Processing math: 100%
Subalgebra A281F14
11 out of 59
Computations done by the calculator project.

Subalgebra type: A281 (click on type for detailed printout).
Centralizer: A81 .
The semisimple part of the centralizer of the semisimple part of my centralizer: G12
Basis of Cartan of centralizer: 1 vectors: (0, 0, 0, 1)
Contained up to conjugation as a direct summand of: A281+A81 .

Elements Cartan subalgebra scaled to act by two by components: A281: (10, 18, 24, 12): 56
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: g1+g2+g13
Positive simple generators: 6g13+6g2+10g1
Cartan symmetric matrix: (1/14)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (56)
Decomposition of ambient Lie algebra: V10ω15V6ω1V2ω13V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). V10ω1V6ω1+2ψV6ω1+ψV6ω1V6ω1ψV6ω12ψV2ω1VψV0Vψ
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.g3+g7h4g7+g3g13+g2+5/3g1g14g19g22+1/2g17g21g20g24
weight0002ω16ω16ω16ω16ω16ω110ω1
weights rel. to Cartan of (centralizer+semisimple s.a.). ψ0ψ2ω16ω12ψ6ω1ψ6ω16ω1+ψ6ω1+2ψ10ω1
Isotypic module decomposition over primal subalgebra (total 10 isotypic components).
Isotypical components + highest weightVψ → (0, -1)V0 → (0, 0)Vψ → (0, 1)V2ω1 → (2, 0)V6ω12ψ → (6, -2)V6ω1ψ → (6, -1)V6ω1 → (6, 0)V6ω1+ψ → (6, 1)V6ω1+2ψ → (6, 2)V10ω1 → (10, 0)
Module label W1W2W3W4W5W6W7W8W9W10
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.
g3+g7
Cartan of centralizer component.
h4
g7+g3
Semisimple subalgebra component.
3/5g133/5g2g1
6/5h4+12/5h3+9/5h2+h1
1/5g1+1/5g2+1/5g13
g14
g11
g9
g4
2g16
2g18
2g20
g19
g8
g6
g3+g7
2g10
2g12
2g21
g22+1/2g17
1/2g15+g5
1/2g13+g2
h4+2h3
2g2+g13
2g5g15
g172g22
g21
g12
g10
g7+g3
2g6
2g8
2g19
g20
g18
g16
g4
2g9
2g11
2g14
g24
g23
g22g17
2g152g5
2g132g2+6g1
4h4+8h3+6h26h1
18g1+10g2+10g13
28g528g15
56g1756g22
168g23
168g24
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0002ω1
0
2ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
10ω1
8ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
8ω1
10ω1
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizerψ0ψ2ω1
0
2ω1
6ω12ψ
4ω12ψ
2ω12ψ
2ψ
2ω12ψ
4ω12ψ
6ω12ψ
6ω1ψ
4ω1ψ
2ω1ψ
ψ
2ω1ψ
4ω1ψ
6ω1ψ
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
6ω1+ψ
4ω1+ψ
2ω1+ψ
ψ
2ω1+ψ
4ω1+ψ
6ω1+ψ
6ω1+2ψ
4ω1+2ψ
2ω1+2ψ
2ψ
2ω1+2ψ
4ω1+2ψ
6ω1+2ψ
10ω1
8ω1
6ω1
4ω1
2ω1
0
2ω1
4ω1
6ω1
8ω1
10ω1
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.MψM0MψM2ω1M0M2ω1M6ω12ψM4ω12ψM2ω12ψM2ψM2ω12ψM4ω12ψM6ω12ψM6ω1ψM4ω1ψM2ω1ψMψM2ω1ψM4ω1ψM6ω1ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M6ω1+ψM4ω1+ψM2ω1+ψMψM2ω1+ψM4ω1+ψM6ω1+ψM6ω1+2ψM4ω1+2ψM2ω1+2ψM2ψM2ω1+2ψM4ω1+2ψM6ω1+2ψM10ω1M8ω1M6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M8ω1M10ω1
Isotypic characterMψM0MψM2ω1M0M2ω1M6ω12ψM4ω12ψM2ω12ψM2ψM2ω12ψM4ω12ψM6ω12ψM6ω1ψM4ω1ψM2ω1ψMψM2ω1ψM4ω1ψM6ω1ψM6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M6ω1+ψM4ω1+ψM2ω1+ψMψM2ω1+ψM4ω1+ψM6ω1+ψM6ω1+2ψM4ω1+2ψM2ω1+2ψM2ψM2ω1+2ψM4ω1+2ψM6ω1+2ψM10ω1M8ω1M6ω1M4ω1M2ω1M0M2ω1M4ω1M6ω1M8ω1M10ω1

Semisimple subalgebra: W_{4}
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, 1.00)
0: (1.00, 0.00): (1600.00, 300.00)
1: (0.00, 1.00): (200.00, 325.00)



Made total 7149101 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_{-2}+x_{3} g_{-6}+x_{4} g_{-9}+x_{5} g_{-10}+x_{6} g_{-13}+x_{7} g_{-16}, x_{14} g_{16}+x_{13} g_{13}+x_{12} g_{10}+x_{11} g_{9}+x_{10} g_{6}+x_{9} g_{2}+x_{8} g_{1})

Unknown splitting cartan of centralizer.
x_{18} h_{4}+x_{17} h_{3}+x_{16} h_{2}+x_{15} h_{1}
h: (10, 18, 24, 12), e = combination of g_{1} g_{2} g_{6} g_{9} g_{10} g_{13} g_{16} , f= combination of g_{-1} g_{-2} g_{-6} g_{-9} g_{-10} g_{-13} g_{-16} Positive weight subsystem: 1 vectors: (1)
Symmetric Cartan default scale: \begin{pmatrix}
2\\
\end{pmatrix}Character ambient Lie algebra: V_{10\omega_{1}}+V_{8\omega_{1}}+6V_{6\omega_{1}}+6V_{4\omega_{1}}+7V_{2\omega_{1}}+10V_{0}+7V_{-2\omega_{1}}+6V_{-4\omega_{1}}+6V_{-6\omega_{1}}+V_{-8\omega_{1}}+V_{-10\omega_{1}}
A necessary system to realize the candidate subalgebra.
x_{18}^{2}x_{19} -x_{17} x_{18} x_{19} +x_{17}^{2}x_{19} -2x_{16} x_{17} x_{19} +2x_{16}^{2}x_{19} -2x_{15} x_{16} x_{19}
+2x_{15}^{2}x_{19} -1= 0
x_{1} x_{8} -10= 0
x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} -18= 0
x_{6} x_{12} +x_{4} x_{10} -x_{3} x_{9} = 0
x_{7} x_{12} -x_{6} x_{10} +x_{5} x_{9} = 0
x_{5} x_{13} +x_{3} x_{11} -x_{2} x_{10} = 0
x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} -12= 0
x_{7} x_{13} -x_{6} x_{11} -x_{5} x_{10} = 0
x_{5} x_{14} -x_{3} x_{13} +x_{2} x_{12} = 0
x_{6} x_{14} -x_{4} x_{13} -x_{3} x_{12} = 0
x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -6= 0
x_{1} x_{16} -2x_{1} x_{15} = 0
x_{2} x_{17} -2x_{2} x_{16} +x_{2} x_{15} = 0
x_{3} x_{18} -2x_{3} x_{16} +2x_{3} x_{15} = 0
x_{4} x_{18} -x_{4} x_{17} +x_{4} x_{15} = 0
x_{5} x_{18} -x_{5} x_{17} +2x_{5} x_{16} -2x_{5} x_{15} = 0
x_{6} x_{17} -2x_{6} x_{15} = 0
x_{7} x_{18} -x_{7} x_{15} = 0
x_{8} x_{16} -2x_{8} x_{15} = 0
x_{9} x_{17} -2x_{9} x_{16} +x_{9} x_{15} = 0
x_{10} x_{18} -2x_{10} x_{16} +2x_{10} x_{15} = 0
x_{11} x_{18} -x_{11} x_{17} +x_{11} x_{15} = 0
x_{12} x_{18} -x_{12} x_{17} +2x_{12} x_{16} -2x_{12} x_{15} = 0
x_{13} x_{17} -2x_{13} x_{15} = 0
x_{14} x_{18} -x_{14} x_{15} = 0
The above system after transformation.
x_{18}^{2}x_{19} -x_{17} x_{18} x_{19} +x_{17}^{2}x_{19} -2x_{16} x_{17} x_{19} +2x_{16}^{2}x_{19} -2x_{15} x_{16} x_{19}
+2x_{15}^{2}x_{19} -1= 0
x_{1} x_{8} -10= 0
x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} -18= 0
x_{6} x_{12} +x_{4} x_{10} -x_{3} x_{9} = 0
x_{7} x_{12} -x_{6} x_{10} +x_{5} x_{9} = 0
x_{5} x_{13} +x_{3} x_{11} -x_{2} x_{10} = 0
x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} -12= 0
x_{7} x_{13} -x_{6} x_{11} -x_{5} x_{10} = 0
x_{5} x_{14} -x_{3} x_{13} +x_{2} x_{12} = 0
x_{6} x_{14} -x_{4} x_{13} -x_{3} x_{12} = 0
x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -6= 0
x_{1} x_{16} -2x_{1} x_{15} = 0
x_{2} x_{17} -2x_{2} x_{16} +x_{2} x_{15} = 0
x_{3} x_{18} -2x_{3} x_{16} +2x_{3} x_{15} = 0
x_{4} x_{18} -x_{4} x_{17} +x_{4} x_{15} = 0
x_{5} x_{18} -x_{5} x_{17} +2x_{5} x_{16} -2x_{5} x_{15} = 0
x_{6} x_{17} -2x_{6} x_{15} = 0
x_{7} x_{18} -x_{7} x_{15} = 0
x_{8} x_{16} -2x_{8} x_{15} = 0
x_{9} x_{17} -2x_{9} x_{16} +x_{9} x_{15} = 0
x_{10} x_{18} -2x_{10} x_{16} +2x_{10} x_{15} = 0
x_{11} x_{18} -x_{11} x_{17} +x_{11} x_{15} = 0
x_{12} x_{18} -x_{12} x_{17} +2x_{12} x_{16} -2x_{12} x_{15} = 0
x_{13} x_{17} -2x_{13} x_{15} = 0
x_{14} x_{18} -x_{14} x_{15} = 0
For the calculator:
(DynkinType =A^{28}_1; ElementsCartan =((10, 18, 24, 12)); generators =(x_{1} g_{-1}+x_{2} g_{-2}+x_{3} g_{-6}+x_{4} g_{-9}+x_{5} g_{-10}+x_{6} g_{-13}+x_{7} g_{-16}, x_{14} g_{16}+x_{13} g_{13}+x_{12} g_{10}+x_{11} g_{9}+x_{10} g_{6}+x_{9} g_{2}+x_{8} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{18}^{2}x_{19} -x_{17} x_{18} x_{19} +x_{17}^{2}x_{19} -2x_{16} x_{17} x_{19} +2x_{16}^{2}x_{19} -2x_{15} x_{16} x_{19} +2x_{15}^{2}x_{19} -1, x_{1} x_{8} -10, x_{7} x_{14} +2x_{6} x_{13} +2x_{5} x_{12} +x_{4} x_{11} +2x_{3} x_{10} +x_{2} x_{9} -18, x_{6} x_{12} +x_{4} x_{10} -x_{3} x_{9} , x_{7} x_{12} -x_{6} x_{10} +x_{5} x_{9} , x_{5} x_{13} +x_{3} x_{11} -x_{2} x_{10} , x_{7} x_{14} +2x_{6} x_{13} +x_{5} x_{12} +x_{4} x_{11} +x_{3} x_{10} -12, x_{7} x_{13} -x_{6} x_{11} -x_{5} x_{10} , x_{5} x_{14} -x_{3} x_{13} +x_{2} x_{12} , x_{6} x_{14} -x_{4} x_{13} -x_{3} x_{12} , x_{7} x_{14} +x_{6} x_{13} +x_{5} x_{12} -6, x_{1} x_{16} -2x_{1} x_{15} , x_{2} x_{17} -2x_{2} x_{16} +x_{2} x_{15} , x_{3} x_{18} -2x_{3} x_{16} +2x_{3} x_{15} , x_{4} x_{18} -x_{4} x_{17} +x_{4} x_{15} , x_{5} x_{18} -x_{5} x_{17} +2x_{5} x_{16} -2x_{5} x_{15} , x_{6} x_{17} -2x_{6} x_{15} , x_{7} x_{18} -x_{7} x_{15} , x_{8} x_{16} -2x_{8} x_{15} , x_{9} x_{17} -2x_{9} x_{16} +x_{9} x_{15} , x_{10} x_{18} -2x_{10} x_{16} +2x_{10} x_{15} , x_{11} x_{18} -x_{11} x_{17} +x_{11} x_{15} , x_{12} x_{18} -x_{12} x_{17} +2x_{12} x_{16} -2x_{12} x_{15} , x_{13} x_{17} -2x_{13} x_{15} , x_{14} x_{18} -x_{14} x_{15} )