrix: (1/14)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (56)
Decomposition of ambient Lie algebra: V10ω1⊕8V6ω1⊕V2ω1⊕8V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V6ω1+2ψ1+2ψ2⊕V10ω1⊕V6ω1−2ψ1+4ψ2⊕V6ω1+4ψ1−2ψ2⊕2V6ω1⊕V2ψ1+2ψ2⊕V6ω1−4ψ1+2ψ2⊕V6ω1+2ψ1−4ψ2⊕V−2ψ1+4ψ2⊕V2ω1⊕V4ψ1−2ψ2⊕V6ω1−2ψ1−2ψ2⊕2V0⊕V−4ψ1+2ψ2⊕V2ψ1−4ψ2⊕V−2ψ1−2ψ2
Made total 9668148 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A301 ↪ E16
15 out of 119
Subalgebra type: A301 (click on type for detailed printout).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: A301: (7, 10, 13, 18, 13, 7): 60
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−2+g−7+1/2g−12+g−15+g−16
Positive simple generators: 7g16+6g15+10g12+g11−10g8+2g7+10g2
Cartan symmetric matrix: (1/15)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (60)
Decomposition of ambient Lie algebra: V10ω1⊕V8ω1⊕2V7ω1⊕2V6ω1⊕2V5ω1⊕V4ω1⊕2V2ω1⊕2Vω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V7ω1+6ψ⊕V5ω1+6ψ⊕V10ω1⊕V8ω1⊕Vω1+6ψ⊕2V6ω1⊕V4ω1⊕2V2ω1⊕V7ω1−6ψ⊕V0⊕V5ω1−6ψ⊕Vω1−6ψ
Made total 2337651 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A351 ↪ E16
16 out of 119
Subalgebra type: A351 (click on type for detailed printout).
Centralizer: A11 .
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: A351+A11 .
Elements Cartan subalgebra scaled to act by two by components: A351: (8, 10, 14, 19, 14, 8): 70
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−6+g−13+g−14+g−15
Positive simple generators: 9g15+5g14+5g13+8g6+8g1
Cartan symmetric matrix: (2/35)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (70)
Decomposition of ambient Lie algebra: V10ω1⊕2V9ω1⊕V8ω1⊕V6ω1⊕2V5ω1⊕V4ω1⊕2V3ω1⊕V2ω1⊕3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V9ω1+2ψ⊕V10ω1⊕V8ω1⊕V5ω1+2ψ⊕V9ω1−2ψ⊕V6ω1⊕V3ω1+2ψ⊕V4ψ⊕V4ω1⊕V5ω1−2ψ⊕V2ω1⊕V3ω1−2ψ⊕V0⊕V−4ψ
Made total 59553 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A361 ↪ E16
17 out of 119
Subalgebra type: A361 (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Elements Cartan subalgebra scaled to act by two by components: A361: (8, 10, 14, 20, 14, 8): 72
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−6+g−8+g−9+g−10+g−19
Positive simple generators: 9g19+5g10+5g9+g8+8g6+8g1
Cartan symmetric matrix: (1/18)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (72)
Decomposition of ambient Lie algebra: 2V10ω1⊕2V8ω1⊕2V6ω1⊕3V4ω1⊕3V2ω1
Made total 14218918 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A601 ↪ E16
18 out of 119
Subalgebra type: A601 (click on type for detailed printout).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: A601: (10, 14, 18, 26, 18, 10): 120
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−2+g−4+g−6+g−15
Positive simple generators: 18g15+10g6+8g4+14g2+10g1
Cartan symmetric matrix: (1/30)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (120)
Decomposition of ambient Lie algebra: V14ω1⊕3V10ω1⊕V8ω1⊕V6ω1⊕2V4ω1⊕V2ω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V10ω1+6ψ⊕V14ω1⊕V4ω1+6ψ⊕V10ω1⊕V8ω1⊕V6ω1⊕V10ω1−6ψ⊕V2ω1⊕V0⊕V4ω1−6ψ
Made total 3102109 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A841 ↪ E16
19 out of 119
Subalgebra type: A841 (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Elements Cartan subalgebra scaled to act by two by components: A841: (12, 16, 22, 30, 22, 12): 168
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−2+g−3+g−6+g−9+g−10
Positive simple generators: 22g10+8g9−8g8+12g6+14g5+14g3+16g2+12g1
Cartan symmetric matrix: (1/42)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (168)
Decomposition of ambient Lie algebra: V16ω1⊕V14ω1⊕2V10ω1⊕V8ω1⊕V6ω1⊕V4ω1⊕V2ω1
Made total 3582365 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A1561 ↪ E16
20 out of 119
Subalgebra type: A1561 (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Elements Cartan subalgebra scaled to act by two by components: A1561: (16, 22, 30, 42, 30, 16): 312
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g−1+g−2+g−3+g−4+g−5+g−6
Positive simple generators: 16g6+30g5+42g4+30g3+22g2+16g1
Cartan symmetric matrix: (1/78)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (312)
Decomposition of ambient Lie algebra: V22ω1⊕V16ω1⊕V14ω1⊕V10ω1⊕V8ω1⊕V2ω1
Made total 129084 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A11 ↪ E16
21 out of 119
Subalgebra type: 2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A11 .
Centralizer: A13 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: 2A11
Basis of Cartan of centralizer: 4 vectors:
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: 3A11 , A21+2A11 , A41+2A11 , A101+2A11 , 4A11 , 2A21+2A11 , A12+2A11 , B12+2A11 , A13+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A11: (1, 2, 2, 3, 2, 1): 2, A11: (1, 0, 1, 1, 1, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−36, g−24
Positive simple generators: g36, g24
Cartan symmetric matrix: (2002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2002)
Decomposition of ambient Lie algebra: V2ω2⊕6Vω1+ω2⊕V2ω1⊕8Vω2⊕8Vω1⊕16V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω2+2ψ3+4ψ4⊕Vω1−2ψ2+2ψ3+4ψ4⊕Vω1−2ψ1+2ψ2+4ψ4⊕Vω1+2ψ1+2ψ4⊕V−2ψ2+4ψ3+2ψ4⊕V−2ψ1+2ψ2+2ψ3+2ψ4⊕Vω1+ω2−2ψ1+2ψ3+2ψ4⊕V2ψ1+2ψ3⊕Vω1+ω2+2ψ1−2ψ2+2ψ3⊕Vω1+ω2+2ψ2⊕Vω2−2ψ1+4ψ4⊕Vω2+2ψ1−2ψ2+2ψ4⊕Vω2+2ψ2−2ψ3+2ψ4⊕V2ω2⊕V2ω1⊕Vω1−2ψ3+2ψ4⊕Vω1+2ψ3−2ψ4⊕V−2ψ1−2ψ2+2ψ3+2ψ4⊕V−4ψ1+2ψ2+2ψ4⊕V2ψ1−4ψ2+2ψ3⊕4V0⊕Vω1+ω2−2ψ2⊕V−2ψ1+4ψ2−2ψ3⊕Vω1+ω2−2ψ1+2ψ2−2ψ3⊕V4ψ1−2ψ2−2ψ4⊕V2ψ1+2ψ2−2ψ3−2ψ4⊕Vω1+ω2+2ψ1−2ψ3−2ψ4⊕Vω2−2ψ2+2ψ3−2ψ4⊕Vω2−2ψ1+2ψ2−2ψ4⊕Vω2+2ψ1−4ψ4⊕Vω1−2ψ1−2ψ4⊕Vω1+2ψ1−2ψ2−4ψ4⊕Vω1+2ψ2−2ψ3−4ψ4⊕V−2ψ1−2ψ3⊕V2ψ1−2ψ2−2ψ3−2ψ4⊕V2ψ2−4ψ3−2ψ4⊕Vω2−2ψ3−4ψ4
Made total 373 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A21+A11 ↪ E16
22 out of 119
Subalgebra type: A21+A11 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A21 .
Centralizer: A21+A11 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A21+A11
Basis of Cartan of centralizer: 3 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A21+2A11 , 2A21+A11 , A31+A21+A11 , 2A21+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 3, 4, 3, 2): 4, A11: (0, 1, 1, 2, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−30+g−34, g−23
Positive simple generators: g34+g30, g23
Cartan symmetric matrix: (1002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4002)
Decomposition of ambient Lie algebra: 2V2ω1+ω2⊕V2ω2⊕4Vω1+ω2⊕4V2ω1⊕6Vω2⊕8Vω1⊕7V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ψ1+2ψ2+2ψ3⊕Vω1+ω2+2ψ2+2ψ3⊕Vω1+2ψ1−2ψ2+4ψ3⊕V2ω1+ω2+2ψ1⊕Vω2+2ψ1+4ψ2−2ψ3⊕Vω1+ω2−2ψ2+4ψ3⊕V4ψ1⊕V2ω1+4ψ2−2ψ3⊕Vω1−2ψ1+2ψ2+2ψ3⊕Vω2+2ψ1⊕V2ω2⊕2V2ω1⊕V4ψ2−2ψ3⊕Vω1−2ψ1−2ψ2+4ψ3⊕Vω2+2ψ1−4ψ2+2ψ3⊕V2ω1+ω2−2ψ1⊕Vω2−2ψ1+4ψ2−2ψ3⊕Vω1+2ψ1+2ψ2−4ψ3⊕V2ω1−4ψ2+2ψ3⊕3V0⊕Vω1+ω2+2ψ2−4ψ3⊕Vω2−2ψ1⊕Vω1+2ψ1−2ψ2−2ψ3⊕V−4ψ2+2ψ3⊕Vω1+ω2−2ψ2−2ψ3⊕Vω2−2ψ1−4ψ2+2ψ3⊕Vω1−2ψ1+2ψ2−4ψ3⊕V−4ψ1⊕Vω1−2ψ1−2ψ2−2ψ3
Made total 630 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A21 ↪ E16
23 out of 119
Subalgebra type: 2A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A21 .
Centralizer: 2A11 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A13
Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: 2A21+A11 , 3A21 , 2A21+2A11 .
Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 3, 4, 3, 2): 4, A21: (0, 2, 1, 2, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−30+g−34, g−8+g−19
Positive simple generators: g34+g30, g19+g8
Cartan symmetric matrix: (1001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (4004)
Decomposition of ambient Lie algebra: V2ω1+2ω2⊕5V2ω2⊕8Vω1+ω2⊕5V2ω1⊕7V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω2+2ψ1+2ψ2+2ψ3⊕Vω1+ω2+2ψ2+2ψ3⊕V4ψ1+4ψ2−2ψ3⊕V2ω2+2ψ1+4ψ2−2ψ3⊕V2ω1+2ψ1+4ψ2−2ψ3⊕Vω1+ω2−2ψ2+4ψ3⊕V2ω2+2ψ1⊕V2ω1+2ψ1⊕V2ω1+2ω2⊕Vω1+ω2−2ψ1−2ψ2+4ψ3⊕V2ω2⊕V2ω1⊕V4ψ2−2ψ3⊕Vω1+ω2+2ψ1+2ψ2−4ψ3⊕3V0⊕V2ω2−2ψ1⊕V2ω1−2ψ1⊕Vω1+ω2+2ψ2−4ψ3⊕V−4ψ2+2ψ3⊕V2ω2−2ψ1−4ψ2+2ψ3⊕V2ω1−2ψ1−4ψ2+2ψ3⊕Vω1+ω2−2ψ2−2ψ3⊕Vω1+ω2−2ψ1−2ψ2−2ψ3⊕V−4ψ1−4ψ2+2ψ3
Made total 3277348 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A31+A11 ↪ E16
24 out of 119
Subalgebra type: A31+A11 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A31 .
Centralizer: A22 .
The semisimple part of the centralizer of the semisimple part of my centralizer: G12
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: A31+A21+A11 , A81+A31+A11 , A22+A31+A11 .
Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4, 6, 4, 2): 6, A11: (0, 1, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−23+g−30+g−34, g−2
Positive simple generators: g34+g30+g23, g2
Cartan symmetric matrix: (2/3002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6002)
Decomposition of ambient Lie algebra: V3ω1+ω2⊕V2ω2⊕8Vω1+ω2⊕9V2ω1⊕8V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω2+2ψ1+2ψ2⊕V2ω1+2ψ1+2ψ2⊕Vω1+ω2−2ψ1+4ψ2⊕V2ω1−2ψ1+4ψ2⊕V2ψ1+2ψ2⊕V3ω1+ω2⊕Vω1+ω2+4ψ1−2ψ2⊕V2ω1+4ψ1−2ψ2⊕V−2ψ1+4ψ2⊕V2ω2⊕2Vω1+ω2⊕3V2ω1⊕V4ψ1−2ψ2⊕Vω1+ω2−4ψ1+2ψ2⊕V2ω1−4ψ1+2ψ2⊕2V0⊕Vω1+ω2+2ψ1−4ψ2⊕V2ω1+2ψ1−4ψ2⊕V−4ψ1+2ψ2⊕Vω1+ω2−2ψ1−2ψ2⊕V2ω1−2ψ1−2ψ2⊕V2ψ1−4ψ2⊕V−2ψ1−2ψ2
Made total 1029 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A31+A21 ↪ E16
25 out of 119
Subalgebra type: A31+A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A31 .
Centralizer: A11 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0, 0, 0), (1, 0, -1, 0, 1, -1)
Contained up to conjugation as a direct summand of: A31+A21+A11 .
Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4, 6, 4, 2): 6, A21: (1, 0, 1, 0, 1, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−29+g−30+g−31, g−7−g−11
Positive simple generators: g31+g30+g29, −g11+g7
Cartan symmetric matrix: (2/3001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6004)
Decomposition of ambient Lie algebra: V2ω1+2ω2⊕2Vω1+2ω2⊕2V2ω1+ω2⊕2V3ω1⊕V2ω2⊕4Vω1+ω2⊕2V2ω1⊕2Vω2⊕2Vω1⊕4V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω2+2ψ1+6ψ2⊕V2ω1+ω2+6ψ2⊕Vω2+6ψ2⊕Vω1+ω2−2ψ1+6ψ2⊕Vω1+2ω2+2ψ1⊕V3ω1+2ψ1⊕V4ψ1⊕V2ω1+2ω2⊕Vω1+2ψ1⊕V2ω2⊕2V2ω1⊕Vω1+2ω2−2ψ1⊕V3ω1−2ψ1⊕2V0⊕Vω1−2ψ1⊕Vω1+ω2+2ψ1−6ψ2⊕V2ω1+ω2−6ψ2⊕V−4ψ1⊕Vω2−6ψ2⊕Vω1+ω2−2ψ1−6ψ2
Made total 825441 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A31 ↪ E16
26 out of 119
Subalgebra type: 2A31 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A31 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: A31: (2, 3, 4, 6, 4, 2): 6, A31: (1, 1, 1, 0, 1, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: g−29+g−30+g−31, g−2+g−7−g−11
Positive simple generators: g31+g30+g29, −g11+g7+g2
Cartan symmetric matrix: (2/3002/3)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6006)
Decomposition of ambient Lie algebra: Vω1+3ω2⊕V2ω1+2ω2⊕V3ω1+ω2⊕2Vω1+2ω2⊕2V2ω1+ω2⊕2V2ω2⊕2Vω1+ω2⊕2V2ω1⊕2Vω2⊕2Vω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+2ω2+6ψ⊕V2ω1+ω2+6ψ⊕Vω2+6ψ⊕Vω1+6ψ⊕Vω1+3ω2⊕V2ω1+2ω2⊕V3ω1+ω2⊕2V2ω2⊕2Vω1+ω2⊕2V2ω1⊕V0⊕Vω1+2ω2−6ψ⊕V2ω1+ω2−6ψ⊕Vω2−6ψ⊕Vω1−6ψ
Made total 1471846 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{4}_1+A^{1}_1 ↪ E^{1}_6
27 out of 119
Subalgebra type: \displaystyle A^{4}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{4}_1 .
Centralizer: \displaystyle A^{1}_2 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle 2A^{1}_2
Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0), (1, 0, 0, 2, 0, -1)
Contained up to conjugation as a direct summand of: \displaystyle A^{4}_1+2A^{1}_1 , \displaystyle 2A^{4}_1+A^{1}_1 , \displaystyle A^{1}_2+A^{4}_1+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{4}_1: (2, 4, 4, 6, 4, 2): 8, \displaystyle A^{1}_1: (1, 0, 1, 1, 1, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-22}+g_{-28}, \displaystyle g_{-24}
Positive simple generators: \displaystyle 2g_{28}+2g_{22}, \displaystyle g_{24}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/2 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}8 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}}\oplus 6V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 7V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 9V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\omega_{1}-2\psi_{1}+2\psi_{2}+8\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{2}+4\psi_{3}}\oplus V_{-2\psi_{1}+4\psi_{2}+6\psi_{3}} \oplus V_{\omega_{2}+6\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}+2\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{3}} \oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}+6\psi_{3}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{3}} \oplus 3V_{0}\oplus V_{2\omega_{1}-2\psi_{1}-2\psi_{3}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{2}-4\psi_{3}}\oplus V_{-2\psi_{1}-2\psi_{2}} \oplus V_{4\psi_{1}-2\psi_{2}-6\psi_{3}}\oplus V_{\omega_{2}-6\psi_{3}}\oplus V_{2\omega_{1}+2\psi_{1}-2\psi_{2}-8\psi_{3}} \oplus V_{2\psi_{1}-4\psi_{2}-6\psi_{3}}
Made total 9595847 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{4}_1+A^{2}_1 ↪ E^{1}_6
28 out of 119
Subalgebra type: \displaystyle A^{4}_1+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{4}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(1, 0, 1, 1, -1, 0), (1, 0, 0, 2, 0, 1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{4}_1: (2, 4, 4, 6, 4, 2): 8, \displaystyle A^{2}_1: (1, 0, 2, 2, 2, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-23}+g_{-27}, \displaystyle g_{-18}+g_{-21}
Positive simple generators: \displaystyle 2g_{27}+2g_{23}, \displaystyle g_{21}+g_{18}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/2 & 0\\
0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}8 & 0\\
0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 4V_{2\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{2}}\oplus 5V_{2\omega_{1}} \oplus 4V_{\omega_{2}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\omega_{1}+4\psi_{1}+8\psi_{2}}\oplus V_{\omega_{2}+6\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{1}+\omega_{2}+4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{2}+6\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}} \oplus 2V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{2}-6\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{2}-6\psi_{1}-6\psi_{2}}
Made total 51475 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{4}_1 ↪ E^{1}_6
29 out of 119
Subalgebra type: \displaystyle 2A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{4}_1 .
Centralizer: \displaystyle A^{1}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle 2A^{1}_2
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0)
Contained up to conjugation as a direct summand of: \displaystyle 2A^{4}_1+A^{1}_1 , \displaystyle 3A^{4}_1 , \displaystyle A^{1}_2+2A^{4}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{4}_1: (2, 4, 4, 6, 4, 2): 8, \displaystyle A^{4}_1: (2, 0, 2, 2, 2, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-25}+g_{-26}, \displaystyle g_{-7}+g_{-16}
Positive simple generators: \displaystyle 2g_{26}+2g_{25}, \displaystyle 2g_{16}+2g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/2 & 0\\
0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}8 & 0\\
0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{2}}\oplus 6V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 8V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\omega_{1}+2\omega_{2}+2\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi_{1}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}+2\psi_{2}} \oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{1}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+2\omega_{2}-2\psi_{2}} \oplus 2V_{0}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}
Made total 32934469 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{5}_1+A^{1}_1 ↪ E^{1}_6
30 out of 119
Subalgebra type: \displaystyle A^{5}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{5}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, -1, -1, 0), (1, 0, 0, 2, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{5}_1: (3, 4, 5, 7, 5, 3): 10, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-22}+g_{-24}+g_{-28}, \displaystyle g_{-15}
Positive simple generators: \displaystyle 2g_{28}+g_{24}+2g_{22}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/5 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}10 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}} \oplus 4V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+8\psi_{2}} \oplus V_{3\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{3\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{2}-6\psi_{1}+6\psi_{2}}\oplus V_{3\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+6\psi_{1}-6\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{1}-6\psi_{2}}
Made total 719 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{5}_1+A^{4}_1 ↪ E^{1}_6
31 out of 119
Subalgebra type: \displaystyle A^{5}_1+A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{5}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 1, 1, -1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{5}_1: (3, 4, 5, 7, 5, 3): 10, \displaystyle A^{4}_1: (0, 0, 2, 2, 2, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-22}+g_{-24}+g_{-28}, \displaystyle g_{-3}+g_{-10}
Positive simple generators: \displaystyle 2g_{28}+g_{24}+2g_{22}, \displaystyle 2g_{10}+2g_{3}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/5 & 0\\
0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}10 & 0\\
0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{3\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{\omega_{1}+2\omega_{2}} \oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\omega_{1}+2\omega_{2}+4\psi}\oplus V_{\omega_{1}+6\psi}\oplus V_{3\omega_{1}+2\omega_{2}+2\psi}\oplus V_{\omega_{1}+2\omega_{2}+2\psi} \oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{3\omega_{1}+2\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{\omega_{1}+2\omega_{2}-2\psi} \oplus V_{0}\oplus V_{2\omega_{1}+2\omega_{2}-4\psi}\oplus V_{\omega_{1}-6\psi}
Made total 34939 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{6}_1 ↪ E^{1}_6
32 out of 119
Subalgebra type: \displaystyle 2A^{6}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{6}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{6}_1: (3, 4, 6, 8, 6, 3): 12, \displaystyle A^{6}_1: (1, 2, 0, 2, 0, 1): 12
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-22}+g_{-23}+g_{-24}+g_{-25}, \displaystyle g_{-1}-2g_{-2}-g_{-4}-g_{-6}
Positive simple generators: \displaystyle g_{25}+2g_{24}+2g_{23}+g_{22}, \displaystyle -g_{6}-2g_{4}-g_{2}+g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/3 & 0\\
0 & 1/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}12 & 0\\
0 & 12\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+3\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+3\omega_{2}+6\psi}\oplus V_{3\omega_{1}+\omega_{2}+6\psi}\oplus V_{2\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+2\omega_{2}} \oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+3\omega_{2}-6\psi}\oplus V_{3\omega_{1}+\omega_{2}-6\psi}
Made total 6418412 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{8}_1+A^{1}_1 ↪ E^{1}_6
33 out of 119
Subalgebra type: \displaystyle A^{8}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{8}_1 .
Centralizer: \displaystyle A^{3}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{2}_2+A^{1}_1
Basis of Cartan of centralizer: 1 vectors:
(0, 1, -1, 0, -1, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{8}_1+A^{3}_1+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{8}_1: (4, 4, 6, 8, 6, 4): 16, \displaystyle A^{1}_1: (0, 1, 1, 2, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-12}+g_{-16}+g_{-22}+g_{-25}, \displaystyle g_{-23}
Positive simple generators: \displaystyle 2g_{25}+2g_{22}+2g_{16}+2g_{12}, \displaystyle g_{23}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/4 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}16 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{4\omega_{1}+\omega_{2}}\oplus 4V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 4V_{2\omega_{1}} \oplus 4V_{\omega_{2}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{4\omega_{1}+4\psi}\oplus V_{\omega_{2}+6\psi}\oplus V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{2\omega_{1}+4\psi}\oplus V_{2\omega_{1}+\omega_{2}+2\psi} \oplus V_{4\psi}\oplus 2V_{4\omega_{1}}\oplus V_{\omega_{2}+2\psi}\oplus V_{4\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}} \oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{4\omega_{1}-4\psi}\oplus V_{\omega_{2}-2\psi}\oplus V_{2\omega_{1}-4\psi}\oplus V_{-4\psi} \oplus V_{\omega_{2}-6\psi}
Made total 3649565 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{8}_1+A^{3}_1 ↪ E^{1}_6
34 out of 119
Subalgebra type: \displaystyle A^{8}_1+A^{3}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{8}_1 .
Centralizer: \displaystyle A^{1}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{8}_1+A^{3}_1+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{8}_1: (4, 4, 6, 8, 6, 4): 16, \displaystyle A^{3}_1: (0, 2, 2, 3, 2, 0): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}, \displaystyle g_{-13}+g_{-14}-g_{-15}
Positive simple generators: \displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}, \displaystyle -g_{15}+g_{14}+g_{13}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/4 & 0\\
0 & 2/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}16 & 0\\
0 & 6\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus 2V_{4\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus 2V_{3\omega_{2}} \oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{4\omega_{1}+\omega_{2}+2\psi}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+2\psi}\oplus V_{2\omega_{1}+\omega_{2}+2\psi} \oplus V_{4\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{4\omega_{1}+\omega_{2}-2\psi}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{3\omega_{2}-2\psi}\oplus V_{2\omega_{1}+\omega_{2}-2\psi}\oplus V_{0}\oplus V_{-4\psi}
Made total 5056886 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{8}_1+A^{4}_1 ↪ E^{1}_6
35 out of 119
Subalgebra type: \displaystyle A^{8}_1+A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{8}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{8}_1: (4, 4, 6, 8, 6, 4): 16, \displaystyle A^{4}_1: (0, 2, 2, 4, 2, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}, \displaystyle g_{-4}+g_{-19}
Positive simple generators: \displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}, \displaystyle 2g_{19}+2g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/4 & 0\\
0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}16 & 0\\
0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{4\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{4\omega_{1}}\oplus 3V_{2\omega_{2}} \oplus 2V_{2\omega_{1}}
Made total 29361990 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{9}_1+A^{3}_1 ↪ E^{1}_6
36 out of 119
Subalgebra type: \displaystyle A^{9}_1+A^{3}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{9}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{9}_1: (4, 5, 7, 10, 7, 4): 18, \displaystyle A^{3}_1: (0, 1, 1, 0, 1, 0): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}+g_{-23}, \displaystyle g_{-2}-g_{-3}+g_{-5}
Positive simple generators: \displaystyle g_{23}+2g_{21}+2g_{20}+2g_{18}+2g_{17}, \displaystyle g_{5}-g_{3}+g_{2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/9 & 0\\
0 & 2/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}18 & 0\\
0 & 6\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+3\omega_{2}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}
Made total 6484741 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{10}_1+A^{1}_1 ↪ E^{1}_6
37 out of 119
Subalgebra type: \displaystyle A^{10}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{10}_1 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2)
Contained up to conjugation as a direct summand of: \displaystyle A^{10}_1+2A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{10}_1: (4, 6, 7, 10, 7, 4): 20, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-13}+g_{-14}+g_{-24}, \displaystyle g_{-15}
Positive simple generators: \displaystyle 4g_{24}+3g_{14}+3g_{13}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/5 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}20 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{6\omega_{1}}\oplus 2V_{4\omega_{1}+\omega_{2}}\oplus 2V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus 4V_{3\omega_{1}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus 2V_{\omega_{2}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{1}+2\psi_{1}+6\psi_{2}}\oplus V_{3\omega_{1}+\omega_{2}+6\psi_{2}}\oplus V_{3\omega_{1}-2\psi_{1}+6\psi_{2}} \oplus V_{4\omega_{1}+\omega_{2}+2\psi_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\psi_{1}}\oplus V_{4\omega_{1}}\oplus V_{\omega_{2}+2\psi_{1}} \oplus V_{4\omega_{1}+\omega_{2}-2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{2}-2\psi_{1}}\oplus V_{3\omega_{1}+2\psi_{1}-6\psi_{2}} \oplus V_{3\omega_{1}+\omega_{2}-6\psi_{2}}\oplus V_{-4\psi_{1}}\oplus V_{3\omega_{1}-2\psi_{1}-6\psi_{2}}
Made total 23596 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{10}_1+A^{2}_1 ↪ E^{1}_6
38 out of 119
Subalgebra type: \displaystyle A^{10}_1+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{10}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{10}_1: (4, 6, 7, 10, 7, 4): 20, \displaystyle A^{2}_1: (0, 0, 1, 2, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-8}+g_{-19}+g_{-24}, \displaystyle -g_{-4}+g_{-15}
Positive simple generators: \displaystyle 4g_{24}+3g_{19}+3g_{8}, \displaystyle g_{15}-g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/5 & 0\\
0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}20 & 0\\
0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{6\omega_{1}}\oplus 4V_{3\omega_{1}+\omega_{2}}\oplus 2V_{4\omega_{1}}\oplus 3V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{3\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{4\omega_{1}+2\omega_{2}} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{2}+4\psi_{1}-2\psi_{2}}\oplus V_{4\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{2\omega_{2}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{3\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}
Made total 262477 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{10}_1 ↪ E^{1}_6
39 out of 119
Subalgebra type: \displaystyle 2A^{10}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{10}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{10}_1: (4, 6, 7, 10, 7, 4): 20, \displaystyle A^{10}_1: (0, 0, 3, 4, 3, 0): 20
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-13}+g_{-14}+g_{-24}, \displaystyle g_{-3}+g_{-4}+g_{-5}
Positive simple generators: \displaystyle 4g_{24}+3g_{14}+3g_{13}, \displaystyle 3g_{5}+4g_{4}+3g_{3}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/5 & 0\\
0 & 1/5\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}20 & 0\\
0 & 20\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}+4\omega_{2}}\oplus V_{6\omega_{2}}\oplus 2V_{3\omega_{1}+3\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{1}+3\omega_{2}+6\psi}\oplus V_{4\omega_{1}+4\omega_{2}}\oplus V_{6\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{3\omega_{1}+3\omega_{2}-6\psi}
Made total 1456960 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{11}_1+A^{1}_1 ↪ E^{1}_6
40 out of 119
Subalgebra type: \displaystyle A^{11}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{11}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{11}_1: (4, 6, 8, 11, 8, 4): 22, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-13}+g_{-14}+g_{-15}+g_{-24}, \displaystyle g_{-4}
Positive simple generators: \displaystyle 4g_{24}+g_{15}+3g_{14}+3g_{13}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/11 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}22 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 3V_{3\omega_{1}+\omega_{2}}\oplus 3V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 4V_{2\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{1}+\omega_{2}+6\psi}\oplus V_{4\omega_{1}+6\psi}\oplus V_{2\omega_{1}+6\psi}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}} \oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0} \oplus V_{3\omega_{1}+\omega_{2}-6\psi}\oplus V_{4\omega_{1}-6\psi}\oplus V_{2\omega_{1}-6\psi}
Made total 956 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{20}_1+A^{1}_1 ↪ E^{1}_6
41 out of 119
Subalgebra type: \displaystyle A^{20}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{20}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 3, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{20}_1: (6, 8, 10, 14, 10, 6): 40, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-7}+g_{-13}+g_{-14}+g_{-16}, \displaystyle g_{-15}
Positive simple generators: \displaystyle 6g_{16}+4g_{14}+4g_{13}+6g_{7}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/10 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}40 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{8\omega_{1}}\oplus 2V_{6\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus 3V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{4\omega_{1}+12\psi}\oplus V_{6\omega_{1}+\omega_{2}+6\psi}\oplus V_{2\omega_{1}+\omega_{2}+6\psi}\oplus V_{8\omega_{1}} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{6\omega_{1}+\omega_{2}-6\psi}\oplus V_{0}\oplus V_{2\omega_{1}+\omega_{2}-6\psi} \oplus V_{4\omega_{1}-12\psi}
Made total 2112126 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{28}_1+A^{2}_1 ↪ E^{1}_6
42 out of 119
Subalgebra type: \displaystyle A^{28}_1+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{28}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{28}_1: (6, 10, 12, 18, 12, 6): 56, \displaystyle A^{2}_1: (1, 0, 1, 0, 1, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-2}+g_{-12}+g_{-15}+g_{-16}, \displaystyle g_{-7}+g_{-11}
Positive simple generators: \displaystyle 6g_{16}+6g_{15}+6g_{12}+10g_{2}, \displaystyle g_{11}+g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/14 & 0\\
0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}56 & 0\\
0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\omega_{2}}\oplus 2V_{6\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus 2V_{\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{6\omega_{1}+\omega_{2}+6\psi}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\omega_{2}}\oplus V_{\omega_{2}+6\psi}\oplus V_{6\omega_{1}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{6\omega_{1}+\omega_{2}-6\psi}\oplus V_{0}\oplus V_{\omega_{2}-6\psi}
Made total 1471223 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{28}_1+A^{8}_1 ↪ E^{1}_6
43 out of 119
Subalgebra type: \displaystyle A^{28}_1+A^{8}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{28}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{28}_1: (6, 10, 12, 18, 12, 6): 56, \displaystyle A^{8}_1: (2, 0, 2, 0, 2, 2): 16
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-2}+g_{-12}+g_{-15}+g_{-16}, \displaystyle g_{-1}+g_{-3}+g_{-5}+g_{-6}
Positive simple generators: \displaystyle 6g_{16}+6g_{15}+6g_{12}+10g_{2}, \displaystyle 2g_{6}+2g_{5}+2g_{3}+2g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/14 & 0\\
0 & 1/4\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}56 & 0\\
0 & 16\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{6\omega_{1}+4\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{6\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
Made total 6394164 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{35}_1+A^{1}_1 ↪ E^{1}_6
44 out of 119
Subalgebra type: \displaystyle A^{35}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{35}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{35}_1: (8, 10, 14, 19, 14, 8): 70, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators: \displaystyle g_{-1}+g_{-6}+g_{-13}+g_{-14}+g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle 9g_{15}+5g_{14}+5g_{13}+8g_{6}+8g_{1}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/35 & 0\\
0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}70 & 0\\
0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{9\omega_{1}+\omega_{2}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
Made total 727 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2 ↪ E^{1}_6
45 out of 119
Subalgebra type: \displaystyle A^{1}_2 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_1 .
Centralizer: \displaystyle 2A^{1}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_2
Basis of Cartan of centralizer: 4 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_2+A^{1}_1 , \displaystyle A^{1}_2+A^{2}_1 , \displaystyle A^{1}_2+A^{4}_1 , \displaystyle A^{1}_2+A^{5}_1 , \displaystyle A^{1}_2+A^{8}_1 , \displaystyle A^{1}_2+2A^{1}_1 , \displaystyle A^{1}_2+A^{4}_1+A^{1}_1 , \displaystyle A^{1}_2+2A^{4}_1 , \displaystyle 2A^{1}_2 , \displaystyle A^{2}_2+A^{1}_2 , \displaystyle A^{2}_2+A^{1}_2 , \displaystyle 2A^{1}_2+A^{1}_1 , \displaystyle 2A^{1}_2+A^{4}_1 , \displaystyle 3A^{1}_2 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1\\
-1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{2}}\oplus 9V_{\omega_{2}}\oplus 9V_{\omega_{1}}\oplus 16V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{3}+2\psi_{4}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{2}-2\psi_{3}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{2}+2\psi_{1}+2\psi_{3}-2\psi_{4}}\oplus V_{-2\psi_{3}+4\psi_{4}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{4\psi_{3}-2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{2}+2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{3}+2\psi_{4}} \oplus V_{\omega_{1}-2\psi_{1}+2\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}-2\psi_{4}} \oplus V_{\omega_{1}+2\psi_{2}-2\psi_{4}}\oplus 4V_{0}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{3}+2\psi_{4}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{2}-2\psi_{2}+2\psi_{3}-2\psi_{4}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-2\psi_{4}}\oplus V_{-4\psi_{3}+2\psi_{4}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{2\psi_{3}-4\psi_{4}}\oplus V_{\omega_{2}-2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{1}-2\psi_{1}-2\psi_{4}}\oplus V_{-2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{3}-2\psi_{4}}
Made total 365 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_2 ↪ E^{1}_6
46 out of 119
Subalgebra type: \displaystyle B^{1}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_1 .
Centralizer: \displaystyle B^{1}_2 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle B^{1}_2
Basis of Cartan of centralizer: 3 vectors:
(0, 0, 1, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \displaystyle B^{1}_2+A^{1}_1 , \displaystyle B^{1}_2+A^{2}_1 , \displaystyle B^{1}_2+A^{10}_1 , \displaystyle B^{1}_2+2A^{1}_1 , \displaystyle 2B^{1}_2 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{19}+g_{8}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-8}+g_{-19}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1\\
-1 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -2\\
-2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{2}}\oplus 8V_{\omega_{2}}\oplus 5V_{\omega_{1}}\oplus 11V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{4\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}+2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}+2\psi_{1}}\oplus V_{2\psi_{1}}\oplus V_{2\omega_{2}}\oplus V_{4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}}\oplus V_{\omega_{2}+2\psi_{1}+2\psi_{2}-4\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{2}-4\psi_{3}} \oplus V_{-4\psi_{2}+2\psi_{3}}\oplus V_{-2\psi_{1}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{2}-2\psi_{3}} \oplus V_{-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{1}-4\psi_{2}+2\psi_{3}}
Made total 2180130 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra G^{1}_2 ↪ E^{1}_6
47 out of 119
Subalgebra type: \displaystyle G^{1}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{3}_1 .
Centralizer: \displaystyle A^{2}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle G^{1}_2
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \displaystyle G^{1}_2+A^{2}_1 , \displaystyle G^{1}_2+A^{8}_1 , \displaystyle A^{2}_2+G^{1}_2 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle G^{1}_2: (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \displaystyle g_{-23}+g_{-30}+g_{-34}, \displaystyle g_{35}
Positive simple generators: \displaystyle g_{34}+g_{30}+g_{23}, \displaystyle g_{-35}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & -1\\
-1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & -3\\
-3 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}}\oplus 8V_{\omega_{1}}\oplus 8V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+2\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus V_{\omega_{1}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}} \oplus V_{-2\psi_{1}-2\psi_{2}}
Made total 1031 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2 ↪ E^{1}_6
48 out of 119
Subalgebra type: \displaystyle A^{2}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_1 .
Centralizer: \displaystyle A^{1}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle 2A^{1}_2
Basis of Cartan of centralizer: 2 vectors:
(0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{2}_2+A^{1}_1 , \displaystyle A^{2}_2+A^{4}_1 , \displaystyle A^{2}_2+A^{1}_2 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2\\
-2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 3V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 3V_{\omega_{2}}\oplus 3V_{\omega_{1}}\oplus 8V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{2}}\oplus V_{2\omega_{1}+2\psi_{1}}\oplus V_{\omega_{1}+2\psi_{2}}\oplus V_{\omega_{2}+2\psi_{1}} \oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}-2\psi_{1}+2\psi_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}} \oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{2\omega_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}}\oplus V_{\omega_{2}-2\psi_{2}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}
Made total 26508 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2 ↪ E^{1}_6
49 out of 119
Subalgebra type: \displaystyle A^{2}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_1 .
Centralizer: \displaystyle G^{1}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{2}_2
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (0, 1, -1, 0, 1, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{2}_2+A^{1}_1 , \displaystyle A^{2}_2+A^{3}_1 , \displaystyle A^{2}_2+A^{4}_1 , \displaystyle A^{2}_2+A^{28}_1 , \displaystyle A^{2}_2+A^{3}_1+A^{1}_1 , \displaystyle A^{2}_2+A^{1}_2 , \displaystyle A^{2}_2+G^{1}_2 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-30}+g_{-34}, \displaystyle g_{25}+g_{16}
Positive simple generators: \displaystyle g_{34}+g_{30}, \displaystyle g_{-16}+g_{-25}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2\\
-2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 8V_{\omega_{1}+\omega_{2}}\oplus 14V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{-2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{2}} \oplus V_{-4\psi_{1}+6\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}}\oplus V_{2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+2\psi_{2}} \oplus V_{2\psi_{1}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}} \oplus V_{-2\psi_{1}}\oplus V_{-2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{4\psi_{1}-6\psi_{2}}\oplus V_{2\psi_{1}-6\psi_{2}}
Made total 12521464 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{2}_2 ↪ E^{1}_6
50 out of 119
Subalgebra type: \displaystyle B^{2}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_1 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 1, 0, 0), (1, 0, -1, 0, 1, -1)
Contained up to conjugation as a direct summand of: \displaystyle B^{2}_2+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{2}_2: (2, 2, 3, 4, 3, 2): 4, (-2, 0, -2, -2, -2, -2): 8
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \displaystyle g_{-30}+g_{-34}, \displaystyle g_{16}+g_{7}
Positive simple generators: \displaystyle g_{34}+g_{30}, \displaystyle 2g_{-7}+2g_{-16}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2\\
-1/2 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -4\\
-4 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 5V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+6\psi_{2}}\oplus V_{2\omega_{2}-2\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{1}}\oplus V_{-4\psi_{1}+6\psi_{2}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}\oplus V_{2\omega_{2}-2\psi_{1}}\oplus V_{4\psi_{1}-6\psi_{2}}\oplus V_{2\omega_{2}+2\psi_{1}-6\psi_{2}} \oplus V_{\omega_{1}-6\psi_{2}}
Made total 9280966 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{3}_2 ↪ E^{1}_6
51 out of 119
Subalgebra type: \displaystyle A^{3}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{3}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{3}_2: (2, 3, 4, 6, 4, 2): 6, (-1, 0, -2, -3, -2, -1): 6
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-23}+g_{-30}+g_{-34}, \displaystyle (1/2\sqrt{-3}+1/2)g_{24}+(-1/2\sqrt{-3}+1/2)g_{15}+g_{4}
Positive simple generators: \displaystyle g_{34}+g_{30}+g_{23}, \displaystyle g_{-4}+(1/2\sqrt{-3}+1/2)g_{-15}+(-1/2\sqrt{-3}+1/2)g_{-24}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & -3\\
-3 & 6\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus 7V_{\omega_{1}+\omega_{2}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}} \oplus V_{3\omega_{2}}\oplus V_{3\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus 2V_{0} \oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}
Made total 212017903 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{3}_2 ↪ E^{1}_6
52 out of 119
Subalgebra type: \displaystyle A^{3}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{3}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{3}_2: (2, 3, 4, 6, 4, 2): 6, (0, -1, -1, -3, -1, 0): 6
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-29}+g_{-30}+g_{-31}, \displaystyle g_{10}+g_{9}+g_{8}
Positive simple generators: \displaystyle g_{31}+g_{30}+g_{29}, \displaystyle g_{-8}+g_{-9}+g_{-10}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & -3\\
-3 & 6\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}}\oplus 3V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}} \oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}
Made total 141240 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{3}_2 ↪ E^{1}_6
53 out of 119
Subalgebra type: \displaystyle B^{3}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{3}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{3}_2: (2, 3, 4, 6, 4, 2): 6, (-1, -2, -3, -6, -3, -1): 12
Dimension of subalgebra generated by predefined or computed generators: 10.
Negative simple generators: \displaystyle g_{-29}+g_{-30}+g_{-31}, \displaystyle (1/2\sqrt{-1}-1/2)g_{16}+g_{15}+(1/2\sqrt{-1}+1/2)g_{12}+g_{8}
Positive simple generators: \displaystyle g_{31}+g_{30}+g_{29}, \displaystyle 2g_{-8}+(-\sqrt{-1}+1)g_{-12}+2g_{-15}+(-\sqrt{-1}-1)g_{-16}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & -1/3\\
-1/3 & 1/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & -6\\
-6 & 12\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{2}+6\psi}\oplus V_{\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}-6\psi}
Made total 7974385 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra G^{3}_2 ↪ E^{1}_6
54 out of 119
Subalgebra type: \displaystyle G^{3}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{9}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle G^{3}_2: (4, 5, 7, 10, 7, 4): 18, (-2, -2, -3, -5, -3, -2): 6
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}+g_{-23}, \displaystyle (-1/4\sqrt{-7}-3/4)g_{28}+g_{26}+(1/2\sqrt{-7}-1/2)g_{24}
Positive simple generators: \displaystyle g_{23}+2g_{21}+2g_{20}+2g_{18}+2g_{17}, \displaystyle (-1/4\sqrt{-7}-1/4)g_{-24}+g_{-26}+(1/4\sqrt{-7}-3/4)g_{-28}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/9 & -1/3\\
-1/3 & 2/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}18 & -9\\
-9 & 6\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}
Made total 16487567 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{5}_2 ↪ E^{1}_6
55 out of 119
Subalgebra type: \displaystyle A^{5}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{5}_1 .
Centralizer: \displaystyle A^{1}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 1, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{5}_2+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{5}_2: (3, 4, 5, 7, 5, 3): 10, (-1, -2, 0, -2, -2, -2): 10
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-22}+g_{-24}+g_{-28}, \displaystyle g_{11}+2g_{8}+g_{1}
Positive simple generators: \displaystyle 2g_{28}+g_{24}+2g_{22}, \displaystyle g_{-1}+g_{-8}+2g_{-11}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/5 & -1/5\\
-1/5 & 2/5\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}10 & -5\\
-5 & 10\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{3\omega_{2}}\oplus 2V_{3\omega_{1}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{2}+2\psi}\oplus V_{3\omega_{1}+2\psi}\oplus V_{4\psi}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}} \oplus V_{3\omega_{2}-2\psi}\oplus V_{3\omega_{1}-2\psi}\oplus V_{0}\oplus V_{-4\psi}
Made total 4669812 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{9}_2 ↪ E^{1}_6
56 out of 119
Subalgebra type: \displaystyle A^{9}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{9}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{9}_2: (4, 5, 7, 10, 7, 4): 18, (-2, -1, -2, -5, -2, -2): 18
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}+g_{-23}, \displaystyle (\sqrt{-1}-1)g_{10}+(-\sqrt{-1}-1)g_{9}+g_{8}+2\sqrt{-1}g_{6}+2\sqrt{-1}g_{1}
Positive simple generators: \displaystyle g_{23}+2g_{21}+2g_{20}+2g_{18}+2g_{17}, \displaystyle -\sqrt{-1}g_{-1}-\sqrt{-1}g_{-6}+g_{-8}+(\sqrt{-1}-1)g_{-9}+(-\sqrt{-1}-1)g_{-10}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/9 & -1/9\\
-1/9 & 2/9\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}18 & -9\\
-9 & 18\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+4\omega_{2}}\oplus V_{4\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}}
Made total 24846702 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 3A^{1}_1 ↪ E^{1}_6
57 out of 119
Subalgebra type: \displaystyle 3A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle 2A^{1}_1 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 3 vectors:
(0, 0, 0, 1, 0, 0), (0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \displaystyle 4A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_1: (1, 2, 2, 3, 2, 1): 2, \displaystyle A^{1}_1: (1, 0, 1, 1, 1, 1): 2, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{-24}, \displaystyle g_{-15}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{24}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{\omega_{3}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}} \oplus 5V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}+4\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi_{1}}\oplus V_{\omega_{3}+2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{2}+4\psi_{3}} \oplus V_{4\psi_{1}}\oplus V_{\omega_{1}+\omega_{2}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{3}+2\psi_{1}-4\psi_{2}+2\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi_{1}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{\omega_{1}+\omega_{2}-4\psi_{2}+2\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{2}+\omega_{3}+2\psi_{2}-4\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}-2\psi_{3}} \oplus V_{\omega_{1}+\omega_{3}-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{3}-2\psi_{1}-4\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}-4\psi_{3}} \oplus V_{-4\psi_{1}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}-2\psi_{3}}
Made total 460 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_1+2A^{1}_1 ↪ E^{1}_6
58 out of 119
Subalgebra type: \displaystyle A^{2}_1+2A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_1+A^{1}_1 .
Centralizer: \displaystyle A^{2}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle B^{1}_3
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \displaystyle 2A^{2}_1+2A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_1: (2, 2, 3, 4, 3, 2): 4, \displaystyle A^{1}_1: (0, 1, 1, 2, 1, 0): 2, \displaystyle A^{1}_1: (0, 1, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-30}+g_{-34}, \displaystyle g_{-23}, \displaystyle g_{-2}
Positive simple generators: \displaystyle g_{34}+g_{30}, \displaystyle g_{23}, \displaystyle g_{2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 3V_{\omega_{2}+\omega_{3}}\oplus 4V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 4V_{2\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}+4\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-4\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}} \oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-2\psi_{2}}
Made total 713 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{2}_1+A^{1}_1 ↪ E^{1}_6
59 out of 119
Subalgebra type: \displaystyle 2A^{2}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle 2A^{2}_1 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2)
Contained up to conjugation as a direct summand of: \displaystyle 2A^{2}_1+2A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_1: (2, 2, 3, 4, 3, 2): 4, \displaystyle A^{2}_1: (0, 2, 1, 2, 1, 0): 4, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{-13}+g_{-14}, \displaystyle g_{-15}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{14}+g_{13}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{2\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus 2V_{2\omega_{1}+\omega_{3}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{2}+2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}+6\psi_{2}} \oplus V_{2\omega_{2}+\omega_{3}+2\psi_{1}}\oplus V_{2\omega_{1}+\omega_{3}+2\psi_{1}}\oplus V_{4\psi_{1}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{2}+\omega_{3}-2\psi_{1}}\oplus V_{2\omega_{1}+\omega_{3}-2\psi_{1}} \oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{2}+2\psi_{1}-6\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi_{2}}\oplus V_{-4\psi_{1}} \oplus V_{\omega_{1}+\omega_{2}-2\psi_{1}-6\psi_{2}}
Made total 679007 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 3A^{2}_1 ↪ E^{1}_6
60 out of 119
Subalgebra type: \displaystyle 3A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle 2A^{2}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_1: (2, 2, 3, 4, 3, 2): 4, \displaystyle A^{2}_1: (0, 2, 1, 2, 1, 0): 4, \displaystyle A^{2}_1: (0, 0, 1, 2, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-30}+g_{-34}, \displaystyle g_{-8}+g_{-19}, \displaystyle g_{-9}+g_{-10}
Positive simple generators: \displaystyle g_{34}+g_{30}, \displaystyle g_{19}+g_{8}, \displaystyle g_{10}+g_{9}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 4V_{\omega_{1}+\omega_{2}+\omega_{3}} \oplus 2V_{2\omega_{3}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{2\omega_{3}}\oplus 2V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}
Made total 73384 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{3}_1+A^{2}_1+A^{1}_1 ↪ E^{1}_6
61 out of 119
Subalgebra type: \displaystyle A^{3}_1+A^{2}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{3}_1+A^{2}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{3}_1: (2, 3, 4, 6, 4, 2): 6, \displaystyle A^{2}_1: (1, 0, 1, 0, 1, 1): 4, \displaystyle A^{1}_1: (0, 1, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-29}+g_{-30}+g_{-31}, \displaystyle g_{-7}-g_{-11}, \displaystyle g_{-2}
Positive simple generators: \displaystyle g_{31}+g_{30}+g_{29}, \displaystyle -g_{11}+g_{7}, \displaystyle g_{2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+2\omega_{2}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}} \oplus 2V_{2\omega_{1}+\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 2V_{\omega_{2}} \oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{1}+\omega_{2}+6\psi}\oplus V_{\omega_{2}+6\psi}\oplus V_{\omega_{1}+2\omega_{2}+\omega_{3}} \oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}} \oplus 2V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}\oplus V_{2\omega_{1}+\omega_{2}-6\psi}\oplus V_{\omega_{2}-6\psi}
Made total 743 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{4}_1+2A^{1}_1 ↪ E^{1}_6
62 out of 119
Subalgebra type: \displaystyle A^{4}_1+2A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{4}_1+A^{1}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, -1, -1, 0), (1, 0, 0, 2, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{4}_1: (2, 4, 4, 6, 4, 2): 8, \displaystyle A^{1}_1: (1, 0, 1, 1, 1, 1): 2, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-22}+g_{-28}, \displaystyle g_{-24}, \displaystyle g_{-15}
Positive simple generators: \displaystyle 2g_{28}+2g_{22}, \displaystyle g_{24}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}8 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{2\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{4\omega_{1}}\oplus 2V_{2\omega_{1}+\omega_{3}}\oplus 2V_{2\omega_{1}+\omega_{2}} \oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus 3V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{1}-4\psi_{1}+8\psi_{2}} \oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{3}-6\psi_{1}+6\psi_{2}}\oplus V_{2\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+6\psi_{1}-6\psi_{2}} \oplus 2V_{0}\oplus V_{2\omega_{1}+\omega_{3}-2\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{1}+4\psi_{1}-8\psi_{2}}\oplus V_{\omega_{2}-6\psi_{2}}
Made total 715 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{4}_1+A^{1}_1 ↪ E^{1}_6
63 out of 119
Subalgebra type: \displaystyle 2A^{4}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle 2A^{4}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 1, 1, -1, 0)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{4}_1: (2, 4, 4, 6, 4, 2): 8, \displaystyle A^{4}_1: (2, 0, 2, 2, 2, 2): 8, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-25}+g_{-26}, \displaystyle g_{-7}+g_{-16}, \displaystyle g_{-15}
Positive simple generators: \displaystyle 2g_{26}+2g_{25}, \displaystyle 2g_{16}+2g_{7}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/2 & 0 & 0\\
0 & 1/2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}8 & 0 & 0\\
0 & 8 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{2\omega_{1}+2\omega_{2}+\omega_{3}}\oplus V_{4\omega_{2}}\oplus 2V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\omega_{1}+2\omega_{2}+4\psi}\oplus V_{\omega_{3}+6\psi}\oplus V_{2\omega_{1}+2\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\omega_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}+2\omega_{2}+\omega_{3}-2\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{0}\oplus V_{2\omega_{1}+2\omega_{2}-4\psi}\oplus V_{\omega_{3}-6\psi}
Made total 954 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 3A^{4}_1 ↪ E^{1}_6
64 out of 119
Subalgebra type: \displaystyle 3A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle 2A^{4}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{4}_1: (2, 4, 4, 6, 4, 2): 8, \displaystyle A^{4}_1: (2, 0, 2, 2, 2, 2): 8, \displaystyle A^{4}_1: (0, 0, 2, 2, 2, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-25}+g_{-26}, \displaystyle g_{-7}+g_{-16}, \displaystyle g_{-5}+g_{-9}
Positive simple generators: \displaystyle 2g_{26}+2g_{25}, \displaystyle 2g_{16}+2g_{7}, \displaystyle 2g_{9}+2g_{5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/2 & 0 & 0\\
0 & 1/2 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}8 & 0 & 0\\
0 & 8 & 0\\
0 & 0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{2\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{4\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}
Made total 45630 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{8}_1+A^{3}_1+A^{1}_1 ↪ E^{1}_6
65 out of 119
Subalgebra type: \displaystyle A^{8}_1+A^{3}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{8}_1+A^{3}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{8}_1: (4, 4, 6, 8, 6, 4): 16, \displaystyle A^{3}_1: (0, 2, 2, 3, 2, 0): 6, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-17}+g_{-18}+g_{-20}+g_{-21}, \displaystyle g_{-13}+g_{-14}-g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle 2g_{21}+2g_{20}+2g_{18}+2g_{17}, \displaystyle -g_{15}+g_{14}+g_{13}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/4 & 0 & 0\\
0 & 2/3 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}16 & 0 & 0\\
0 & 6 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{4\omega_{1}+2\omega_{2}}\oplus V_{3\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+\omega_{2}+\omega_{3}} \oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}
Made total 925 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{10}_1+2A^{1}_1 ↪ E^{1}_6
66 out of 119
Subalgebra type: \displaystyle A^{10}_1+2A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{10}_1+A^{1}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{10}_1: (4, 6, 7, 10, 7, 4): 20, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: \displaystyle g_{-13}+g_{-14}+g_{-24}, \displaystyle g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle 4g_{24}+3g_{14}+3g_{13}, \displaystyle g_{15}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1/5 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}20 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{6\omega_{1}}\oplus 2V_{3\omega_{1}+\omega_{3}}\oplus 2V_{3\omega_{1}+\omega_{2}} \oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{1}+\omega_{3}+6\psi}\oplus V_{3\omega_{1}+\omega_{2}+6\psi}\oplus V_{4\omega_{1}+\omega_{2}+\omega_{3}} \oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0} \oplus V_{3\omega_{1}+\omega_{3}-6\psi}\oplus V_{3\omega_{1}+\omega_{2}-6\psi}
Made total 638 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+A^{1}_1 ↪ E^{1}_6
67 out of 119
Subalgebra type: \displaystyle A^{1}_2+A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: \displaystyle A^{1}_2 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle 2A^{1}_2
Basis of Cartan of centralizer: 3 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, -1)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_2+2A^{1}_1 , \displaystyle A^{1}_2+A^{4}_1+A^{1}_1 , \displaystyle 2A^{1}_2+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 0, 1, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-11}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{3}}\oplus 3V_{\omega_{2}+\omega_{3}}\oplus 3V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{3}} \oplus 3V_{\omega_{2}}\oplus 3V_{\omega_{1}}\oplus 9V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+6\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}+4\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{2}+2\psi_{3}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}-2\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}+2\psi_{3}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+2\psi_{1}-2\psi_{3}} \oplus 3V_{0}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}+2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{2}-4\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}} \oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{2}-2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{1}-2\psi_{2}-4\psi_{3}} \oplus V_{-2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{3}}\oplus V_{\omega_{3}-6\psi_{3}}
Made total 450 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+A^{2}_1 ↪ E^{1}_6
68 out of 119
Subalgebra type: \displaystyle A^{1}_2+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(1, 0, -1, 0, 0, 0), (0, 0, 0, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{2}_1: (1, 0, 1, 0, 1, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-7}+g_{-11}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}+g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}}\oplus 2V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}} \oplus V_{\omega_{1}+\omega_{2}}\oplus 4V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+\omega_{3}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+6\psi_{2}} \oplus V_{\omega_{3}+6\psi_{1}}\oplus V_{\omega_{1}+2\omega_{3}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+2\omega_{3}+2\psi_{1}-2\psi_{2}} \oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}-4\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-4\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{2}+\omega_{3}-4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{3}-6\psi_{1}}\oplus V_{\omega_{3}-6\psi_{2}}
Made total 2784 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+A^{4}_1 ↪ E^{1}_6
69 out of 119
Subalgebra type: \displaystyle A^{1}_2+A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: \displaystyle A^{1}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle 2A^{1}_2
Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_2+A^{4}_1+A^{1}_1 , \displaystyle A^{1}_2+2A^{4}_1 , \displaystyle 2A^{1}_2+A^{4}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{4}_1: (0, 0, 0, 0, 2, 2): 8
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-5}+g_{-6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle 2g_{6}+2g_{5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{3}}\oplus 3V_{\omega_{2}+2\omega_{3}}\oplus 3V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 8V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+2\omega_{3}+2\psi_{2}}\oplus V_{\omega_{2}+2\omega_{3}+2\psi_{1}}\oplus V_{2\psi_{1}+2\psi_{2}}\oplus V_{4\omega_{3}} \oplus V_{\omega_{2}+2\omega_{3}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+2\omega_{3}+2\psi_{1}-2\psi_{2}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+2\omega_{3}-2\psi_{1}}\oplus V_{\omega_{2}+2\omega_{3}-2\psi_{2}} \oplus 2V_{0}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}
Made total 2780 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+A^{5}_1 ↪ E^{1}_6
70 out of 119
Subalgebra type: \displaystyle A^{1}_2+A^{5}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{5}_1: (1, 0, 1, 0, 2, 2): 10
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-5}+g_{-6}+g_{-7}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{7}+2g_{6}+2g_{5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2/5\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 10\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{3}}\oplus V_{\omega_{2}+3\omega_{3}}\oplus V_{\omega_{1}+3\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}} \oplus 2V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{3}} \oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+6\psi}\oplus V_{\omega_{1}+2\omega_{3}+4\psi}\oplus V_{\omega_{2}+3\omega_{3}+2\psi}\oplus V_{\omega_{2}+\omega_{3}+2\psi} \oplus V_{4\omega_{3}}\oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+3\omega_{3}-2\psi}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi} \oplus V_{\omega_{2}+2\omega_{3}-4\psi}\oplus V_{\omega_{3}-6\psi}
Made total 10697 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+A^{8}_1 ↪ E^{1}_6
71 out of 119
Subalgebra type: \displaystyle A^{1}_2+A^{8}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{8}_1: (2, 0, 2, 0, 2, 2): 16
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-1}+g_{-3}+g_{-5}+g_{-6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle 2g_{6}+2g_{5}+2g_{3}+2g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/4\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 16\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+4\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus 2V_{4\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}} \oplus 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}
Made total 30250 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_2+A^{1}_1 ↪ E^{1}_6
72 out of 119
Subalgebra type: \displaystyle B^{1}_2+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_2 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 0, 1, 0, 0), (2, 0, 1, 0, -1, -2)
Contained up to conjugation as a direct summand of: \displaystyle B^{1}_2+2A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{14}+g_{13}, \displaystyle g_{-15}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-13}+g_{-14}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{3}}\oplus 4V_{\omega_{2}} \oplus V_{\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+2\psi_{1}+6\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+6\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}+6\psi_{2}} \oplus V_{4\psi_{1}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}}\oplus V_{\omega_{3}+2\psi_{1}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}}\oplus V_{\omega_{3}-2\psi_{1}}\oplus V_{\omega_{2}+2\psi_{1}-6\psi_{2}} \oplus V_{-4\psi_{1}}\oplus V_{\omega_{2}+\omega_{3}-6\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}-6\psi_{2}}
Made total 22148 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_2+A^{2}_1 ↪ E^{1}_6
73 out of 119
Subalgebra type: \displaystyle B^{1}_2+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_2 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \displaystyle A^{2}_1: (0, 0, 1, 2, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{19}+g_{8}, \displaystyle -g_{-4}+g_{-15}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-8}+g_{-19}, \displaystyle g_{15}-g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+2\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus 4V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{2\omega_{3}+4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{3}-4\psi_{1}+2\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{2}+\omega_{3}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}-2\psi_{2}}
Made total 262431 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_2+A^{10}_1 ↪ E^{1}_6
74 out of 119
Subalgebra type: \displaystyle B^{1}_2+A^{10}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_2 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \displaystyle A^{10}_1: (0, 0, 3, 4, 3, 0): 20
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{14}+g_{13}, \displaystyle g_{-3}+g_{-4}+g_{-5}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-13}+g_{-14}, \displaystyle 3g_{5}+4g_{4}+3g_{3}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 1 & 0\\
0 & 0 & 1/5\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 20\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{6\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus 2V_{\omega_{2}+3\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+3\omega_{3}+6\psi}\oplus V_{6\omega_{3}}\oplus V_{\omega_{1}+4\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{0}\oplus V_{\omega_{2}+3\omega_{3}-6\psi}
Made total 1458590 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra G^{1}_2+A^{2}_1 ↪ E^{1}_6
75 out of 119
Subalgebra type: \displaystyle G^{1}_2+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle G^{1}_2 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle G^{1}_2: (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2, \displaystyle A^{2}_1: (1, 0, 1, 0, 1, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 17.
Negative simple generators: \displaystyle g_{-29}+g_{-30}+g_{-31}, \displaystyle g_{35}, \displaystyle g_{-7}-g_{-11}
Positive simple generators: \displaystyle g_{31}+g_{30}+g_{29}, \displaystyle g_{-35}, \displaystyle -g_{11}+g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & -3 & 0\\
-3 & 2 & 0\\
0 & 0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}} \oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{3}+6\psi}\oplus V_{\omega_{3}+6\psi}\oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}} \oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-6\psi}\oplus V_{\omega_{3}-6\psi}
Made total 1471286 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra G^{1}_2+A^{8}_1 ↪ E^{1}_6
76 out of 119
Subalgebra type: \displaystyle G^{1}_2+A^{8}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle G^{1}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle G^{1}_2: (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2, \displaystyle A^{8}_1: (2, 0, 2, 0, 2, 2): 16
Dimension of subalgebra generated by predefined or computed generators: 17.
Negative simple generators: \displaystyle g_{-29}+g_{-30}+g_{-31}, \displaystyle g_{35}, \displaystyle g_{-1}+g_{-3}-g_{-5}+g_{-6}
Positive simple generators: \displaystyle g_{31}+g_{30}+g_{29}, \displaystyle g_{-35}, \displaystyle 2g_{6}-2g_{5}+2g_{3}+2g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/3 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 1/4\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}6 & -3 & 0\\
-3 & 2 & 0\\
0 & 0 & 16\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+4\omega_{3}}\oplus V_{4\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}}
Made total 5318551 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{1}_1 ↪ E^{1}_6
77 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, -1, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \displaystyle A^{1}_1: (0, 1, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle g_{-8}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{8}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{2}+\omega_{3}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{2\omega_{2}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+6\psi}\oplus V_{2\omega_{2}+4\psi}\oplus V_{\omega_{1}+4\psi}\oplus V_{2\omega_{1}+\omega_{3}+2\psi}\oplus V_{\omega_{2}+\omega_{3}+2\psi} \oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}+\omega_{3}-2\psi}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi} \oplus V_{2\omega_{1}-4\psi}\oplus V_{\omega_{2}-4\psi}\oplus V_{\omega_{3}-6\psi}
Made total 630 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{1}_1 ↪ E^{1}_6
78 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: \displaystyle A^{3}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{2}_2+A^{1}_1
Basis of Cartan of centralizer: 1 vectors:
(0, 1, -1, 0, 1, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{2}_2+A^{3}_1+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle A^{1}_1: (0, 1, 1, 2, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-30}+g_{-34}, \displaystyle g_{25}+g_{16}, \displaystyle g_{-23}
Positive simple generators: \displaystyle g_{34}+g_{30}, \displaystyle g_{-16}+g_{-25}, \displaystyle g_{23}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus 4V_{\omega_{1}+\omega_{2}}\oplus 4V_{\omega_{3}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+6\psi}\oplus V_{\omega_{1}+\omega_{2}+4\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\psi} \oplus V_{\omega_{3}+2\psi}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi} \oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{\omega_{1}+\omega_{2}-4\psi}\oplus V_{-4\psi}\oplus V_{\omega_{3}-6\psi}
Made total 958 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{3}_1 ↪ E^{1}_6
79 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{3}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: \displaystyle A^{1}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{2}_2+A^{3}_1+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle A^{3}_1: (0, 2, 2, 3, 2, 0): 6
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{21}+g_{20}, \displaystyle g_{-13}+g_{-14}-g_{-15}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-20}+g_{-21}, \displaystyle -g_{15}+g_{14}+g_{13}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 2/3\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 6\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus 2V_{3\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}} \oplus V_{\omega_{1}+\omega_{2}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{3\omega_{3}+2\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+2\psi}\oplus V_{4\psi}\oplus V_{\omega_{1}+\omega_{2}+2\omega_{3}} \oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{3\omega_{3}-2\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-2\psi} \oplus V_{0}\oplus V_{-4\psi}
Made total 5136524 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{4}_1 ↪ E^{1}_6
80 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \displaystyle A^{4}_1: (0, 2, 0, 2, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle g_{-2}+g_{-4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle 2g_{4}+2g_{2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{3}}\oplus V_{2\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}}\oplus V_{\omega_{1}+2\omega_{3}} \oplus V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}
Made total 3810 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{4}_1 ↪ E^{1}_6
81 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle A^{4}_1: (0, 2, 2, 4, 2, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-30}+g_{-34}, \displaystyle g_{25}+g_{16}, \displaystyle g_{-8}+g_{-9}-g_{-10}+g_{-19}
Positive simple generators: \displaystyle g_{34}+g_{30}, \displaystyle g_{-16}+g_{-25}, \displaystyle g_{19}-g_{10}+g_{9}+g_{8}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus 3V_{2\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}
Made total 6862360 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{28}_1 ↪ E^{1}_6
82 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{28}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle A^{28}_1: (0, 6, 6, 10, 6, 0): 56
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{21}+g_{20}, \displaystyle g_{-2}-g_{-3}+g_{-4}+g_{-5}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-20}+g_{-21}, \displaystyle 6g_{5}+10g_{4}-6g_{3}+6g_{2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & 0\\
0 & 0 & 1/14\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & 0\\
0 & 0 & 56\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{10\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}+6\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}
Made total 6570789 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{2}_2+A^{1}_1 ↪ E^{1}_6
83 out of 119
Subalgebra type: \displaystyle B^{2}_2+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{2}_2 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 3, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{2}_2: (2, 2, 3, 4, 3, 2): 4, (-2, 0, -2, -2, -2, -2): 8, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{16}+g_{7}, \displaystyle g_{-15}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle 2g_{-7}+2g_{-16}, \displaystyle g_{15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1/2 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -4 & 0\\
-4 & 8 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{2\omega_{2}+\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+12\psi}\oplus V_{2\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0} \oplus V_{2\omega_{2}+\omega_{3}-6\psi}\oplus V_{\omega_{1}-12\psi}
Made total 2168207 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{5}_2+A^{1}_1 ↪ E^{1}_6
84 out of 119
Subalgebra type: \displaystyle A^{5}_2+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{5}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{5}_2: (3, 4, 5, 7, 5, 3): 10, (-1, -2, 0, -2, -2, -2): 10, \displaystyle A^{1}_1: (0, 0, 0, 1, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators: \displaystyle g_{-22}+g_{-24}+g_{-28}, \displaystyle g_{11}+2g_{8}+g_{1}, \displaystyle g_{-10}
Positive simple generators: \displaystyle 2g_{28}+g_{24}+2g_{22}, \displaystyle g_{-1}+g_{-8}+2g_{-11}, \displaystyle g_{10}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2/5 & -1/5 & 0\\
-1/5 & 2/5 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}10 & -5 & 0\\
-5 & 10 & 0\\
0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{3\omega_{2}+\omega_{3}}\oplus V_{3\omega_{1}+\omega_{3}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}
Made total 812 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_3 ↪ E^{1}_6
85 out of 119
Subalgebra type: \displaystyle A^{1}_3 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: \displaystyle 2A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_3
Basis of Cartan of centralizer: 3 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0), (0, 0, 0, 0, 0, 1)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_3+A^{1}_1 , \displaystyle A^{1}_3+A^{2}_1 , \displaystyle A^{1}_3+2A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_3: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{3}}\oplus 4V_{\omega_{3}}\oplus 4V_{\omega_{2}}\oplus 4V_{\omega_{1}}\oplus 7V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+4\psi_{2}+2\psi_{3}}\oplus V_{2\psi_{2}+4\psi_{3}}\oplus V_{\omega_{2}+2\psi_{1}+2\psi_{3}}\oplus V_{\omega_{3}+2\psi_{1}+2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}+2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{2}+2\psi_{3}}\oplus V_{\omega_{1}+2\psi_{2}-2\psi_{3}}\oplus 3V_{0}\oplus V_{\omega_{1}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{2}+2\psi_{1}-2\psi_{2}-2\psi_{3}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-2\psi_{2}} \oplus V_{\omega_{2}-2\psi_{1}-2\psi_{3}}\oplus V_{\omega_{3}-4\psi_{2}-2\psi_{3}}\oplus V_{-2\psi_{2}-4\psi_{3}}
Made total 448 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_3 ↪ E^{1}_6
86 out of 119
Subalgebra type: \displaystyle B^{1}_3 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2 .
Centralizer: \displaystyle A^{2}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle B^{1}_3
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Contained up to conjugation as a direct summand of: \displaystyle B^{1}_3+A^{2}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_3: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, -1, -2, -1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{15}+g_{4}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}+g_{-15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0\\
-1 & 2 & -2\\
0 & -2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 4V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 3V_{\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 2V_{0}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+2\psi_{1}-4\psi_{2}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}-2\psi_{2}}
Made total 201627 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra C^{1}_3 ↪ E^{1}_6
87 out of 119
Subalgebra type: \displaystyle C^{1}_3 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: \displaystyle A^{1}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 1 vectors:
(0, 0, 0, 1, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle C^{1}_3+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle C^{1}_3: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, -1, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle g_{15}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{-15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & -1\\
0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & -2\\
0 & -2 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{1}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{4\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{1}+2\psi}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}-2\psi} \oplus V_{\omega_{1}-2\psi}\oplus V_{-4\psi}
Made total 632 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_3 ↪ E^{1}_6
88 out of 119
Subalgebra type: \displaystyle A^{2}_3 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2 .
Centralizer: \displaystyle A^{1}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 1 vectors:
(0, 1, 0, 0, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{2}_3+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_3: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, 0, -1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 15.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle -g_{5}+g_{3}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{-3}-g_{-5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0\\
-1/2 & 1 & -1/2\\
0 & -1/2 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0\\
-2 & 4 & -2\\
0 & -2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus 2V_{2\omega_{1}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{4\psi}\oplus V_{2\omega_{3}+2\psi}\oplus V_{2\omega_{1}+2\psi}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{2\omega_{3}-2\psi} \oplus V_{2\omega_{1}-2\psi}\oplus V_{-4\psi}
Made total 8556 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 4A^{1}_1 ↪ E^{1}_6
89 out of 119
Subalgebra type: \displaystyle 4A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle 3A^{1}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_1: (1, 2, 2, 3, 2, 1): 2, \displaystyle A^{1}_1: (1, 0, 1, 1, 1, 1): 2, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{-24}, \displaystyle g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{24}, \displaystyle g_{15}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}} \oplus 2V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}} \oplus 2V_{\omega_{1}+\omega_{2}}\oplus V_{2\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+\omega_{4}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{4}-2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{2}+\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}+4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+\omega_{2}+4\psi_{1}-2\psi_{2}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}} \oplus V_{\omega_{3}+\omega_{4}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{2}-4\psi_{1}+2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{4}+2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{2}+\omega_{3}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{2}+\omega_{4}-2\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}-2\psi_{2}}
Made total 543 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{2}_1+2A^{1}_1 ↪ E^{1}_6
90 out of 119
Subalgebra type: \displaystyle 2A^{2}_1+2A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle 2A^{2}_1+A^{1}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_1: (2, 2, 3, 4, 3, 2): 4, \displaystyle A^{2}_1: (0, 2, 1, 2, 1, 0): 4, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 12.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{-13}+g_{-14}, \displaystyle g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{14}+g_{13}, \displaystyle g_{15}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & 0 & 0 & 0\\
0 & 4 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+2\omega_{2}} \oplus 2V_{\omega_{1}+\omega_{2}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}} \oplus V_{2\omega_{2}}\oplus V_{2\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{2}+\omega_{4}+6\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+6\psi}\oplus V_{2\omega_{2}+\omega_{3}+\omega_{4}} \oplus V_{2\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+2\omega_{2}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}} \oplus V_{2\omega_{1}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{2}+\omega_{4}-6\psi}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}-6\psi}
Made total 731 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+2A^{1}_1 ↪ E^{1}_6
91 out of 119
Subalgebra type: \displaystyle A^{1}_2+2A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2+A^{1}_1 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(1, 0, -1, 0, 0, 0), (0, 0, 0, 0, 1, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 0, 1, 1): 2, \displaystyle A^{1}_1: (1, 0, 1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-11}, \displaystyle g_{-7}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}, \displaystyle g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}+\omega_{4}} \oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 2V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+\omega_{4}+2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+6\psi_{2}} \oplus V_{\omega_{4}+6\psi_{1}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{4}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{3}+\omega_{4}+2\psi_{1}-2\psi_{2}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}-4\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-4\psi_{2}} \oplus 2V_{0}\oplus V_{\omega_{2}+\omega_{3}-4\psi_{1}-2\psi_{2}}\oplus V_{\omega_{1}+\omega_{4}-2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{4}-6\psi_{1}} \oplus V_{\omega_{3}-6\psi_{2}}
Made total 535 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+A^{4}_1+A^{1}_1 ↪ E^{1}_6
92 out of 119
Subalgebra type: \displaystyle A^{1}_2+A^{4}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2+A^{4}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{4}_1: (0, 0, 0, 0, 2, 2): 8, \displaystyle A^{1}_1: (1, 0, 1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-5}+g_{-6}, \displaystyle g_{-7}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle 2g_{6}+2g_{5}, \displaystyle g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 1/2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 8 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+2\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+2\omega_{3}+\omega_{4}}\oplus V_{4\omega_{3}}\oplus V_{\omega_{2}+2\omega_{3}} \oplus V_{\omega_{1}+2\omega_{3}}\oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{4}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{4}+6\psi}\oplus V_{\omega_{1}+2\omega_{3}+4\psi}\oplus V_{\omega_{2}+2\omega_{3}+\omega_{4}+2\psi}\oplus V_{4\omega_{3}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+2\omega_{3}+\omega_{4}-2\psi}\oplus V_{0} \oplus V_{\omega_{2}+2\omega_{3}-4\psi}\oplus V_{\omega_{4}-6\psi}
Made total 624 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_2+2A^{4}_1 ↪ E^{1}_6
93 out of 119
Subalgebra type: \displaystyle A^{1}_2+2A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2+A^{4}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{4}_1: (0, 0, 0, 0, 2, 2): 8, \displaystyle A^{4}_1: (2, 0, 2, 0, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-5}+g_{-6}, \displaystyle g_{-1}+g_{-3}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle 2g_{6}+2g_{5}, \displaystyle 2g_{3}+2g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 1/2 & 0\\
0 & 0 & 0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 8 & 0\\
0 & 0 & 0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+2\omega_{3}+2\omega_{4}}\oplus V_{\omega_{1}+2\omega_{3}+2\omega_{4}}\oplus V_{4\omega_{4}}\oplus V_{4\omega_{3}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}
Made total 3798 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{1}_2 ↪ E^{1}_6
94 out of 119
Subalgebra type: \displaystyle 2A^{1}_2 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_2+A^{1}_1 .
Centralizer: \displaystyle A^{1}_2 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle 2A^{1}_2
Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle 2A^{1}_2+A^{1}_1 , \displaystyle 2A^{1}_2+A^{4}_1 , \displaystyle 3A^{1}_2 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{1}_2: (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-11}, \displaystyle g_{6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}, \displaystyle g_{-6}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{3}+\omega_{4}}\oplus 3V_{\omega_{2}+\omega_{4}}\oplus 3V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}} \oplus 8V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{1}+\omega_{3}+2\psi_{2}}\oplus V_{\omega_{2}+\omega_{4}+2\psi_{1}}\oplus V_{-2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{2}+\omega_{4}-2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{1}+\omega_{3}+2\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}+\omega_{3}-2\psi_{1}}\oplus V_{\omega_{2}+\omega_{4}-2\psi_{2}} \oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{2\psi_{1}-4\psi_{2}}\oplus V_{-2\psi_{1}-2\psi_{2}}
Made total 533 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_2+2A^{1}_1 ↪ E^{1}_6
95 out of 119
Subalgebra type: \displaystyle B^{1}_2+2A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_2+A^{1}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \displaystyle A^{1}_1: (0, 0, 1, 1, 1, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{14}+g_{13}, \displaystyle g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-13}+g_{-14}, \displaystyle g_{15}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}} \oplus V_{2\omega_{3}}\oplus 2V_{\omega_{2}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+\omega_{4}+6\psi}\oplus V_{\omega_{2}+\omega_{3}+6\psi}\oplus V_{\omega_{1}+\omega_{3}+\omega_{4}}\oplus V_{2\omega_{4}} \oplus V_{\omega_{3}+\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-6\psi} \oplus V_{\omega_{2}+\omega_{3}-6\psi}
Made total 628 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2B^{1}_2 ↪ E^{1}_6
96 out of 119
Subalgebra type: \displaystyle 2B^{1}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_2+A^{1}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -2, -1, -2, -1, 0): 4, \displaystyle B^{1}_2: (0, 0, 1, 1, 1, 0): 2, (0, 0, -1, 0, -1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 20.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{14}+g_{13}, \displaystyle g_{-15}, \displaystyle g_{5}+g_{3}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-13}+g_{-14}, \displaystyle g_{15}, \displaystyle g_{-3}+g_{-5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 1 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{2}+\omega_{4}+6\psi}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-6\psi}
Made total 8550 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{3}_1+A^{1}_1 ↪ E^{1}_6
97 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{3}_1+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2+A^{3}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle A^{3}_1: (0, 2, 2, 3, 2, 0): 6, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{21}+g_{20}, \displaystyle g_{-13}+g_{-14}-g_{-15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-20}+g_{-21}, \displaystyle -g_{15}+g_{14}+g_{13}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2/3 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 6 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{3\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}+2\omega_{3}} \oplus V_{2\omega_{4}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}
Made total 913 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{1}_2 ↪ E^{1}_6
98 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{1}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2+A^{1}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle A^{1}_2: (0, 1, 1, 2, 1, 0): 2, (0, 0, 0, -1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{21}+g_{20}, \displaystyle g_{-23}, \displaystyle g_{4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-20}+g_{-21}, \displaystyle g_{23}, \displaystyle g_{-4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{2}} \oplus V_{\omega_{4}}\oplus V_{\omega_{3}}
Made total 130682 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+A^{1}_2 ↪ E^{1}_6
99 out of 119
Subalgebra type: \displaystyle A^{2}_2+A^{1}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2+A^{1}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, \displaystyle A^{1}_2: (0, 1, 0, 1, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 16.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle g_{-8}, \displaystyle g_{4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{8}, \displaystyle g_{-4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{2}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{3}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}} \oplus V_{\omega_{2}+\omega_{3}}\oplus 2V_{\omega_{1}+\omega_{2}}
Made total 721 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_2+G^{1}_2 ↪ E^{1}_6
100 out of 119
Subalgebra type: \displaystyle A^{2}_2+G^{1}_2 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_2+A^{3}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_2: (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4, \displaystyle G^{1}_2: (0, 2, 2, 3, 2, 0): 6, (0, -1, -1, -1, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 22.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{21}+g_{20}, \displaystyle g_{-13}+g_{-14}-g_{-15}, \displaystyle g_{19}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-20}+g_{-21}, \displaystyle -g_{15}+g_{14}+g_{13}, \displaystyle g_{-19}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & 0 & 0\\
0 & 0 & 2/3 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & 0 & 0\\
0 & 0 & 6 & -3\\
0 & 0 & -3 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{2}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{4}}
Made total 921 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_3+A^{1}_1 ↪ E^{1}_6
101 out of 119
Subalgebra type: \displaystyle A^{1}_3+A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_3 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 2, 0, -2, -1)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_3+2A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_3: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 0, 0, 1): 2
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{-6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{6}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}} \oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+\omega_{4}+6\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}+8\psi_{2}}\oplus V_{\omega_{3}+2\psi_{1}+4\psi_{2}} \oplus V_{\omega_{2}+\omega_{4}-2\psi_{1}+2\psi_{2}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}+\omega_{4}+2\psi_{1}-2\psi_{2}} \oplus V_{-4\psi_{1}+4\psi_{2}}\oplus 2V_{0}\oplus V_{4\psi_{1}-4\psi_{2}}\oplus V_{\omega_{3}+\omega_{4}-6\psi_{2}}\oplus V_{\omega_{1}-2\psi_{1}-4\psi_{2}} \oplus V_{\omega_{1}+2\psi_{1}-8\psi_{2}}
Made total 531 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_3+A^{2}_1 ↪ E^{1}_6
102 out of 119
Subalgebra type: \displaystyle A^{1}_3+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_3 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_3: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, \displaystyle A^{2}_1: (1, 0, 0, 0, 0, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{-1}+g_{-6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{6}+g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+2\omega_{4}}\oplus 2V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus 2V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}} \oplus V_{\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+\omega_{4}+6\psi}\oplus V_{\omega_{1}+\omega_{4}+6\psi}\oplus V_{\omega_{2}+2\omega_{4}}\oplus 2V_{2\omega_{4}} \oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}+\omega_{4}-6\psi}\oplus V_{\omega_{1}+\omega_{4}-6\psi}
Made total 3288 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_3+A^{2}_1 ↪ E^{1}_6
103 out of 119
Subalgebra type: \displaystyle B^{1}_3+A^{2}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_3 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_3: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, -1, -2, -1, 0): 4, \displaystyle A^{2}_1: (0, 0, 1, 0, 1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{10}+g_{9}, \displaystyle g_{-3}+g_{-5}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-9}+g_{-10}, \displaystyle g_{5}+g_{3}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -2 & 0\\
0 & -2 & 4 & 0\\
0 & 0 & 0 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+2\omega_{4}}\oplus V_{2\omega_{4}}\oplus 2V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+\omega_{4}+6\psi}\oplus V_{\omega_{1}+2\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{3}+\omega_{4}-6\psi}
Made total 1466930 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra C^{1}_3+A^{1}_1 ↪ E^{1}_6
104 out of 119
Subalgebra type: \displaystyle C^{1}_3+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle C^{1}_3 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle C^{1}_3: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, -1, -1, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle g_{15}, \displaystyle g_{-4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{-15}, \displaystyle g_{4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & -1 & 0\\
0 & -1 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & -2 & 0\\
0 & -2 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{2\omega_{1}}\oplus V_{\omega_{2}}
Made total 715 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{2}_3+A^{1}_1 ↪ E^{1}_6
105 out of 119
Subalgebra type: \displaystyle A^{2}_3+A^{1}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_3 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{2}_3: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, 0, -1, 0): 4, \displaystyle A^{1}_1: (0, 1, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 18.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle -g_{5}+g_{3}, \displaystyle g_{-2}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{-3}-g_{-5}, \displaystyle g_{2}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & -1/2 & 0\\
0 & -1/2 & 1 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & -2 & 0\\
0 & -2 & 4 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{3}+\omega_{4}}\oplus V_{2\omega_{1}+\omega_{4}}\oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{2\omega_{2}}
Made total 796 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_4 ↪ E^{1}_6
106 out of 119
Subalgebra type: \displaystyle A^{1}_4 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_3 .
Centralizer: \displaystyle A^{1}_1 + \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 2 vectors:
(1, 0, 0, 0, 0, 0), (0, 0, 1, 0, -1, -2)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_4+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_4: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus 2V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus 4V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+6\psi_{2}}\oplus V_{\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+\omega_{4}} \oplus V_{4\psi_{1}-2\psi_{2}}\oplus 2V_{0}\oplus V_{\omega_{2}+2\psi_{1}-4\psi_{2}}\oplus V_{-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{2}-2\psi_{1}-2\psi_{2}} \oplus V_{\omega_{4}-6\psi_{2}}
Made total 531 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra D^{1}_4 ↪ E^{1}_6
107 out of 119
Subalgebra type: \displaystyle D^{1}_4 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_3 .
Centralizer: \displaystyle T_{2} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 2 vectors:
(0, 0, 1, 0, -1, 0), (1, 0, 0, 0, 0, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle D^{1}_4: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, -1, -1, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 28.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{15}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-15}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & -1\\
0 & -1 & 2 & 0\\
0 & -1 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & -1\\
0 & -1 & 2 & 0\\
0 & -1 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{\omega_{4}}\oplus 2V_{\omega_{3}}\oplus V_{\omega_{2}}\oplus 2V_{\omega_{1}}\oplus 2V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+2\psi_{1}+2\psi_{2}}\oplus V_{\omega_{4}-2\psi_{1}+4\psi_{2}}\oplus V_{\omega_{1}+4\psi_{1}-2\psi_{2}} \oplus V_{\omega_{2}}\oplus 2V_{0}\oplus V_{\omega_{1}-4\psi_{1}+2\psi_{2}}\oplus V_{\omega_{4}+2\psi_{1}-4\psi_{2}}\oplus V_{\omega_{3}-2\psi_{1}-2\psi_{2}}
Made total 535 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra B^{1}_4 ↪ E^{1}_6
108 out of 119
Subalgebra type: \displaystyle B^{1}_4 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_3 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle B^{1}_4: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, -1, 0, -1, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 36.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}+g_{3}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-3}+g_{-5}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle 2V_{\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{4}+6\psi}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\oplus V_{0}\oplus V_{\omega_{4}-6\psi}
Made total 3288 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra C^{1}_4 ↪ E^{1}_6
109 out of 119
Subalgebra type: \displaystyle C^{1}_4 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle A^{2}_3 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle C^{1}_4: (2, 2, 3, 4, 3, 2): 4, (-1, 0, 0, 0, 0, -1): 4, (0, 0, -1, 0, -1, 0): 4, (0, 0, 0, -1, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 36.
Negative simple generators: \displaystyle g_{-32}+g_{-33}, \displaystyle g_{6}+g_{1}, \displaystyle -g_{5}+g_{3}, \displaystyle g_{4}
Positive simple generators: \displaystyle g_{33}+g_{32}, \displaystyle g_{-1}+g_{-6}, \displaystyle g_{-3}-g_{-5}, \displaystyle g_{-4}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}1 & -1/2 & 0 & 0\\
-1/2 & 1 & -1/2 & 0\\
0 & -1/2 & 1 & -1\\
0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}4 & -2 & 0 & 0\\
-2 & 4 & -2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & -2 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{1}}\oplus V_{\omega_{4}}
Made total 796 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra F^{1}_4 ↪ E^{1}_6
110 out of 119
Subalgebra type: \displaystyle F^{1}_4 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle B^{1}_3 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle F^{1}_4: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, -1, -2, -1, 0): 4, (-1, 0, 0, 0, 0, -1): 4
Dimension of subalgebra generated by predefined or computed generators: 52.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{10}+g_{9}, \displaystyle g_{6}+g_{1}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-9}+g_{-10}, \displaystyle g_{-1}+g_{-6}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 1 & -1/2\\
0 & 0 & -1/2 & 1\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0\\
-1 & 2 & -2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & -2 & 4\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{4}}\oplus V_{\omega_{1}}
Made total 133227 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{1}_2+A^{1}_1 ↪ E^{1}_6
111 out of 119
Subalgebra type: \displaystyle 2A^{1}_2+A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle 2A^{1}_2 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, -1, 0, 0, 0)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{1}_2: (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, \displaystyle A^{1}_1: (1, 0, 1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 19.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-11}, \displaystyle g_{6}, \displaystyle g_{-7}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}, \displaystyle g_{-6}, \displaystyle g_{7}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{5}}\oplus V_{2\omega_{5}}\oplus V_{\omega_{3}+\omega_{4}} \oplus V_{\omega_{2}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{\omega_{5}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{5}+6\psi}\oplus V_{\omega_{1}+\omega_{3}+4\psi}\oplus V_{\omega_{2}+\omega_{4}+\omega_{5}+2\psi}\oplus V_{2\omega_{5}} \oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{1}+\omega_{3}+\omega_{5}-2\psi}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{4}-4\psi} \oplus V_{\omega_{5}-6\psi}
Made total 618 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 2A^{1}_2+A^{4}_1 ↪ E^{1}_6
112 out of 119
Subalgebra type: \displaystyle 2A^{1}_2+A^{4}_1 (click on type for detailed printout).
Subalgebra is (parabolically) induced from \displaystyle 2A^{1}_2 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{1}_2: (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, \displaystyle A^{4}_1: (2, 0, 2, 0, 0, 0): 8
Dimension of subalgebra generated by predefined or computed generators: 19.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-11}, \displaystyle g_{6}, \displaystyle g_{-1}+g_{-3}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}, \displaystyle g_{-6}, \displaystyle 2g_{3}+2g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 1/2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 8\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{4\omega_{5}}\oplus V_{\omega_{2}+\omega_{4}+2\omega_{5}}\oplus V_{\omega_{1}+\omega_{3}+2\omega_{5}}\oplus V_{2\omega_{5}} \oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{2}}
Made total 3798 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_3+2A^{1}_1 ↪ E^{1}_6
113 out of 119
Subalgebra type: \displaystyle A^{1}_3+2A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_3+A^{1}_1 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -1)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_3: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, \displaystyle A^{1}_1: (0, 0, 0, 0, 0, 1): 2, \displaystyle A^{1}_1: (1, 0, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 21.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{-6}, \displaystyle g_{-1}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{6}, \displaystyle g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{2\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{5}} \oplus V_{2\omega_{4}}\oplus V_{\omega_{3}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{3}+\omega_{5}+6\psi}\oplus V_{\omega_{1}+\omega_{4}+6\psi}\oplus V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{2\omega_{5}} \oplus V_{2\omega_{4}}\oplus V_{\omega_{1}+\omega_{3}}\oplus V_{0}\oplus V_{\omega_{1}+\omega_{5}-6\psi}\oplus V_{\omega_{3}+\omega_{4}-6\psi}
Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_4+A^{1}_1 ↪ E^{1}_6
114 out of 119
Subalgebra type: \displaystyle A^{1}_4+A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_4 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 2, 0, -2, -4)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_4: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, \displaystyle A^{1}_1: (1, 0, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 27.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}, \displaystyle g_{-1}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-5}, \displaystyle g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{5}}\oplus V_{\omega_{3}+\omega_{5}}\oplus V_{\omega_{2}+\omega_{5}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{\omega_{4}} \oplus V_{\omega_{1}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{1}+12\psi}\oplus V_{\omega_{3}+\omega_{5}+6\psi}\oplus V_{2\omega_{5}}\oplus V_{\omega_{1}+\omega_{4}}\oplus V_{0}\oplus V_{\omega_{2}+\omega_{5}-6\psi} \oplus V_{\omega_{4}-12\psi}
Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_5 ↪ E^{1}_6
115 out of 119
Subalgebra type: \displaystyle A^{1}_5 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_4 .
Centralizer: \displaystyle A^{1}_1 .
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle A^{1}_5
Basis of Cartan of centralizer: 1 vectors:
(1, 0, 0, 0, 0, 0)
Contained up to conjugation as a direct summand of: \displaystyle A^{1}_5+A^{1}_1 .
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_5: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, 0, 0, 0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 35.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}, \displaystyle g_{6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-5}, \displaystyle g_{-6}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & 0\\
0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{5}}\oplus 2V_{\omega_{3}}\oplus 3V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{4\psi}\oplus V_{\omega_{3}+2\psi}\oplus V_{\omega_{1}+\omega_{5}}\oplus V_{0}\oplus V_{\omega_{3}-2\psi}\oplus V_{-4\psi}
Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra D^{1}_5 ↪ E^{1}_6
116 out of 119
Subalgebra type: \displaystyle D^{1}_5 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_4 .
Centralizer: \displaystyle T_{1} (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Basis of Cartan of centralizer: 1 vectors:
(2, 0, 1, 0, -1, -2)
Elements Cartan subalgebra scaled to act by two by components: \displaystyle D^{1}_5: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, -1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 45.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}, \displaystyle g_{3}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-5}, \displaystyle g_{-3}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & -1\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & -1 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0\\
0 & -1 & 2 & -1 & -1\\
0 & 0 & -1 & 2 & 0\\
0 & 0 & -1 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{5}}\oplus V_{\omega_{4}}\oplus V_{\omega_{2}}\oplus V_{0}
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). \displaystyle V_{\omega_{5}+6\psi}\oplus V_{\omega_{2}}\oplus V_{0}\oplus V_{\omega_{4}-6\psi}
Made total 614 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra 3A^{1}_2 ↪ E^{1}_6
117 out of 119
Subalgebra type: \displaystyle 3A^{1}_2 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle 2A^{1}_2+A^{1}_1 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_2: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, \displaystyle A^{1}_2: (0, 0, 0, 0, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, \displaystyle A^{1}_2: (1, 0, 1, 0, 0, 0): 2, (0, 0, -1, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 24.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{-11}, \displaystyle g_{6}, \displaystyle g_{-7}, \displaystyle g_{3}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{11}, \displaystyle g_{-6}, \displaystyle g_{7}, \displaystyle g_{-3}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 0 & 2 & -1\\
0 & 0 & 0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & 0 & 0\\
0 & 0 & 0 & 0 & 2 & -1\\
0 & 0 & 0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{1}+\omega_{3}+\omega_{6}}\oplus V_{\omega_{2}+\omega_{4}+\omega_{5}}\oplus V_{\omega_{5}+\omega_{6}}\oplus V_{\omega_{3}+\omega_{4}} \oplus V_{\omega_{1}+\omega_{2}}
Made total 701 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra A^{1}_5+A^{1}_1 ↪ E^{1}_6
118 out of 119
Subalgebra type: \displaystyle A^{1}_5+A^{1}_1 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle A^{1}_5 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle A^{1}_5: (1, 2, 2, 3, 2, 1): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, 0, 0, 0, -1): 2, \displaystyle A^{1}_1: (1, 0, 0, 0, 0, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 38.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}, \displaystyle g_{6}, \displaystyle g_{-1}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-5}, \displaystyle g_{-6}, \displaystyle g_{1}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{2\omega_{6}}\oplus V_{\omega_{3}+\omega_{6}}\oplus V_{\omega_{1}+\omega_{5}}
Made total 697 arithmetic operations while solving the Serre relations polynomial system.
Subalgebra E^{1}_6 ↪ E^{1}_6
119 out of 119
Subalgebra type: \displaystyle E^{1}_6 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \displaystyle D^{1}_5 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \displaystyle E^{1}_6
Elements Cartan subalgebra scaled to act by two by components: \displaystyle E^{1}_6: (1, 2, 2, 3, 2, 1): 2, (0, 0, -1, 0, 0, 0): 2, (0, -1, 0, 0, 0, 0): 2, (0, 0, 0, -1, 0, 0): 2, (0, 0, 0, 0, -1, 0): 2, (0, 0, 0, 0, 0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 78.
Negative simple generators: \displaystyle g_{-36}, \displaystyle g_{3}, \displaystyle g_{2}, \displaystyle g_{4}, \displaystyle g_{5}, \displaystyle g_{6}
Positive simple generators: \displaystyle g_{36}, \displaystyle g_{-3}, \displaystyle g_{-2}, \displaystyle g_{-4}, \displaystyle g_{-5}, \displaystyle g_{-6}
Cartan symmetric matrix: \displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0\\
0 & 2 & 0 & -1 & 0 & 0\\
-1 & 0 & 2 & -1 & 0 & 0\\
0 & -1 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & 0 & -1 & 2\\
\end{pmatrix}
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0\\
0 & 2 & 0 & -1 & 0 & 0\\
-1 & 0 & 2 & -1 & 0 & 0\\
0 & -1 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & 0 & -1 & 2\\
\end{pmatrix}
Decomposition of ambient Lie algebra: \displaystyle V_{\omega_{2}}
Made total 697 arithmetic operations while solving the Serre relations polynomial system.
Nilpotent orbit computation summary.
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