Subalgebra A12+2A21A16
52 out of 61
Computations done by the calculator project.

Subalgebra type: A12+2A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from A12+A21 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: A16
Basis of Cartan of centralizer: 1 vectors: (4, 1, -2, -5, -8, -4)

Elements Cartan subalgebra scaled to act by two by components: A12: (1, 1, 1, 1, 1, 1): 2, (0, 0, 0, 0, 0, -1): 2, A21: (0, 1, 2, 1, 0, 0): 4, A21: (0, 1, 0, 1, 0, 0): 4
Dimension of subalgebra generated by predefined or computed generators: 14.
Negative simple generators: g21, g6, g8+g9, g2+g4
Positive simple generators: g21, g6, g9+g8, g4+g2
Cartan symmetric matrix: (2100120000100001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (2100120000400004)
Decomposition of ambient Lie algebra: V2ω3+2ω4Vω2+ω3+ω4Vω1+ω3+ω4V2ω4V2ω3Vω1+ω2V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω3+ω4+14ψV2ω3+2ω4V2ω4V2ω3Vω1+ω2V0Vω2+ω3+ω414ψ
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 7) ; the vectors are over the primal subalgebra.h62h55/4h41/2h3+1/4h2+h1g19g9+g8g4+g2g16g17g13
weight0ω1+ω22ω32ω4ω1+ω3+ω4ω2+ω3+ω42ω3+2ω4
weights rel. to Cartan of (centralizer+semisimple s.a.). 0ω1+ω22ω32ω4ω1+ω3+ω4+14ψω2+ω3+ω414ψ2ω3+2ω4
Isotypic module decomposition over primal subalgebra (total 7 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0, 0)Vω1+ω2 → (1, 1, 0, 0, 0)V2ω3 → (0, 0, 2, 0, 0)V2ω4 → (0, 0, 0, 2, 0)Vω1+ω3+ω4+14ψ → (1, 0, 1, 1, 14)Vω2+ω3+ω414ψ → (0, 1, 1, 1, -14)V2ω3+2ω4 → (0, 0, 2, 2, 0)
Module label W1W2W3W4W5W6W7
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
h62h55/4h41/2h3+1/4h2+h1
Semisimple subalgebra component.
g19
g6
g21
h6
h6h5h4h3h2h1
g21
2g6
g19
Semisimple subalgebra component.
g9g8
h4+2h3+h2
2g8+2g9
Semisimple subalgebra component.
g4g2
h4+h2
2g2+2g4
g16
g11
g7
g12
g5
g18
g15
g1
g14
g10
g20
g17
g17
g20
g10
g14
g1
g15
g18
g5
g12
g7
g11
g16
g13
g4g2
g9g8
2g3
h4+h2
2g3
2g82g9
2g22g4
4g13
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1+ω2
ω1+2ω2
2ω1ω2
0
0
2ω1+ω2
ω12ω2
ω1ω2
2ω3
0
2ω3
2ω4
0
2ω4
ω1+ω3+ω4
ω1+ω2+ω3+ω4
ω1ω3+ω4
ω1+ω3ω4
ω2+ω3+ω4
ω1+ω2ω3+ω4
ω1+ω2+ω3ω4
ω1ω3ω4
ω2ω3+ω4
ω2+ω3ω4
ω1+ω2ω3ω4
ω2ω3ω4
ω2+ω3+ω4
ω1ω2+ω3+ω4
ω2ω3+ω4
ω2+ω3ω4
ω1+ω3+ω4
ω1ω2ω3+ω4
ω1ω2+ω3ω4
ω2ω3ω4
ω1ω3+ω4
ω1+ω3ω4
ω1ω2ω3ω4
ω1ω3ω4
2ω3+2ω4
2ω4
2ω3
2ω3+2ω4
0
2ω32ω4
2ω3
2ω4
2ω32ω4
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1+ω2
ω1+2ω2
2ω1ω2
0
0
2ω1+ω2
ω12ω2
ω1ω2
2ω3
0
2ω3
2ω4
0
2ω4
ω1+ω3+ω4+14ψ
ω1+ω2+ω3+ω4+14ψ
ω1ω3+ω4+14ψ
ω1+ω3ω4+14ψ
ω2+ω3+ω4+14ψ
ω1+ω2ω3+ω4+14ψ
ω1+ω2+ω3ω4+14ψ
ω1ω3ω4+14ψ
ω2ω3+ω4+14ψ
ω2+ω3ω4+14ψ
ω1+ω2ω3ω4+14ψ
ω2ω3ω4+14ψ
ω2+ω3+ω414ψ
ω1ω2+ω3+ω414ψ
ω2ω3+ω414ψ
ω2+ω3ω414ψ
ω1+ω3+ω414ψ
ω1ω2ω3+ω414ψ
ω1ω2+ω3ω414ψ
ω2ω3ω414ψ
ω1ω3+ω414ψ
ω1+ω3ω414ψ
ω1ω2ω3ω414ψ
ω1ω3ω414ψ
2ω3+2ω4
2ω4
2ω3
2ω3+2ω4
0
2ω32ω4
2ω3
2ω4
2ω32ω4
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1+ω2Mω1+2ω2M2ω1ω22M0M2ω1+ω2Mω12ω2Mω1ω2M2ω3M0M2ω3M2ω4M0M2ω4Mω1+ω3+ω4+14ψMω1+ω2+ω3+ω4+14ψMω2+ω3+ω4+14ψMω1ω3+ω4+14ψMω1+ω3ω4+14ψMω1+ω2ω3+ω4+14ψMω1+ω2+ω3ω4+14ψMω2ω3+ω4+14ψMω2+ω3ω4+14ψMω1ω3ω4+14ψMω1+ω2ω3ω4+14ψMω2ω3ω4+14ψMω2+ω3+ω414ψMω1ω2+ω3+ω414ψMω1+ω3+ω414ψMω2ω3+ω414ψMω2+ω3ω414ψMω1ω2ω3+ω414ψMω1ω2+ω3ω414ψMω1ω3+ω414ψMω1+ω3ω414ψMω2ω3ω414ψMω1ω2ω3ω414ψMω1ω3ω414ψM2ω3+2ω4M2ω4M2ω3M2ω3+2ω4M0M2ω32ω4M2ω3M2ω4M2ω32ω4
Isotypic characterM0Mω1+ω2Mω1+2ω2M2ω1ω22M0M2ω1+ω2Mω12ω2Mω1ω2M2ω3M0M2ω3M2ω4M0M2ω4Mω1+ω3+ω4+14ψMω1+ω2+ω3+ω4+14ψMω2+ω3+ω4+14ψMω1ω3+ω4+14ψMω1+ω3ω4+14ψMω1+ω2ω3+ω4+14ψMω1+ω2+ω3ω4+14ψMω2ω3+ω4+14ψMω2+ω3ω4+14ψMω1ω3ω4+14ψMω1+ω2ω3ω4+14ψMω2ω3ω4+14ψMω2+ω3+ω414ψMω1ω2+ω3+ω414ψMω1+ω3+ω414ψMω2ω3+ω414ψMω2+ω3ω414ψMω1ω2ω3+ω414ψMω1ω2+ω3ω414ψMω1ω3+ω414ψMω1+ω3ω414ψMω2ω3ω414ψMω1ω2ω3ω414ψMω1ω3ω414ψM2ω3+2ω4M2ω4M2ω3M2ω3+2ω4M0M2ω32ω4M2ω3M2ω4M2ω32ω4

Semisimple subalgebra: W_{2}+W_{3}+W_{4}
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, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00, 0.00): (266.67, 333.33)
1: (0.00, 1.00, 0.00, 0.00, 0.00): (233.33, 366.67)
2: (0.00, 0.00, 1.00, 0.00, 0.00): (200.00, 300.00)
3: (0.00, 0.00, 0.00, 1.00, 0.00): (200.00, 300.00)
4: (0.00, 0.00, 0.00, 0.00, 1.00): (200.00, 300.00)




Made total 1466930 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.
4*2 (unknown) gens:
(
x_{1} g_{-21}, x_{7} g_{21},
x_{2} g_{6}, x_{8} g_{-6},
x_{3} g_{-8}+x_{4} g_{-9}, x_{10} g_{9}+x_{9} g_{8},
x_{5} g_{-2}+x_{6} g_{-4}, x_{12} g_{4}+x_{11} g_{2})
h: (1, 1, 1, 1, 1, 1), e = combination of g_{21} , f= combination of g_{-21} h: (0, 0, 0, 0, 0, -1), e = combination of g_{-6} , f= combination of g_{6} h: (0, 1, 2, 1, 0, 0), e = combination of g_{8} g_{9} , f= combination of g_{-8} g_{-9} h: (0, 1, 0, 1, 0, 0), e = combination of g_{2} g_{4} , f= combination of g_{-2} g_{-4} Positive weight subsystem: 5 vectors: (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 1, 0, 0)
Symmetric Cartan default scale: \begin{pmatrix}
2 & -1 & 0 & 0\\
-1 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{2\omega_{3}+2\omega_{4}}+V_{\omega_{2}+\omega_{3}+\omega_{4}}+V_{\omega_{1}+\omega_{3}+\omega_{4}}+2V_{2\omega_{4}}+V_{-\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}}+V_{\omega_{1}-\omega_{2}+\omega_{3}+\omega_{4}}+2V_{2\omega_{3}}+V_{\omega_{1}+\omega_{2}}+V_{-\omega_{1}+\omega_{3}+\omega_{4}}+V_{-\omega_{2}+\omega_{3}+\omega_{4}}+V_{\omega_{2}-\omega_{3}+\omega_{4}}+V_{\omega_{1}-\omega_{3}+\omega_{4}}+V_{-\omega_{1}+2\omega_{2}}+V_{2\omega_{1}-\omega_{2}}+V_{\omega_{2}+\omega_{3}-\omega_{4}}+V_{\omega_{1}+\omega_{3}-\omega_{4}}+V_{-2\omega_{3}+2\omega_{4}}+V_{-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}}+V_{\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}}+6V_{0}+V_{-\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}}+V_{\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}}+V_{2\omega_{3}-2\omega_{4}}+V_{-\omega_{1}-\omega_{3}+\omega_{4}}+V_{-\omega_{2}-\omega_{3}+\omega_{4}}+V_{-2\omega_{1}+\omega_{2}}+V_{\omega_{1}-2\omega_{2}}+V_{-\omega_{1}+\omega_{3}-\omega_{4}}+V_{-\omega_{2}+\omega_{3}-\omega_{4}}+V_{\omega_{2}-\omega_{3}-\omega_{4}}+V_{\omega_{1}-\omega_{3}-\omega_{4}}+V_{-\omega_{1}-\omega_{2}}+2V_{-2\omega_{3}}+V_{-\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}}+V_{\omega_{1}-\omega_{2}-\omega_{3}-\omega_{4}}+2V_{-2\omega_{4}}+V_{-\omega_{1}-\omega_{3}-\omega_{4}}+V_{-\omega_{2}-\omega_{3}-\omega_{4}}+V_{-2\omega_{3}-2\omega_{4}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{7} -1= 0
x_{2} x_{8} -1= 0
x_{3} x_{9} -1= 0
x_{4} x_{10} +x_{3} x_{9} -2= 0
x_{4} x_{10} -1= 0
x_{4} x_{12} -x_{3} x_{11} = 0
x_{9} x_{12} -x_{10} x_{11} = 0
x_{3} x_{6} -x_{4} x_{5} = 0
x_{5} x_{11} -1= 0
x_{6} x_{12} -1= 0
x_{6} x_{10} -x_{5} x_{9} = 0
The above system after transformation.
x_{1} x_{7} -1= 0
x_{2} x_{8} -1= 0
x_{3} x_{9} -1= 0
x_{4} x_{10} +x_{3} x_{9} -2= 0
x_{4} x_{10} -1= 0
x_{4} x_{12} -x_{3} x_{11} = 0
x_{9} x_{12} -x_{10} x_{11} = 0
x_{3} x_{6} -x_{4} x_{5} = 0
x_{5} x_{11} -1= 0
x_{6} x_{12} -1= 0
x_{6} x_{10} -x_{5} x_{9} = 0
For the calculator:
(DynkinType =A^{1}_2+2A^{2}_1; ElementsCartan =((1, 1, 1, 1, 1, 1), (0, 0, 0, 0, 0, -1), (0, 1, 2, 1, 0, 0), (0, 1, 0, 1, 0, 0)); generators =(x_{1} g_{-21}, x_{7} g_{21}, x_{2} g_{6}, x_{8} g_{-6}, x_{3} g_{-8}+x_{4} g_{-9}, x_{10} g_{9}+x_{9} g_{8}, x_{5} g_{-2}+x_{6} g_{-4}, x_{12} g_{4}+x_{11} g_{2}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{7} -1, x_{2} x_{8} -1, x_{3} x_{9} -1, x_{4} x_{10} +x_{3} x_{9} -2, x_{4} x_{10} -1, x_{4} x_{12} -x_{3} x_{11} , x_{9} x_{12} -x_{10} x_{11} , x_{3} x_{6} -x_{4} x_{5} , x_{5} x_{11} -1, x_{6} x_{12} -1, x_{6} x_{10} -x_{5} x_{9} )