Subalgebra A111A15
8 out of 37
Computations done by the calculator project.

Subalgebra type: A111 (click on type for detailed printout).
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A15
Basis of Cartan of centralizer: 1 vectors: (1, 2, 0, -2, -1)

Elements Cartan subalgebra scaled to act by two by components: A111: (3, 4, 5, 4, 3): 22
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g1+g3+g5+g11
Positive simple generators: 4g11+3g5+g3+3g1
Cartan symmetric matrix: (2/11)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (22)
Decomposition of ambient Lie algebra: V6ω13V4ω14V2ω1V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V4ω1+6ψV2ω1+6ψV6ω1V4ω12V2ω1V0V4ω16ψV2ω16ψ
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 9) ; the vectors are over the primal subalgebra.h52h4+2h2+h1g9+1/3g8g11+3/4g5+3/4g1g3g7+3g6g12g14+g13g10g15
weight02ω12ω12ω12ω14ω14ω14ω16ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). 02ω16ψ2ω12ω12ω1+6ψ4ω16ψ4ω14ω1+6ψ6ω1
Isotypic module decomposition over primal subalgebra (total 9 isotypic components).
Isotypical components + highest weightV0 → (0, 0)V2ω16ψ → (2, -6)V2ω1 → (2, 0)V2ω1+6ψ → (2, 6)V4ω16ψ → (4, -6)V4ω1 → (4, 0)V4ω1+6ψ → (4, 6)V6ω1 → (6, 0)
Module label W1W2W3W4W5W6W7W8W9
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
h52h4+2h2+h1
g9+1/3g8
2/3g41/3g2
1/3g6+1/3g7
Semisimple subalgebra component.
4/3g11g51/3g3g1
h5+4/3h4+5/3h3+4/3h2+h1
2/3g1+2/3g3+2/3g5+2/3g11
g11+3/4g5+3/4g1
3/4h5h4h3h23/4h1
1/2g11/2g51/2g11
g7+3g6
2g2+g4
g8+g9
g12
g9g8
2g4+g2
g6+3g7
4g10
g14+g13
g5g1
h5+h1
2g12g5
2g13+2g14
g10
g7g6
2g2+g4
3g8+g9
4g12
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 above02ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
2ω1
0
2ω1
4ω1
2ω1
0
2ω1
4ω1
4ω1
2ω1
0
2ω1
4ω1
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 centralizer02ω16ψ
6ψ
2ω16ψ
2ω1
0
2ω1
2ω1
0
2ω1
2ω1+6ψ
6ψ
2ω1+6ψ
4ω16ψ
2ω16ψ
6ψ
2ω16ψ
4ω16ψ
4ω1
2ω1
0
2ω1
4ω1
4ω1+6ψ
2ω1+6ψ
6ψ
2ω1+6ψ
4ω1+6ψ
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0M2ω16ψM6ψM2ω16ψM2ω1M0M2ω1M2ω1M0M2ω1M2ω1+6ψM6ψM2ω1+6ψM4ω16ψM2ω16ψM6ψM2ω16ψM4ω16ψM4ω1M2ω1M0M2ω1M4ω1M4ω1+6ψM2ω1+6ψM6ψM2ω1+6ψM4ω1+6ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1
Isotypic characterM0M2ω16ψM6ψM2ω16ψM2ω1M0M2ω1M2ω1M0M2ω1M2ω1+6ψM6ψM2ω1+6ψM4ω16ψM2ω16ψM6ψM2ω16ψM4ω16ψM4ω1M2ω1M0M2ω1M4ω1M4ω1+6ψM2ω1+6ψM6ψM2ω1+6ψM4ω1+6ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1

Semisimple subalgebra: W_{3}
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.
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, 1.00)
0: (1.00, 0.00): (750.00, 300.00)
1: (0.00, 1.00): (200.00, 302.08)




Made total 1960919 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
1*2 (unknown) gens:
(
x_{1} g_{-1}+x_{2} g_{-3}+x_{3} g_{-5}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-8}+x_{7} g_{-9}+x_{8} g_{-11}, x_{16} g_{11}+x_{15} g_{9}+x_{14} g_{8}+x_{13} g_{7}+x_{12} g_{6}+x_{11} g_{5}+x_{10} g_{3}+x_{9} g_{1})
h: (3, 4, 5, 4, 3), e = combination of g_{1} g_{3} g_{5} g_{6} g_{7} g_{8} g_{9} g_{11} , f= combination of g_{-1} g_{-3} g_{-5} g_{-6} g_{-7} g_{-8} g_{-9} g_{-11} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{6\omega_{1}}+4V_{4\omega_{1}}+8V_{2\omega_{1}}+9V_{0}+8V_{-2\omega_{1}}+4V_{-4\omega_{1}}+V_{-6\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{4} x_{12} +x_{1} x_{9} -3= 0
x_{8} x_{14} +x_{5} x_{10} -x_{4} x_{9} = 0
x_{8} x_{16} +x_{6} x_{14} +x_{5} x_{13} +x_{2} x_{10} -5= 0
x_{8} x_{13} -x_{7} x_{11} +x_{6} x_{10} = 0
x_{7} x_{15} +x_{3} x_{11} -3= 0
x_{6} x_{16} +x_{2} x_{13} -x_{1} x_{12} = 0
x_{8} x_{16} +x_{5} x_{13} +x_{4} x_{12} -4= 0
x_{5} x_{16} -x_{3} x_{15} +x_{2} x_{14} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} -4= 0
The above system after transformation.
x_{4} x_{12} +x_{1} x_{9} -3= 0
x_{8} x_{14} +x_{5} x_{10} -x_{4} x_{9} = 0
x_{8} x_{16} +x_{6} x_{14} +x_{5} x_{13} +x_{2} x_{10} -5= 0
x_{8} x_{13} -x_{7} x_{11} +x_{6} x_{10} = 0
x_{7} x_{15} +x_{3} x_{11} -3= 0
x_{6} x_{16} +x_{2} x_{13} -x_{1} x_{12} = 0
x_{8} x_{16} +x_{5} x_{13} +x_{4} x_{12} -4= 0
x_{5} x_{16} -x_{3} x_{15} +x_{2} x_{14} = 0
x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} -4= 0
For the calculator:
(DynkinType =A^{11}_1; ElementsCartan =((3, 4, 5, 4, 3)); generators =(x_{1} g_{-1}+x_{2} g_{-3}+x_{3} g_{-5}+x_{4} g_{-6}+x_{5} g_{-7}+x_{6} g_{-8}+x_{7} g_{-9}+x_{8} g_{-11}, x_{16} g_{11}+x_{15} g_{9}+x_{14} g_{8}+x_{13} g_{7}+x_{12} g_{6}+x_{11} g_{5}+x_{10} g_{3}+x_{9} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{4} x_{12} +x_{1} x_{9} -3, x_{8} x_{14} +x_{5} x_{10} -x_{4} x_{9} , x_{8} x_{16} +x_{6} x_{14} +x_{5} x_{13} +x_{2} x_{10} -5, x_{8} x_{13} -x_{7} x_{11} +x_{6} x_{10} , x_{7} x_{15} +x_{3} x_{11} -3, x_{6} x_{16} +x_{2} x_{13} -x_{1} x_{12} , x_{8} x_{16} +x_{5} x_{13} +x_{4} x_{12} -4, x_{5} x_{16} -x_{3} x_{15} +x_{2} x_{14} , x_{8} x_{16} +x_{7} x_{15} +x_{6} x_{14} -4 )