Processing math: 100%
Subalgebra A21+2A11B13
14 out of 16
Computations done by the calculator project.

Subalgebra type: A21+2A11 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A21+A11 .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: B13

Elements Cartan subalgebra scaled to act by two by components: A21: (2, 2, 2): 4, A11: (0, 1, 2): 2, A11: (0, 1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 9.
Negative simple generators: g6, g7, g2
Positive simple generators: g6, g7, g2
Cartan symmetric matrix: (100020002)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (400020002)
Decomposition of ambient Lie algebra: V2ω1+ω2+ω3V2ω3V2ω2V2ω1
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.
Highest vectors of representations (total 4) ; the vectors are over the primal subalgebra.g6g7g2g9
weight2ω12ω22ω32ω1+ω2+ω3
Isotypic module decomposition over primal subalgebra (total 4 isotypic components).
Isotypical components + highest weightV2ω1 → (2, 0, 0)V2ω2 → (0, 2, 0)V2ω3 → (0, 0, 2)V2ω1+ω2+ω3 → (2, 1, 1)
Module label W1W2W3W4
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
g6
2h3+2h2+2h1
2g6
Semisimple subalgebra component.
g7
2h3+h2
2g7
Semisimple subalgebra component.
g2
h2
2g2
g9
g5
g4
g8
2g1
g3
g3
g1
2g8
2g4
g5
2g9
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above2ω1
0
2ω1
2ω2
0
2ω2
2ω3
0
2ω3
2ω1+ω2+ω3
ω2+ω3
2ω1ω2+ω3
2ω1+ω2ω3
2ω1+ω2+ω3
ω2+ω3
ω2ω3
2ω1ω2ω3
2ω1ω2+ω3
2ω1+ω2ω3
ω2ω3
2ω1ω2ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer2ω1
0
2ω1
2ω2
0
2ω2
2ω3
0
2ω3
2ω1+ω2+ω3
ω2+ω3
2ω1ω2+ω3
2ω1+ω2ω3
2ω1+ω2+ω3
ω2+ω3
ω2ω3
2ω1ω2ω3
2ω1ω2+ω3
2ω1+ω2ω3
ω2ω3
2ω1ω2ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M2ω1M0M2ω1M2ω2M0M2ω2M2ω3M0M2ω3M2ω1+ω2+ω3Mω2+ω3M2ω1ω2+ω3M2ω1+ω2ω3M2ω1+ω2+ω3Mω2+ω3Mω2ω3M2ω1ω2ω3M2ω1ω2+ω3M2ω1+ω2ω3Mω2ω3M2ω1ω2ω3
Isotypic characterM2ω1M0M2ω1M2ω2M0M2ω2M2ω3M0M2ω3M2ω1+ω2+ω3Mω2+ω3M2ω1ω2+ω3M2ω1+ω2ω3M2ω1+ω2+ω3Mω2+ω3Mω2ω3M2ω1ω2ω3M2ω1ω2+ω3M2ω1+ω2ω3Mω2ω3M2ω1ω2ω3

Semisimple subalgebra: W_{1}+W_{2}+W_{3}
Centralizer extension: 0

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): (300.00, 300.00)
1: (0.00, 1.00, 0.00): (200.00, 350.00)
2: (0.00, 0.00, 1.00): (200.00, 300.00)




Made total 452 arithmetic operations while solving the Serre relations polynomial system.