Subalgebra A61A16
6 out of 61
Computations done by the calculator project.

Subalgebra type: A61 (click on type for detailed printout).
Centralizer: A21 + T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A12+A21
Basis of Cartan of centralizer: 2 vectors: (0, 1, 0, 0, -1, 0), (2, 0, -1, 1, 0, -2)
Contained up to conjugation as a direct summand of: A61+A21 .

Elements Cartan subalgebra scaled to act by two by components: A61: (2, 3, 4, 4, 3, 2): 12
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g9+g12+g15+g17
Positive simple generators: g17+2g15+2g12+g9
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: V4ω14V3ω15V2ω14Vω14V0
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+6ψ2V3ω12ψ1+8ψ2Vω1+2ψ1+6ψ2Vω12ψ1+8ψ2V4ω1V2ω1+4ψ12ψ23V2ω1V4ψ12ψ2V2ω14ψ1+2ψ22V0V4ψ1+2ψ2V3ω1+2ψ18ψ2V3ω12ψ16ψ2Vω1+2ψ18ψ2Vω12ψ16ψ2
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra. As the centralizer is well-chosen and the centralizer of our subalgebra is non-trivial, we may in addition split highest weight vectors with the same weight over the semisimple part over the centralizer (recall that the centralizer preserves the weights over the subalgebra and in particular acts on the highest weight vectors). Therefore we have chosen our highest weight vectors to be, in addition, weight vectors over the Cartan of the centralizer of the starting subalgebra. Their weight over the sum of the Cartans of the semisimple subalgebra and its centralizer is indicated in the third row. The weights corresponding to the Cartan of the centralizer are again indicated with the letter \omega. As there is no preferred way of chosing a basis of the Cartan of the centralizer (unlike the starting semisimple Lie algebra: there we have a preferred basis induced by the fundamental weights), our centralizer weights are simply given by the constant by which the k^th basis element of the Cartan of the centralizer acts on the highest weight vector. Here, we use the choice for basis of the Cartan of the centralizer given at the start of the page.

Highest vectors of representations (total 18) ; the vectors are over the primal subalgebra.g5+g2h6+1/2h41/2h3+h1h5+h2g2+g5g11+1/2g3g8+2g6g10+2g1g7+1/2g4g14g17g9g15+g12g13g18g20g19g16g21
weight0000ω1ω1ω1ω12ω12ω12ω12ω12ω13ω13ω13ω13ω14ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). 4ψ1+2ψ2004ψ12ψ2ω12ψ16ψ2ω1+2ψ18ψ2ω12ψ1+8ψ2ω1+2ψ1+6ψ22ω14ψ1+2ψ22ω12ω12ω12ω1+4ψ12ψ23ω12ψ16ψ23ω1+2ψ18ψ23ω12ψ1+8ψ23ω1+2ψ1+6ψ24ω1
Isotypic module decomposition over primal subalgebra (total 16 isotypic components).
Isotypical components + highest weightV4ψ1+2ψ2 → (0, -4, 2)V0 → (0, 0, 0)V4ψ12ψ2 → (0, 4, -2)Vω12ψ16ψ2 → (1, -2, -6)Vω1+2ψ18ψ2 → (1, 2, -8)Vω12ψ1+8ψ2 → (1, -2, 8)Vω1+2ψ1+6ψ2 → (1, 2, 6)V2ω14ψ1+2ψ2 → (2, -4, 2)V2ω1 → (2, 0, 0)V2ω1+4ψ12ψ2 → (2, 4, -2)V3ω12ψ16ψ2 → (3, -2, -6)V3ω1+2ψ18ψ2 → (3, 2, -8)V3ω12ψ1+8ψ2 → (3, -2, 8)V3ω1+2ψ1+6ψ2 → (3, 2, 6)V4ω1 → (4, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g5+g2
Cartan of centralizer component.
h6+1/2h41/2h3+h1
h5+h2
g2+g5
g11+1/2g3
1/2g41/2g7
g8+2g6
g1g10
g10+2g1
g6+g8
g7+1/2g4
1/2g3+1/2g11
g14
g5g2
2g13
Semisimple subalgebra component.
g172g152g12g9
2h6+3h5+4h4+4h3+3h2+2h1
2g9+2g12+2g15+2g17
g9
h4h3
2g9
g17
h5h4h3h2
2g17
g13
g2+g5
2g14
g18
g11g3
2g4+g7
3g16
g20
g8+g6
g12g10
3g19
g19
g10g1
g62g8
3g20
g16
g7+g4
2g3+g11
3g18
g21
g15g12
h6h5h4+h3+h2+h1
3g123g15
6g21
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above000ω1
ω1
ω1
ω1
ω1
ω1
ω1
ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
4ω1
2ω1
0
2ω1
4ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer4ψ1+2ψ204ψ12ψ2ω12ψ16ψ2
ω12ψ16ψ2
ω1+2ψ18ψ2
ω1+2ψ18ψ2
ω12ψ1+8ψ2
ω12ψ1+8ψ2
ω1+2ψ1+6ψ2
ω1+2ψ1+6ψ2
2ω14ψ1+2ψ2
4ψ1+2ψ2
2ω14ψ1+2ψ2
2ω1
0
2ω1
2ω1
0
2ω1
2ω1+4ψ12ψ2
4ψ12ψ2
2ω1+4ψ12ψ2
3ω12ψ16ψ2
ω12ψ16ψ2
ω12ψ16ψ2
3ω12ψ16ψ2
3ω1+2ψ18ψ2
ω1+2ψ18ψ2
ω1+2ψ18ψ2
3ω1+2ψ18ψ2
3ω12ψ1+8ψ2
ω12ψ1+8ψ2
ω12ψ1+8ψ2
3ω12ψ1+8ψ2
3ω1+2ψ1+6ψ2
ω1+2ψ1+6ψ2
ω1+2ψ1+6ψ2
3ω1+2ψ1+6ψ2
4ω1
2ω1
0
2ω1
4ω1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M4ψ1+2ψ2M0M4ψ12ψ2Mω12ψ16ψ2Mω12ψ16ψ2Mω1+2ψ18ψ2Mω1+2ψ18ψ2Mω12ψ1+8ψ2Mω12ψ1+8ψ2Mω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2M2ω14ψ1+2ψ2M4ψ1+2ψ2M2ω14ψ1+2ψ2M2ω1M0M2ω1M2ω1M0M2ω1M2ω1+4ψ12ψ2M4ψ12ψ2M2ω1+4ψ12ψ2M3ω12ψ16ψ2Mω12ψ16ψ2Mω12ψ16ψ2M3ω12ψ16ψ2M3ω1+2ψ18ψ2Mω1+2ψ18ψ2Mω1+2ψ18ψ2M3ω1+2ψ18ψ2M3ω12ψ1+8ψ2Mω12ψ1+8ψ2Mω12ψ1+8ψ2M3ω12ψ1+8ψ2M3ω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2M3ω1+2ψ1+6ψ2M4ω1M2ω1M0M2ω1M4ω1
Isotypic characterM4ψ1+2ψ22M0M4ψ12ψ2Mω12ψ16ψ2Mω12ψ16ψ2Mω1+2ψ18ψ2Mω1+2ψ18ψ2Mω12ψ1+8ψ2Mω12ψ1+8ψ2Mω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2M2ω14ψ1+2ψ2M4ψ1+2ψ2M2ω14ψ1+2ψ2M2ω1M0M2ω12M2ω12M02M2ω1M2ω1+4ψ12ψ2M4ψ12ψ2M2ω1+4ψ12ψ2M3ω12ψ16ψ2Mω12ψ16ψ2Mω12ψ16ψ2M3ω12ψ16ψ2M3ω1+2ψ18ψ2Mω1+2ψ18ψ2Mω1+2ψ18ψ2M3ω1+2ψ18ψ2M3ω12ψ1+8ψ2Mω12ψ1+8ψ2Mω12ψ1+8ψ2M3ω12ψ1+8ψ2M3ω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2M3ω1+2ψ1+6ψ2M4ω1M2ω1M0M2ω1M4ω1

Semisimple subalgebra: W_{9}
Centralizer extension: W_{1}+W_{2}+W_{3}

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.
Canvas not supported




Mouse position: (0.00, 0.00)
Selected index: -1
Coordinate center in screen coordinates:
(200.00, 300.00)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00)
(0.00, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (500.00, 300.00)
1: (0.00, 1.00, 0.00): (200.00, 306.55)
2: (0.00, 0.00, 1.00): (200.00, 300.60)




Made total 1042511 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
1*2 (unknown) gens:
(
x_{1} g_{-9}+x_{2} g_{-12}+x_{3} g_{-13}+x_{4} g_{-14}+x_{5} g_{-15}+x_{6} g_{-17}, x_{12} g_{17}+x_{11} g_{15}+x_{10} g_{14}+x_{9} g_{13}+x_{8} g_{12}+x_{7} g_{9})
h: (2, 3, 4, 4, 3, 2), e = combination of g_{9} g_{12} g_{13} g_{14} g_{15} g_{17} , f= combination of g_{-9} g_{-12} g_{-13} g_{-14} g_{-15} g_{-17} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{4\omega_{1}}+4V_{3\omega_{1}}+6V_{2\omega_{1}}+8V_{\omega_{1}}+10V_{0}+8V_{-\omega_{1}}+6V_{-2\omega_{1}}+4V_{-3\omega_{1}}+V_{-4\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4= 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -4= 0
x_{6} x_{10} +x_{3} x_{7} = 0
x_{6} x_{9} +x_{4} x_{7} = 0
x_{2} x_{8} -2= 0
x_{6} x_{12} +x_{3} x_{9} +x_{2} x_{8} -3= 0
x_{4} x_{12} +x_{1} x_{9} = 0
x_{3} x_{12} +x_{1} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -3= 0
x_{5} x_{11} -2= 0
The above system after transformation.
x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4= 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -4= 0
x_{6} x_{10} +x_{3} x_{7} = 0
x_{6} x_{9} +x_{4} x_{7} = 0
x_{2} x_{8} -2= 0
x_{6} x_{12} +x_{3} x_{9} +x_{2} x_{8} -3= 0
x_{4} x_{12} +x_{1} x_{9} = 0
x_{3} x_{12} +x_{1} x_{10} = 0
x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -3= 0
x_{5} x_{11} -2= 0
For the calculator:
(DynkinType =A^{6}_1; ElementsCartan =((2, 3, 4, 4, 3, 2)); generators =(x_{1} g_{-9}+x_{2} g_{-12}+x_{3} g_{-13}+x_{4} g_{-14}+x_{5} g_{-15}+x_{6} g_{-17}, x_{12} g_{17}+x_{11} g_{15}+x_{10} g_{14}+x_{9} g_{13}+x_{8} g_{12}+x_{7} g_{9}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{6} x_{12} +x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4, x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} +x_{3} x_{9} +x_{1} x_{7} -4, x_{6} x_{10} +x_{3} x_{7} , x_{6} x_{9} +x_{4} x_{7} , x_{2} x_{8} -2, x_{6} x_{12} +x_{3} x_{9} +x_{2} x_{8} -3, x_{4} x_{12} +x_{1} x_{9} , x_{3} x_{12} +x_{1} x_{10} , x_{6} x_{12} +x_{5} x_{11} +x_{4} x_{10} -3, x_{5} x_{11} -2 )