Processing math: 100%
Subalgebra B12+A81 ↪ C15
84 out of 119
Computations done by the calculator project.

Subalgebra type: B12+A81 (click on type for detailed printout).
Subalgebra is (parabolically) induced from B12 .
Centralizer: A31 .
The semisimple part of the centralizer of the semisimple part of my centralizer: B12+A81
Basis of Cartan of centralizer: 1 vectors: (0, 0, 2, 0, -1)
Contained up to conjugation as a direct summand of: B12+A81+A31 .

Elements Cartan subalgebra scaled to act by two by components: B12: (2, 2, 2, 2, 1): 2, (-2, 0, 0, 0, 0): 4, A81: (0, 0, 4, 8, 4): 16
Dimension of subalgebra generated by predefined or computed generators: 13.
Negative simple generators: g−25, g1, g−4+g−12
Positive simple generators: g25, g−1, 2g12+2g4
Cartan symmetric matrix: (2−10−110001/4)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2−20−2400016)
Decomposition of ambient Lie algebra: 3V4ω3⊕2Vω2+2ω3⊕V2ω3⊕V2ω2⊕3V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω3+4ψ⊕Vω2+2ω3+2ψ⊕V4ψ⊕V4ω3⊕V2ω3⊕V2ω2⊕Vω2+2ω3−2ψ⊕V0⊕V4ω3−4ψ⊕V−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 10) ; the vectors are over the primal subalgebra.g5+g−3−h5+2h3g3+g−5g23g12+g4g18g21g13g16g19
weight0002ω22ω3ω2+2ω3ω2+2ω34ω34ω34ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). −4ψ04ψ2ω22ω3ω2+2ω3−2ψω2+2ω3+2ψ4ω3−4ψ4ω34ω3+4ψ
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weightV−4ψ → (0, 0, 0, -4)V0 → (0, 0, 0, 0)V4ψ → (0, 0, 0, 4)V2ω2 → (0, 2, 0, 0)V2ω3 → (0, 0, 2, 0)Vω2+2ω3−2ψ → (0, 1, 2, -2)Vω2+2ω3+2ψ → (0, 1, 2, 2)V4ω3−4ψ → (0, 0, 4, -4)V4ω3 → (0, 0, 4, 0)V4ω3+4ψ → (0, 0, 4, 4)
Module label W1W2W3W4W5W6W7W8W9W10
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g5+g−3
Cartan of centralizer component.
−h5+2h3
g3+g−5
Semisimple subalgebra component.
−g23
−g24
g−1
−2g25
2h1
2h5+4h4+4h3+4h2+4h1
2g−25
−2g1
−2g−24
4g−23
Semisimple subalgebra component.
−g12−g4
2h5+4h4+2h3
g−4+g−12
g18
g20
g15
−g−10
g17
−g2
g−7
−g−14
−g6
g−11
−g−22
g−21
g21
g22
−g11
−g−6
−g14
g7
g−2
−g−17
g10
g−15
g−20
−g−18
g13
g9
2g5−g−3
−3g−8
−6g−19
g16
g12−g4
−2h5−2h3
3g−4−3g−12
6g−16
g19
−g8
g3−2g−5
3g−9
−6g−13
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0002ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		2ω3
0
−2ω3		ω2+2ω3
ω1−ω2+2ω3
ω2
−ω1+ω2+2ω3
ω1−ω2
ω2−2ω3
−ω2+2ω3
−ω1+ω2
ω1−ω2−2ω3
−ω2
−ω1+ω2−2ω3
−ω2−2ω3		ω2+2ω3
ω1−ω2+2ω3
ω2
−ω1+ω2+2ω3
ω1−ω2
ω2−2ω3
−ω2+2ω3
−ω1+ω2
ω1−ω2−2ω3
−ω2
−ω1+ω2−2ω3
−ω2−2ω3		4ω3
2ω3
0
−2ω3
−4ω3		4ω3
2ω3
0
−2ω3
−4ω3		4ω3
2ω3
0
−2ω3
−4ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer−4ψ04ψ2ω2
ω1
−ω1+2ω2
2ω1−2ω2
0
0
−2ω1+2ω2
ω1−2ω2
−ω1
−2ω2		2ω3
0
−2ω3		ω2+2ω3−2ψ
ω1−ω2+2ω3−2ψ
ω2−2ψ
−ω1+ω2+2ω3−2ψ
ω1−ω2−2ψ
ω2−2ω3−2ψ
−ω2+2ω3−2ψ
−ω1+ω2−2ψ
ω1−ω2−2ω3−2ψ
−ω2−2ψ
−ω1+ω2−2ω3−2ψ
−ω2−2ω3−2ψ		ω2+2ω3+2ψ
ω1−ω2+2ω3+2ψ
ω2+2ψ
−ω1+ω2+2ω3+2ψ
ω1−ω2+2ψ
ω2−2ω3+2ψ
−ω2+2ω3+2ψ
−ω1+ω2+2ψ
ω1−ω2−2ω3+2ψ
−ω2+2ψ
−ω1+ω2−2ω3+2ψ
−ω2−2ω3+2ψ		4ω3−4ψ
2ω3−4ψ
−4ψ
−2ω3−4ψ
−4ω3−4ψ		4ω3
2ω3
0
−2ω3
−4ω3		4ω3+4ψ
2ω3+4ψ
4ψ
−2ω3+4ψ
−4ω3+4ψ
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M−4ψM0M4ψM2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M2ω3⊕M0⊕M−2ω3Mω2+2ω3−2ψ⊕M−ω1+ω2+2ω3−2ψ⊕Mω1−ω2+2ω3−2ψ⊕M−ω2+2ω3−2ψ⊕Mω2−2ψ⊕M−ω1+ω2−2ψ⊕Mω1−ω2−2ψ⊕M−ω2−2ψ⊕Mω2−2ω3−2ψ⊕M−ω1+ω2−2ω3−2ψ⊕Mω1−ω2−2ω3−2ψ⊕M−ω2−2ω3−2ψMω2+2ω3+2ψ⊕M−ω1+ω2+2ω3+2ψ⊕Mω1−ω2+2ω3+2ψ⊕M−ω2+2ω3+2ψ⊕Mω2+2ψ⊕M−ω1+ω2+2ψ⊕Mω1−ω2+2ψ⊕M−ω2+2ψ⊕Mω2−2ω3+2ψ⊕M−ω1+ω2−2ω3+2ψ⊕Mω1−ω2−2ω3+2ψ⊕M−ω2−2ω3+2ψM4ω3−4ψ⊕M2ω3−4ψ⊕M−4ψ⊕M−2ω3−4ψ⊕M−4ω3−4ψM4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3M4ω3+4ψ⊕M2ω3+4ψ⊕M4ψ⊕M−2ω3+4ψ⊕M−4ω3+4ψ
Isotypic characterM−4ψM0M4ψM2ω2⊕M−ω1+2ω2⊕Mω1⊕M−2ω1+2ω2⊕2M0⊕M2ω1−2ω2⊕M−ω1⊕Mω1−2ω2⊕M−2ω2M2ω3⊕M0⊕M−2ω3Mω2+2ω3−2ψ⊕M−ω1+ω2+2ω3−2ψ⊕Mω1−ω2+2ω3−2ψ⊕M−ω2+2ω3−2ψ⊕Mω2−2ψ⊕M−ω1+ω2−2ψ⊕Mω1−ω2−2ψ⊕M−ω2−2ψ⊕Mω2−2ω3−2ψ⊕M−ω1+ω2−2ω3−2ψ⊕Mω1−ω2−2ω3−2ψ⊕M−ω2−2ω3−2ψMω2+2ω3+2ψ⊕M−ω1+ω2+2ω3+2ψ⊕Mω1−ω2+2ω3+2ψ⊕M−ω2+2ω3+2ψ⊕Mω2+2ψ⊕M−ω1+ω2+2ψ⊕Mω1−ω2+2ψ⊕M−ω2+2ψ⊕Mω2−2ω3+2ψ⊕M−ω1+ω2−2ω3+2ψ⊕Mω1−ω2−2ω3+2ψ⊕M−ω2−2ω3+2ψM4ω3−4ψ⊕M2ω3−4ψ⊕M−4ψ⊕M−2ω3−4ψ⊕M−4ω3−4ψM4ω3⊕M2ω3⊕M0⊕M−2ω3⊕M−4ω3M4ω3+4ψ⊕M2ω3+4ψ⊕M4ψ⊕M−2ω3+4ψ⊕M−4ω3+4ψ

Semisimple subalgebra: W_{4}+W_{5}
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)
(0.00, 1.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00): (300.00, 350.00)
1: (0.00, 1.00, 0.00, 0.00): (250.00, 350.00)
2: (0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00): (200.00, 300.00)




Made total 206640 arithmetic operations while solving the Serre relations polynomial system.