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Subalgebra A13 ↪ E16
85 out of 119
Computations done by the calculator project.

Subalgebra type: A13 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A12 .
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: (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: A13+A11 , A13+A21 , A13+2A11 .

Elements Cartan subalgebra scaled to act by two by components: A13: (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: g−36, g2, g4
Positive simple generators: g36, g−2, g−4
Cartan symmetric matrix: (2−10−12−10−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−10−12−10−12)
Decomposition of ambient Lie algebra: Vω1+ω3⊕4Vω3⊕4Vω2⊕4Vω1⊕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+4ψ2+2ψ3⊕V2ψ2+4ψ3⊕Vω2+2ψ1+2ψ3⊕Vω3+2ψ1+2ψ2⊕Vω2−2ψ1+2ψ2+2ψ3⊕Vω3−2ψ1+4ψ2⊕Vω1+ω3⊕V4ψ1−2ψ2⊕Vω3−2ψ2+2ψ3⊕Vω1+2ψ2−2ψ3⊕3V0⊕Vω1+2ψ1−4ψ2⊕Vω2+2ψ1−2ψ2−2ψ3⊕V−4ψ1+2ψ2⊕Vω1−2ψ1−2ψ2⊕Vω2−2ψ1−2ψ3⊕Vω3−4ψ2−2ψ3⊕V−2ψ2−4ψ3
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 20) ; the vectors are over the primal subalgebra.g−6g−1−h5+h3h6h1g1g6g31g33g29g32g15g18g21g24g5g11g3g7g34
weight0000000ω1ω1ω1ω1ω2ω2ω2ω2ω3ω3ω3ω3ω1+ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). −2ψ2−4ψ3−4ψ1+2ψ20004ψ1−2ψ22ψ2+4ψ3ω1−2ψ1−2ψ2ω1+2ψ1−4ψ2ω1+2ψ2−2ψ3ω1+4ψ2+2ψ3ω2−2ψ1−2ψ3ω2+2ψ1−2ψ2−2ψ3ω2−2ψ1+2ψ2+2ψ3ω2+2ψ1+2ψ3ω3−4ψ2−2ψ3ω3−2ψ2+2ψ3ω3−2ψ1+4ψ2ω3+2ψ1+2ψ2ω1+ω3
Isotypic module decomposition over primal subalgebra (total 18 isotypic components).
Isotypical components + highest weightV−2ψ2−4ψ3 → (0, 0, 0, 0, -2, -4)V−4ψ1+2ψ2 → (0, 0, 0, -4, 2, 0)V0 → (0, 0, 0, 0, 0, 0)V4ψ1−2ψ2 → (0, 0, 0, 4, -2, 0)V2ψ2+4ψ3 → (0, 0, 0, 0, 2, 4)Vω1−2ψ1−2ψ2 → (1, 0, 0, -2, -2, 0)Vω1+2ψ1−4ψ2 → (1, 0, 0, 2, -4, 0)Vω1+2ψ2−2ψ3 → (1, 0, 0, 0, 2, -2)Vω1+4ψ2+2ψ3 → (1, 0, 0, 0, 4, 2)Vω2−2ψ1−2ψ3 → (0, 1, 0, -2, 0, -2)Vω2+2ψ1−2ψ2−2ψ3 → (0, 1, 0, 2, -2, -2)Vω2−2ψ1+2ψ2+2ψ3 → (0, 1, 0, -2, 2, 2)Vω2+2ψ1+2ψ3 → (0, 1, 0, 2, 0, 2)Vω3−4ψ2−2ψ3 → (0, 0, 1, 0, -4, -2)Vω3−2ψ2+2ψ3 → (0, 0, 1, 0, -2, 2)Vω3−2ψ1+4ψ2 → (0, 0, 1, -2, 4, 0)Vω3+2ψ1+2ψ2 → (0, 0, 1, 2, 2, 0)Vω1+ω3 → (1, 0, 1, 0, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10W11W12W13W14W15W16W17W18
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g−6
g−1
Cartan of centralizer component.
−h5+h3
h6
h1
g1
g6
g31
−g−17
g−12
g−7
g33
g−13
−g−9
−g−3
g29
g−20
−g−16
g−11
g32
−g−14
g−10
−g−5
g15
g19
−g−30
g23
g−27
−g−24
g18
g22
g−28
g26
−g−25
g−21
g21
g25
g−26
g28
−g−22
g−18
g24
g27
−g−23
g30
g−19
−g−15
g5
g10
g14
g−32
g11
g16
g20
−g−29
g3
−g9
−g13
g−33
g7
−g12
−g17
−g−31
Semisimple subalgebra component.
−g34
g−8
−g35
−g−4
g−2
−g36
−h4
h2
h6+2h5+3h4+2h3+2h2+h1
−g2
2g4
g−36
−g−35
g8
g−34
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above00000ω1
−ω1+ω2
−ω2+ω3
−ω3		ω1
−ω1+ω2
−ω2+ω3
−ω3		ω1
−ω1+ω2
−ω2+ω3
−ω3		ω1
−ω1+ω2
−ω2+ω3
−ω3		ω2
ω1−ω2+ω3
−ω1+ω3
ω1−ω3
−ω1+ω2−ω3
−ω2		ω2
ω1−ω2+ω3
−ω1+ω3
ω1−ω3
−ω1+ω2−ω3
−ω2		ω2
ω1−ω2+ω3
−ω1+ω3
ω1−ω3
−ω1+ω2−ω3
−ω2		ω2
ω1−ω2+ω3
−ω1+ω3
ω1−ω3
−ω1+ω2−ω3
−ω2		ω3
ω2−ω3
ω1−ω2
−ω1		ω3
ω2−ω3
ω1−ω2
−ω1		ω3
ω2−ω3
ω1−ω2
−ω1		ω3
ω2−ω3
ω1−ω2
−ω1		ω1+ω3
−ω1+ω2+ω3
ω1+ω2−ω3
−ω2+2ω3
−ω1+2ω2−ω3
2ω1−ω2
0
0
0
ω1−2ω2+ω3
ω2−2ω3
−2ω1+ω2
−ω1−ω2+ω3
ω1−ω2−ω3
−ω1−ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer−2ψ2−4ψ3−4ψ1+2ψ204ψ1−2ψ22ψ2+4ψ3ω1−2ψ1−2ψ2
−ω1+ω2−2ψ1−2ψ2
−ω2+ω3−2ψ1−2ψ2
−ω3−2ψ1−2ψ2		ω1+2ψ1−4ψ2
−ω1+ω2+2ψ1−4ψ2
−ω2+ω3+2ψ1−4ψ2
−ω3+2ψ1−4ψ2		ω1+2ψ2−2ψ3
−ω1+ω2+2ψ2−2ψ3
−ω2+ω3+2ψ2−2ψ3
−ω3+2ψ2−2ψ3		ω1+4ψ2+2ψ3
−ω1+ω2+4ψ2+2ψ3
−ω2+ω3+4ψ2+2ψ3
−ω3+4ψ2+2ψ3		ω2−2ψ1−2ψ3
ω1−ω2+ω3−2ψ1−2ψ3
−ω1+ω3−2ψ1−2ψ3
ω1−ω3−2ψ1−2ψ3
−ω1+ω2−ω3−2ψ1−2ψ3
−ω2−2ψ1−2ψ3		ω2+2ψ1−2ψ2−2ψ3
ω1−ω2+ω3+2ψ1−2ψ2−2ψ3
−ω1+ω3+2ψ1−2ψ2−2ψ3
ω1−ω3+2ψ1−2ψ2−2ψ3
−ω1+ω2−ω3+2ψ1−2ψ2−2ψ3
−ω2+2ψ1−2ψ2−2ψ3		ω2−2ψ1+2ψ2+2ψ3
ω1−ω2+ω3−2ψ1+2ψ2+2ψ3
−ω1+ω3−2ψ1+2ψ2+2ψ3
ω1−ω3−2ψ1+2ψ2+2ψ3
−ω1+ω2−ω3−2ψ1+2ψ2+2ψ3
−ω2−2ψ1+2ψ2+2ψ3		ω2+2ψ1+2ψ3
ω1−ω2+ω3+2ψ1+2ψ3
−ω1+ω3+2ψ1+2ψ3
ω1−ω3+2ψ1+2ψ3
−ω1+ω2−ω3+2ψ1+2ψ3
−ω2+2ψ1+2ψ3		ω3−4ψ2−2ψ3
ω2−ω3−4ψ2−2ψ3
ω1−ω2−4ψ2−2ψ3
−ω1−4ψ2−2ψ3		ω3−2ψ2+2ψ3
ω2−ω3−2ψ2+2ψ3
ω1−ω2−2ψ2+2ψ3
−ω1−2ψ2+2ψ3		ω3−2ψ1+4ψ2
ω2−ω3−2ψ1+4ψ2
ω1−ω2−2ψ1+4ψ2
−ω1−2ψ1+4ψ2		ω3+2ψ1+2ψ2
ω2−ω3+2ψ1+2ψ2
ω1−ω2+2ψ1+2ψ2
−ω1+2ψ1+2ψ2		ω1+ω3
−ω1+ω2+ω3
ω1+ω2−ω3
−ω2+2ω3
−ω1+2ω2−ω3
2ω1−ω2
0
0
0
ω1−2ω2+ω3
ω2−2ω3
−2ω1+ω2
−ω1−ω2+ω3
ω1−ω2−ω3
−ω1−ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M−2ψ2−4ψ3M−4ψ1+2ψ2M0M4ψ1−2ψ2M2ψ2+4ψ3Mω1−2ψ1−2ψ2⊕M−ω2+ω3−2ψ1−2ψ2⊕M−ω1+ω2−2ψ1−2ψ2⊕M−ω3−2ψ1−2ψ2Mω1+2ψ1−4ψ2⊕M−ω2+ω3+2ψ1−4ψ2⊕M−ω1+ω2+2ψ1−4ψ2⊕M−ω3+2ψ1−4ψ2Mω1+2ψ2−2ψ3⊕M−ω2+ω3+2ψ2−2ψ3⊕M−ω1+ω2+2ψ2−2ψ3⊕M−ω3+2ψ2−2ψ3Mω1+4ψ2+2ψ3⊕M−ω2+ω3+4ψ2+2ψ3⊕M−ω1+ω2+4ψ2+2ψ3⊕M−ω3+4ψ2+2ψ3Mω1−ω2+ω3−2ψ1−2ψ3⊕Mω2−2ψ1−2ψ3⊕M−ω1+ω3−2ψ1−2ψ3⊕Mω1−ω3−2ψ1−2ψ3⊕M−ω2−2ψ1−2ψ3⊕M−ω1+ω2−ω3−2ψ1−2ψ3Mω1−ω2+ω3+2ψ1−2ψ2−2ψ3⊕Mω2+2ψ1−2ψ2−2ψ3⊕M−ω1+ω3+2ψ1−2ψ2−2ψ3⊕Mω1−ω3+2ψ1−2ψ2−2ψ3⊕M−ω2+2ψ1−2ψ2−2ψ3⊕M−ω1+ω2−ω3+2ψ1−2ψ2−2ψ3Mω1−ω2+ω3−2ψ1+2ψ2+2ψ3⊕Mω2−2ψ1+2ψ2+2ψ3⊕M−ω1+ω3−2ψ1+2ψ2+2ψ3⊕Mω1−ω3−2ψ1+2ψ2+2ψ3⊕M−ω2−2ψ1+2ψ2+2ψ3⊕M−ω1+ω2−ω3−2ψ1+2ψ2+2ψ3Mω1−ω2+ω3+2ψ1+2ψ3⊕Mω2+2ψ1+2ψ3⊕M−ω1+ω3+2ψ1+2ψ3⊕Mω1−ω3+2ψ1+2ψ3⊕M−ω2+2ψ1+2ψ3⊕M−ω1+ω2−ω3+2ψ1+2ψ3Mω3−4ψ2−2ψ3⊕Mω1−ω2−4ψ2−2ψ3⊕Mω2−ω3−4ψ2−2ψ3⊕M−ω1−4ψ2−2ψ3Mω3−2ψ2+2ψ3⊕Mω1−ω2−2ψ2+2ψ3⊕Mω2−ω3−2ψ2+2ψ3⊕M−ω1−2ψ2+2ψ3Mω3−2ψ1+4ψ2⊕Mω1−ω2−2ψ1+4ψ2⊕Mω2−ω3−2ψ1+4ψ2⊕M−ω1−2ψ1+4ψ2Mω3+2ψ1+2ψ2⊕Mω1−ω2+2ψ1+2ψ2⊕Mω2−ω3+2ψ1+2ψ2⊕M−ω1+2ψ1+2ψ2Mω1+ω3⊕M−ω2+2ω3⊕M−ω1+ω2+ω3⊕M2ω1−ω2⊕Mω1+ω2−ω3⊕Mω1−2ω2+ω3⊕3M0⊕M−ω1+2ω2−ω3⊕M−ω1−ω2+ω3⊕M−2ω1+ω2⊕Mω1−ω2−ω3⊕Mω2−2ω3⊕M−ω1−ω3
Isotypic characterM−2ψ2−4ψ3M−4ψ1+2ψ23M0M4ψ1−2ψ2M2ψ2+4ψ3Mω1−2ψ1−2ψ2⊕M−ω2+ω3−2ψ1−2ψ2⊕M−ω1+ω2−2ψ1−2ψ2⊕M−ω3−2ψ1−2ψ2Mω1+2ψ1−4ψ2⊕M−ω2+ω3+2ψ1−4ψ2⊕M−ω1+ω2+2ψ1−4ψ2⊕M−ω3+2ψ1−4ψ2Mω1+2ψ2−2ψ3⊕M−ω2+ω3+2ψ2−2ψ3⊕M−ω1+ω2+2ψ2−2ψ3⊕M−ω3+2ψ2−2ψ3Mω1+4ψ2+2ψ3⊕M−ω2+ω3+4ψ2+2ψ3⊕M−ω1+ω2+4ψ2+2ψ3⊕M−ω3+4ψ2+2ψ3Mω1−ω2+ω3−2ψ1−2ψ3⊕Mω2−2ψ1−2ψ3⊕M−ω1+ω3−2ψ1−2ψ3⊕Mω1−ω3−2ψ1−2ψ3⊕M−ω2−2ψ1−2ψ3⊕M−ω1+ω2−ω3−2ψ1−2ψ3Mω1−ω2+ω3+2ψ1−2ψ2−2ψ3⊕Mω2+2ψ1−2ψ2−2ψ3⊕M−ω1+ω3+2ψ1−2ψ2−2ψ3⊕Mω1−ω3+2ψ1−2ψ2−2ψ3⊕M−ω2+2ψ1−2ψ2−2ψ3⊕M−ω1+ω2−ω3+2ψ1−2ψ2−2ψ3Mω1−ω2+ω3−2ψ1+2ψ2+2ψ3⊕Mω2−2ψ1+2ψ2+2ψ3⊕M−ω1+ω3−2ψ1+2ψ2+2ψ3⊕Mω1−ω3−2ψ1+2ψ2+2ψ3⊕M−ω2−2ψ1+2ψ2+2ψ3⊕M−ω1+ω2−ω3−2ψ1+2ψ2+2ψ3Mω1−ω2+ω3+2ψ1+2ψ3⊕Mω2+2ψ1+2ψ3⊕M−ω1+ω3+2ψ1+2ψ3⊕Mω1−ω3+2ψ1+2ψ3⊕M−ω2+2ψ1+2ψ3⊕M−ω1+ω2−ω3+2ψ1+2ψ3Mω3−4ψ2−2ψ3⊕Mω1−ω2−4ψ2−2ψ3⊕Mω2−ω3−4ψ2−2ψ3⊕M−ω1−4ψ2−2ψ3Mω3−2ψ2+2ψ3⊕Mω1−ω2−2ψ2+2ψ3⊕Mω2−ω3−2ψ2+2ψ3⊕M−ω1−2ψ2+2ψ3Mω3−2ψ1+4ψ2⊕Mω1−ω2−2ψ1+4ψ2⊕Mω2−ω3−2ψ1+4ψ2⊕M−ω1−2ψ1+4ψ2Mω3+2ψ1+2ψ2⊕Mω1−ω2+2ψ1+2ψ2⊕Mω2−ω3+2ψ1+2ψ2⊕M−ω1+2ψ1+2ψ2Mω1+ω3⊕M−ω2+2ω3⊕M−ω1+ω2+ω3⊕M2ω1−ω2⊕Mω1+ω2−ω3⊕Mω1−2ω2+ω3⊕3M0⊕M−ω1+2ω2−ω3⊕M−ω1−ω2+ω3⊕M−2ω1+ω2⊕Mω1−ω2−ω3⊕Mω2−2ω3⊕M−ω1−ω3

Semisimple subalgebra: W_{18}
Centralizer extension: W_{1}+W_{2}+W_{3}+W_{4}+W_{5}

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.
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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, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00, 0.00): (275.00, 350.00)
1: (0.00, 1.00, 0.00, 0.00, 0.00, 0.00): (250.00, 400.00)
2: (0.00, 0.00, 1.00, 0.00, 0.00, 0.00): (225.00, 350.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 300.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
5: (0.00, 0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 300.00)



Made total 448 arithmetic operations while solving the Serre relations polynomial system.