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Subalgebra A101A15
7 out of 37
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Subalgebra type: A101 (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: A13
Basis of Cartan of centralizer: 2 vectors: (0, 0, 1, 0, 0), (1, 2, 0, -2, -1)
Contained up to conjugation as a direct summand of: A101+A11 .

Elements Cartan subalgebra scaled to act by two by components: A101: (3, 4, 4, 4, 3): 20
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g1+g5+g11
Positive simple generators: 4g11+3g5+3g1
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: V6ω1V4ω14V3ω1V2ω14V0
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+6ψ2V6ω1V4ψ1V4ω1V2ω12V0V3ω1+2ψ16ψ2V4ψ1V3ω12ψ16ψ2
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 11) ; the vectors are over the primal subalgebra.g3h52h4+2h2+h1h3g3g11+3/4g5+3/4g1g9g12g6g10g14+g13g15
weight00002ω13ω13ω13ω13ω14ω16ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). 4ψ1004ψ12ω13ω12ψ16ψ23ω1+2ψ16ψ23ω12ψ1+6ψ23ω1+2ψ1+6ψ24ω16ω1
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weightV4ψ1 → (0, -4, 0)V0 → (0, 0, 0)V4ψ1 → (0, 4, 0)V2ω1 → (2, 0, 0)V3ω12ψ16ψ2 → (3, -2, -6)V3ω1+2ψ16ψ2 → (3, 2, -6)V3ω12ψ1+6ψ2 → (3, -2, 6)V3ω1+2ψ1+6ψ2 → (3, 2, 6)V4ω1 → (4, 0, 0)V6ω1 → (6, 0, 0)
Module label W1W2W3W4W5W6W7W8W9W10
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g3
Cartan of centralizer component.
h52h4+2h2+h1
h3
g3
Semisimple subalgebra component.
4/3g11g5g1
h5+4/3h4+4/3h3+4/3h2+h1
2/3g1+2/3g5+2/3g11
g9
g4
g7
g10
g12
g8
g2
g6
g6
g2
g8
g12
g10
g7
g4
g9
g14+g13
g5g1
h5+h1
2g12g5
2g13+2g14
g15
g14g13
2g11+g5+g1
h5+2h4+2h3+2h2h1
4g14g5+6g11
10g13+10g14
20g15
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0002ω1
0
2ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
3ω1
ω1
ω1
3ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer4ψ104ψ12ω1
0
2ω1
3ω12ψ16ψ2
ω12ψ16ψ2
ω12ψ16ψ2
3ω12ψ16ψ2
3ω1+2ψ16ψ2
ω1+2ψ16ψ2
ω1+2ψ16ψ2
3ω1+2ψ16ψ2
3ω12ψ1+6ψ2
ω12ψ1+6ψ2
ω12ψ1+6ψ2
3ω12ψ1+6ψ2
3ω1+2ψ1+6ψ2
ω1+2ψ1+6ψ2
ω1+2ψ1+6ψ2
3ω1+2ψ1+6ψ2
4ω1
2ω1
0
2ω1
4ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M4ψ1M0M4ψ1M2ω1M0M2ω1M3ω12ψ16ψ2Mω12ψ16ψ2Mω12ψ16ψ2M3ω12ψ16ψ2M3ω1+2ψ16ψ2Mω1+2ψ16ψ2Mω1+2ψ16ψ2M3ω1+2ψ16ψ2M3ω12ψ1+6ψ2Mω12ψ1+6ψ2Mω12ψ1+6ψ2M3ω12ψ1+6ψ2M3ω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2M3ω1+2ψ1+6ψ2M4ω1M2ω1M0M2ω1M4ω1M6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1
Isotypic characterM4ψ12M0M4ψ1M2ω1M0M2ω1M3ω12ψ16ψ2Mω12ψ16ψ2Mω12ψ16ψ2M3ω12ψ16ψ2M3ω1+2ψ16ψ2Mω1+2ψ16ψ2Mω1+2ψ16ψ2M3ω1+2ψ16ψ2M3ω12ψ1+6ψ2Mω12ψ1+6ψ2Mω12ψ1+6ψ2M3ω12ψ1+6ψ2M3ω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2Mω1+2ψ1+6ψ2M3ω1+2ψ1+6ψ2M4ω1M2ω1M0M2ω1M4ω1M6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1

Semisimple subalgebra: W_{4}
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.
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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, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (700.00, 300.00)
1: (0.00, 1.00, 0.00): (200.00, 312.50)
2: (0.00, 0.00, 1.00): (200.00, 300.00)




Made total 7189 arithmetic operations while solving the Serre relations polynomial system.