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Subalgebra

$A^{18}_1+A^{3}_1$ ↪

$C^{1}_5$
51 out of 119

Computations done by the calculator project.

Subalgebra type:

$\displaystyle A^{18}_1+A^{3}_1$ (click on type for detailed printout).
Subalgebra is (parabolically) induced from

$\displaystyle A^{18}_1$ .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer:

$\displaystyle C^{1}_5$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{18}_1$ : (6, 10, 14, 16, 8): 36,

$\displaystyle A^{3}_1$ : (0, 2, 0, 0, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 6.
Negative simple generators:

$\displaystyle g_{-10}+g_{-11}+g_{-12}+g_{-13}$ ,

$\displaystyle g_{-2}+g_{-5}$
Positive simple generators:

$\displaystyle 4g_{13}+2g_{12}+2g_{11}+3g_{10}$ ,

$\displaystyle g_{5}+g_{2}$
Cartan symmetric matrix:

$\displaystyle \begin{pmatrix}1/9 & 0\\ 0 & 2/3\\ \end{pmatrix}$
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix):

$\displaystyle \begin{pmatrix}36 & 0\\ 0 & 6\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{4\omega_{1}+2\omega_{2}}\oplus V_{5\omega_{1}+\omega_{2}}\oplus V_{6\omega_{1}}\oplus V_{3\omega_{1}+\omega_{2}}\oplus V_{2\omega_{2}} \oplus V_{\omega_{1}+\omega_{2}}\oplus 2V_{2\omega_{1}}$
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.

Highest vectors of representations (total 8) ; the vectors are over the primal subalgebra.	$g_{13}+3/4g_{10}$	$g_{12}+g_{11}$	$g_{9}+1/2g_{7}+3/2g_{6}$	$g_{5}+g_{2}$	$-g_{18}+3/2g_{17}$	$g_{25}$	$g_{24}$	$g_{23}$
weight	$2\omega_{1}$	$2\omega_{1}$	$\omega_{1}+\omega_{2}$	$2\omega_{2}$	$3\omega_{1}+\omega_{2}$	$6\omega_{1}$	$5\omega_{1}+\omega_{2}$	$4\omega_{1}+2\omega_{2}$

Isotypic module decomposition over primal subalgebra (total 8 isotypic components).

Isotypical components + highest weight

$\displaystyle V_{2\omega_{1}}$ → (2, 0)

$\displaystyle V_{\omega_{1}+\omega_{2}}$ → (1, 1)

$\displaystyle V_{2\omega_{2}}$ → (0, 2)

$\displaystyle V_{3\omega_{1}+\omega_{2}}$ → (3, 1)

$\displaystyle V_{6\omega_{1}}$ → (6, 0)

$\displaystyle V_{5\omega_{1}+\omega_{2}}$ → (5, 1)

$\displaystyle V_{4\omega_{1}+2\omega_{2}}$ → (4, 2)

Module label

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

$W_{5}$

$W_{6}$

$W_{7}$

$W_{8}$

Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.

Semisimple subalgebra component.

$-4/3g_{13}-2/3g_{12}-2/3g_{11}-g_{10}$

$8/3h_{5}+16/3h_{4}+14/3h_{3}+10/3h_{2}+2h_{1}$

$2/3g_{-10}+2/3g_{-11}+2/3g_{-12}+2/3g_{-13}$

$g_{13}+3/4g_{10}$

$-h_{5}-2h_{4}-3/2h_{3}-3/2h_{2}-3/2h_{1}$

$-1/2g_{-10}-1/2g_{-13}$

$g_{9}+1/2g_{7}+3/2g_{6}$

$-1/2g_{-1}+1/2g_{-3}-1/2g_{-4}$

$-g_{4}+1/2g_{3}-3/2g_{1}$

$-1/2g_{-6}-1/2g_{-7}-1/2g_{-9}$

Semisimple subalgebra component.

$-g_{5}-g_{2}$

$h_{5}+2h_{2}$

$2g_{-2}+2g_{-5}$

$-g_{18}+3/2g_{17}$

$1/2g_{9}+g_{7}-3/2g_{6}$

$-g_{16}-3/2g_{14}$

$-g_{-1}-2g_{-3}+1/2g_{-4}$

$-1/2g_{4}+g_{3}+3/2g_{1}$

$-3/2g_{-14}-3/2g_{-16}$

$-g_{-6}+2g_{-7}+1/2g_{-9}$

$-3/2g_{-17}+3/2g_{-18}$

$g_{25}$

$g_{20}$

$2g_{13}-g_{10}$

$-2h_{5}-4h_{4}+2h_{3}+2h_{2}+2h_{1}$

$4g_{-10}-6g_{-13}$

$10g_{-20}$

$-20g_{-25}$

$g_{24}$

$g_{18}+g_{17}$

$g_{22}$

$2g_{9}-g_{7}-g_{6}$

$g_{16}-g_{14}$

$g_{-1}-3g_{-3}-3g_{-4}$

$-2g_{4}-g_{3}+g_{1}$

$4g_{-14}-6g_{-16}$

$g_{-6}+3g_{-7}-3g_{-9}$

$10g_{-22}$

$4g_{-17}+6g_{-18}$

$-10g_{-24}$

$g_{23}$

$g_{15}$

$g_{21}$

$2g_{5}-g_{2}$

$g_{12}-g_{11}$

$2g_{19}$

$-3g_{-8}$

$-2h_{5}+2h_{2}$

$-2g_{8}$

$-6g_{-19}$

$3g_{-11}-3g_{-12}$

$2g_{-2}-4g_{-5}$

$6g_{-21}$

$6g_{-15}$

$-12g_{-23}$

Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above

$2\omega_{1}$

$0$

$-2\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$\omega_{1}+\omega_{2}$

$-\omega_{1}+\omega_{2}$

$\omega_{1}-\omega_{2}$

$-\omega_{1}-\omega_{2}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$3\omega_{1}+\omega_{2}$

$\omega_{1}+\omega_{2}$

$3\omega_{1}-\omega_{2}$

$-\omega_{1}+\omega_{2}$

$\omega_{1}-\omega_{2}$

$-3\omega_{1}+\omega_{2}$

$-\omega_{1}-\omega_{2}$

$-3\omega_{1}-\omega_{2}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$5\omega_{1}+\omega_{2}$

$3\omega_{1}+\omega_{2}$

$5\omega_{1}-\omega_{2}$

$\omega_{1}+\omega_{2}$

$3\omega_{1}-\omega_{2}$

$-\omega_{1}+\omega_{2}$

$\omega_{1}-\omega_{2}$

$-3\omega_{1}+\omega_{2}$

$-\omega_{1}-\omega_{2}$

$-5\omega_{1}+\omega_{2}$

$-3\omega_{1}-\omega_{2}$

$-5\omega_{1}-\omega_{2}$

$4\omega_{1}+2\omega_{2}$

$2\omega_{1}+2\omega_{2}$

$4\omega_{1}$

$2\omega_{2}$

$2\omega_{1}$

$4\omega_{1}-2\omega_{2}$

$-2\omega_{1}+2\omega_{2}$

$0$

$2\omega_{1}-2\omega_{2}$

$-4\omega_{1}+2\omega_{2}$

$-2\omega_{1}$

$-2\omega_{2}$

$-4\omega_{1}$

$-2\omega_{1}-2\omega_{2}$

$-4\omega_{1}-2\omega_{2}$

Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer

$2\omega_{1}$

$0$

$-2\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$\omega_{1}+\omega_{2}$

$-\omega_{1}+\omega_{2}$

$\omega_{1}-\omega_{2}$

$-\omega_{1}-\omega_{2}$

$2\omega_{2}$

$0$

$-2\omega_{2}$

$3\omega_{1}+\omega_{2}$

$\omega_{1}+\omega_{2}$

$3\omega_{1}-\omega_{2}$

$-\omega_{1}+\omega_{2}$

$\omega_{1}-\omega_{2}$

$-3\omega_{1}+\omega_{2}$

$-\omega_{1}-\omega_{2}$

$-3\omega_{1}-\omega_{2}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$5\omega_{1}+\omega_{2}$

$3\omega_{1}+\omega_{2}$

$5\omega_{1}-\omega_{2}$

$\omega_{1}+\omega_{2}$

$3\omega_{1}-\omega_{2}$

$-\omega_{1}+\omega_{2}$

$\omega_{1}-\omega_{2}$

$-3\omega_{1}+\omega_{2}$

$-\omega_{1}-\omega_{2}$

$-5\omega_{1}+\omega_{2}$

$-3\omega_{1}-\omega_{2}$

$-5\omega_{1}-\omega_{2}$

$4\omega_{1}+2\omega_{2}$

$2\omega_{1}+2\omega_{2}$

$4\omega_{1}$

$2\omega_{2}$

$2\omega_{1}$

$4\omega_{1}-2\omega_{2}$

$-2\omega_{1}+2\omega_{2}$

$0$

$2\omega_{1}-2\omega_{2}$

$-4\omega_{1}+2\omega_{2}$

$-2\omega_{1}$

$-2\omega_{2}$

$-4\omega_{1}$

$-2\omega_{1}-2\omega_{2}$

$-4\omega_{1}-2\omega_{2}$

Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.

$\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}$

$\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}$

$\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}$

$\displaystyle M_{3\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}} \oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}$

$\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}$

$\displaystyle M_{5\omega_{1}+\omega_{2}}\oplus M_{3\omega_{1}+\omega_{2}}\oplus M_{5\omega_{1}-\omega_{2}}\oplus M_{\omega_{1}+\omega_{2}} \oplus M_{3\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-3\omega_{1}+\omega_{2}} \oplus M_{-\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}+\omega_{2}}\oplus M_{-3\omega_{1}-\omega_{2}}\oplus M_{-5\omega_{1}-\omega_{2}}$

$\displaystyle M_{4\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}+2\omega_{2}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{2}}\oplus M_{2\omega_{1}}\oplus M_{4\omega_{1}-2\omega_{2}} \oplus M_{-2\omega_{1}+2\omega_{2}}\oplus M_{0}\oplus M_{2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}+2\omega_{2}}\oplus M_{-2\omega_{1}} \oplus M_{-2\omega_{2}}\oplus M_{-4\omega_{1}}\oplus M_{-2\omega_{1}-2\omega_{2}}\oplus M_{-4\omega_{1}-2\omega_{2}}$

Isotypic character

$\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}$

$\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}$

$\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}$

$\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}$

Semisimple subalgebra: W_{1}+W_{4}
Centralizer extension: 0

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.

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): (1100.00, 300.00)
1: (0.00, 1.00): (200.00, 450.00)

Made total 1462843 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.
2*2 (unknown) gens:
(
x_{1} g_{-10}+x_{2} g_{-11}+x_{3} g_{-12}+x_{4} g_{-13}, x_{10} g_{13}+x_{9} g_{12}+x_{8} g_{11}+x_{7} g_{10},
x_{5} g_{-2}+x_{6} g_{-5}, x_{12} g_{5}+x_{11} g_{2})
h: (6, 10, 14, 16, 8), e = combination of g_{10} g_{11} g_{12} g_{13} , f= combination of g_{-10} g_{-11} g_{-12} g_{-13} h: (0, 2, 0, 0, 1), e = combination of g_{2} g_{5} , f= combination of g_{-2} g_{-5} Positive weight subsystem: 2 vectors: (1, 0), (0, 1)
Symmetric Cartan default scale: \begin{pmatrix}

2 &	0\\
0 &	2\\

\end{pmatrix}Character ambient Lie algebra: V_{4\omega_{1}+2\omega_{2}}+V_{5\omega_{1}+\omega_{2}}+V_{6\omega_{1}}+V_{2\omega_{1}+2\omega_{2}}+2V_{3\omega_{1}+\omega_{2}}+2V_{4\omega_{1}}+V_{5\omega_{1}-\omega_{2}}+2V_{2\omega_{2}}+3V_{\omega_{1}+\omega_{2}}+4V_{2\omega_{1}}+2V_{3\omega_{1}-\omega_{2}}+V_{4\omega_{1}-2\omega_{2}}+V_{-2\omega_{1}+2\omega_{2}}+3V_{-\omega_{1}+\omega_{2}}+5V_{0}+3V_{\omega_{1}-\omega_{2}}+V_{2\omega_{1}-2\omega_{2}}+V_{-4\omega_{1}+2\omega_{2}}+2V_{-3\omega_{1}+\omega_{2}}+4V_{-2\omega_{1}}+3V_{-\omega_{1}-\omega_{2}}+2V_{-2\omega_{2}}+V_{-5\omega_{1}+\omega_{2}}+2V_{-4\omega_{1}}+2V_{-3\omega_{1}-\omega_{2}}+V_{-2\omega_{1}-2\omega_{2}}+V_{-6\omega_{1}}+V_{-5\omega_{1}-\omega_{2}}+V_{-4\omega_{1}-2\omega_{2}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{7} -3= 0
x_{2} x_{8} +x_{1} x_{7} -5= 0
x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -7= 0
x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} -8= 0
x_{4} x_{10} +2x_{3} x_{9} -8= 0
x_{3} x_{12} -x_{2} x_{11} = 0
x_{8} x_{12} -x_{9} x_{11} = 0
x_{2} x_{6} -x_{3} x_{5} = 0
x_{5} x_{11} -1= 0
x_{6} x_{12} -1= 0
x_{6} x_{9} -x_{5} x_{8} = 0
The above system after transformation.
x_{1} x_{7} -3= 0
x_{2} x_{8} +x_{1} x_{7} -5= 0
x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -7= 0
x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} -8= 0
x_{4} x_{10} +2x_{3} x_{9} -8= 0
x_{3} x_{12} -x_{2} x_{11} = 0
x_{8} x_{12} -x_{9} x_{11} = 0
x_{2} x_{6} -x_{3} x_{5} = 0
x_{5} x_{11} -1= 0
x_{6} x_{12} -1= 0
x_{6} x_{9} -x_{5} x_{8} = 0
For the calculator:
(DynkinType =A^{18}_1+A^{3}_1; ElementsCartan =((6, 10, 14, 16, 8), (0, 2, 0, 0, 1)); generators =(x_{1} g_{-10}+x_{2} g_{-11}+x_{3} g_{-12}+x_{4} g_{-13}, x_{10} g_{13}+x_{9} g_{12}+x_{8} g_{11}+x_{7} g_{10}, x_{5} g_{-2}+x_{6} g_{-5}, x_{12} g_{5}+x_{11} g_{2}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{7} -3, x_{2} x_{8} +x_{1} x_{7} -5, x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -7, x_{4} x_{10} +x_{3} x_{9} +x_{2} x_{8} -8, x_{4} x_{10} +2x_{3} x_{9} -8, x_{3} x_{12} -x_{2} x_{11} , x_{8} x_{12} -x_{9} x_{11} , x_{2} x_{6} -x_{3} x_{5} , x_{5} x_{11} -1, x_{6} x_{12} -1, x_{6} x_{9} -x_{5} x_{8} )