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Subalgebra A12 ↪ B13
11 out of 16
Computations done by the calculator project.

Subalgebra type: A12 (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from A11 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: B13
Basis of Cartan of centralizer: 1 vectors: (1, 0, -1)

Elements Cartan subalgebra scaled to act by two by components: A12: (1, 2, 2): 2, (0, -1, 0): 2
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: g−9, g2
Positive simple generators: g9, g−2
Cartan symmetric matrix: (2−1−12)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−1−12)
Decomposition of ambient Lie algebra: Vω1+ω2⊕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). Vω2+4ψ⊕Vω1+2ψ⊕Vω1+ω2⊕V0⊕Vω2−2ψ⊕Vω1−4ψ
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 6) ; the vectors are over the primal subalgebra.−h3+h1g7g6g3g1g8
weight0ω1ω1ω2ω2ω1+ω2
weights rel. to Cartan of (centralizer+semisimple s.a.). 0ω1−4ψω1+2ψω2−2ψω2+4ψω1+ω2
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0)Vω1−4ψ → (1, 0, -4)Vω1+2ψ → (1, 0, 2)Vω2−2ψ → (0, 1, -2)Vω2+4ψ → (0, 1, 4)Vω1+ω2 → (1, 1, 0)
Module label W1W2W3W4W5W6
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
−h3+h1
g7
g−4
g−1
g6
−g−5
g−3
g3
g5
g−6
g1
−g4
g−7
Semisimple subalgebra component.
−g8
g−2
−g9
h2
2h3+2h2+h1
g−9
−2g2
−g−8
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1
−ω1+ω2
−ω2		ω1
−ω1+ω2
−ω2		ω2
ω1−ω2
−ω1		ω2
ω1−ω2
−ω1		ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1−4ψ
−ω1+ω2−4ψ
−ω2−4ψ		ω1+2ψ
−ω1+ω2+2ψ
−ω2+2ψ		ω2−2ψ
ω1−ω2−2ψ
−ω1−2ψ		ω2+4ψ
ω1−ω2+4ψ
−ω1+4ψ		ω1+ω2
−ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
ω1−2ω2
−ω1−ω2
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1−4ψ⊕M−ω1+ω2−4ψ⊕M−ω2−4ψMω1+2ψ⊕M−ω1+ω2+2ψ⊕M−ω2+2ψMω2−2ψ⊕Mω1−ω2−2ψ⊕M−ω1−2ψMω2+4ψ⊕Mω1−ω2+4ψ⊕M−ω1+4ψMω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2
Isotypic characterM0Mω1−4ψ⊕M−ω1+ω2−4ψ⊕M−ω2−4ψMω1+2ψ⊕M−ω1+ω2+2ψ⊕M−ω2+2ψMω2−2ψ⊕Mω1−ω2−2ψ⊕M−ω1−2ψMω2+4ψ⊕Mω1−ω2+4ψ⊕M−ω1+4ψMω1+ω2⊕M−ω1+2ω2⊕M2ω1−ω2⊕2M0⊕M−2ω1+ω2⊕Mω1−2ω2⊕M−ω1−ω2

Semisimple subalgebra: W_{6}
Centralizer extension: W_{1}

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): (266.67, 333.33)
1: (0.00, 1.00, 0.00): (233.33, 366.67)
2: (0.00, 0.00, 1.00): (200.00, 300.00)




Made total 359 arithmetic operations while solving the Serre relations polynomial system.