Subalgebra

$A^{2}_2+A^{3}_1$ ↪

$A^{1}_5$
30 out of 37

Computations done by the calculator project.

Subalgebra type:

$\displaystyle A^{2}_2+A^{3}_1$ (click on type for detailed printout).
Subalgebra is (parabolically) induced from

$\displaystyle A^{2}_2$ .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer:

$\displaystyle A^{1}_5$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{2}_2$ : (1, 2, 2, 2, 1): 4, (0, 0, -1, -2, -1): 4,

$\displaystyle A^{3}_1$ : (1, 0, 1, 0, 1): 6
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators:

$\displaystyle g_{-13}+g_{-14}$ ,

$\displaystyle g_{9}+g_{8}$ ,

$\displaystyle g_{-1}+g_{-3}+g_{-5}$
Positive simple generators:

$\displaystyle g_{14}+g_{13}$ ,

$\displaystyle g_{-8}+g_{-9}$ ,

$\displaystyle g_{5}+g_{3}+g_{1}$
Cartan symmetric matrix:

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

$\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 6\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}$
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 3) ; the vectors are over the primal subalgebra.	$g_{7}+g_{6}$	$g_{5}+g_{3}+g_{1}$	$g_{10}$
weight	$\omega_{1}+\omega_{2}$	$2\omega_{3}$	$\omega_{1}+\omega_{2}+2\omega_{3}$

Isotypic module decomposition over primal subalgebra (total 3 isotypic components).

Isotypical components + highest weight

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

$\displaystyle V_{2\omega_{3}}$ → (0, 0, 2)

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

Module label

$W_{1}$

$W_{2}$

$W_{3}$

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

Semisimple subalgebra component.

$-g_{7}-g_{6}$

$-g_{-8}-g_{-9}$

$g_{14}+g_{13}$

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

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

$-g_{-13}-g_{-14}$

$2g_{9}+2g_{8}$

$-g_{-6}-g_{-7}$

Semisimple subalgebra component.

$-g_{5}-g_{3}-g_{1}$

$h_{5}+h_{3}+h_{1}$

$2g_{-1}+2g_{-3}+2g_{-5}$

$g_{10}$

$g_{-4}$

$-g_{15}$

$g_{7}-g_{6}$

$-g_{5}+g_{3}$

$-g_{-8}+g_{-9}$

$-g_{5}+g_{1}$

$-g_{14}+g_{13}$

$-2g_{2}$

$g_{-11}$

$-2g_{12}$

$h_{5}-h_{3}$

$-2g_{-12}$

$h_{5}-h_{1}$

$2g_{11}$

$g_{-2}$

$-g_{-13}+g_{-14}$

$-2g_{9}+2g_{8}$

$-2g_{-3}+2g_{-5}$

$-2g_{-1}+2g_{-5}$

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

$-2g_{-15}$

$4g_{4}$

$-2g_{-10}$

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

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

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

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

$0$

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

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

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

$2\omega_{3}$

$0$

$-2\omega_{3}$

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

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

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

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

$2\omega_{3}$

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

$2\omega_{3}$

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

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

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

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

$0$

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

$0$

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

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

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

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

$-2\omega_{3}$

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

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

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

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

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

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

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

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

$0$

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

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

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

$2\omega_{3}$

$0$

$-2\omega_{3}$

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

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

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

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

$2\omega_{3}$

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

$2\omega_{3}$

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

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

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

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

$0$

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

$0$

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

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

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

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

$-2\omega_{3}$

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

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

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

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

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

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

$\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}$

$\displaystyle M_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+2\omega_{3}} \oplus 2M_{2\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}} \oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}+2\omega_{3}}\oplus 2M_{0}\oplus M_{\omega_{1}+\omega_{2}-2\omega_{3}} \oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-2\omega_{3}} \oplus M_{-\omega_{1}-\omega_{2}}\oplus 2M_{-2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}-2\omega_{3}} \oplus M_{-\omega_{1}-\omega_{2}-2\omega_{3}}$

Isotypic character

$\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}$

Semisimple subalgebra: W_{1}+W_{2}
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, 420.00)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00)
(0.00, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (333.33, 486.67)
1: (0.00, 1.00, 0.00): (266.67, 553.33)
2: (0.00, 0.00, 1.00): (200.00, 420.00)

Made total 5038281 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.
3*2 (unknown) gens:
(
x_{1} g_{-13}+x_{2} g_{-14}, x_{9} g_{14}+x_{8} g_{13},
x_{3} g_{9}+x_{4} g_{8}, x_{11} g_{-8}+x_{10} g_{-9},
x_{5} g_{-1}+x_{6} g_{-3}+x_{7} g_{-5}, x_{14} g_{5}+x_{13} g_{3}+x_{12} g_{1})
h: (1, 2, 2, 2, 1), e = combination of g_{13} g_{14} , f= combination of g_{-13} g_{-14} h: (0, 0, -1, -2, -1), e = combination of g_{-9} g_{-8} , f= combination of g_{9} g_{8} h: (1, 0, 1, 0, 1), e = combination of g_{1} g_{3} g_{5} , f= combination of g_{-1} g_{-3} g_{-5} Positive weight subsystem: 4 vectors: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0)
Symmetric Cartan default scale: \begin{pmatrix}

2 &	-1 &	0\\
-1 &	2 &	0\\
0 &	0 &	2\\

\end{pmatrix}Character ambient Lie algebra: V_{\omega_{1}+\omega_{2}+2\omega_{3}}+V_{-\omega_{1}+2\omega_{2}+2\omega_{3}}+V_{2\omega_{1}-\omega_{2}+2\omega_{3}}+3V_{2\omega_{3}}+2V_{\omega_{1}+\omega_{2}}+V_{-2\omega_{1}+\omega_{2}+2\omega_{3}}+V_{\omega_{1}-2\omega_{2}+2\omega_{3}}+2V_{-\omega_{1}+2\omega_{2}}+2V_{2\omega_{1}-\omega_{2}}+V_{-\omega_{1}-\omega_{2}+2\omega_{3}}+5V_{0}+V_{\omega_{1}+\omega_{2}-2\omega_{3}}+2V_{-2\omega_{1}+\omega_{2}}+2V_{\omega_{1}-2\omega_{2}}+V_{-\omega_{1}+2\omega_{2}-2\omega_{3}}+V_{2\omega_{1}-\omega_{2}-2\omega_{3}}+2V_{-\omega_{1}-\omega_{2}}+3V_{-2\omega_{3}}+V_{-2\omega_{1}+\omega_{2}-2\omega_{3}}+V_{\omega_{1}-2\omega_{2}-2\omega_{3}}+V_{-\omega_{1}-\omega_{2}-2\omega_{3}}
A necessary system to realize the candidate subalgebra.
x_{1} x_{8} -1= 0
x_{2} x_{9} +x_{1} x_{8} -2= 0
x_{2} x_{9} -1= 0
x_{2} x_{14} -x_{1} x_{12} = 0
x_{8} x_{14} -x_{9} x_{12} = 0
x_{1} x_{7} -x_{2} x_{5} = 0
x_{4} x_{11} +x_{3} x_{10} -2= 0
x_{3} x_{10} -1= 0
x_{4} x_{11} -1= 0
x_{4} x_{14} -x_{3} x_{13} = 0
x_{10} x_{14} -x_{11} x_{13} = 0
x_{3} x_{7} -x_{4} x_{6} = 0
x_{5} x_{12} -1= 0
x_{6} x_{13} -1= 0
x_{7} x_{14} -1= 0
x_{7} x_{9} -x_{5} x_{8} = 0
x_{7} x_{11} -x_{6} x_{10} = 0
The above system after transformation.
x_{1} x_{8} -1= 0
x_{2} x_{9} +x_{1} x_{8} -2= 0
x_{2} x_{9} -1= 0
x_{2} x_{14} -x_{1} x_{12} = 0
x_{8} x_{14} -x_{9} x_{12} = 0
x_{1} x_{7} -x_{2} x_{5} = 0
x_{4} x_{11} +x_{3} x_{10} -2= 0
x_{3} x_{10} -1= 0
x_{4} x_{11} -1= 0
x_{4} x_{14} -x_{3} x_{13} = 0
x_{10} x_{14} -x_{11} x_{13} = 0
x_{3} x_{7} -x_{4} x_{6} = 0
x_{5} x_{12} -1= 0
x_{6} x_{13} -1= 0
x_{7} x_{14} -1= 0
x_{7} x_{9} -x_{5} x_{8} = 0
x_{7} x_{11} -x_{6} x_{10} = 0
For the calculator:
(DynkinType =A^{2}_2+A^{3}_1; ElementsCartan =((1, 2, 2, 2, 1), (0, 0, -1, -2, -1), (1, 0, 1, 0, 1)); generators =(x_{1} g_{-13}+x_{2} g_{-14}, x_{9} g_{14}+x_{8} g_{13}, x_{3} g_{9}+x_{4} g_{8}, x_{11} g_{-8}+x_{10} g_{-9}, x_{5} g_{-1}+x_{6} g_{-3}+x_{7} g_{-5}, x_{14} g_{5}+x_{13} g_{3}+x_{12} g_{1}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{1} x_{8} -1, x_{2} x_{9} +x_{1} x_{8} -2, x_{2} x_{9} -1, x_{2} x_{14} -x_{1} x_{12} , x_{8} x_{14} -x_{9} x_{12} , x_{1} x_{7} -x_{2} x_{5} , x_{4} x_{11} +x_{3} x_{10} -2, x_{3} x_{10} -1, x_{4} x_{11} -1, x_{4} x_{14} -x_{3} x_{13} , x_{10} x_{14} -x_{11} x_{13} , x_{3} x_{7} -x_{4} x_{6} , x_{5} x_{12} -1, x_{6} x_{13} -1, x_{7} x_{14} -1, x_{7} x_{9} -x_{5} x_{8} , x_{7} x_{11} -x_{6} x_{10} )