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Subalgebra

$A^{20}_1$ ↪

$A^{1}_4$
6 out of 15

Computations done by the calculator project.

Subalgebra type:

$\displaystyle A^{20}_1$ (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer:

$\displaystyle A^{1}_4$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{20}_1$ : (4, 6, 6, 4): 40
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators:

$\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}$
Positive simple generators:

$\displaystyle 4g_{4}+6g_{3}+6g_{2}+4g_{1}$
Cartan symmetric matrix:

$\displaystyle \begin{pmatrix}1/10\\ \end{pmatrix}$
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix):

$\displaystyle \begin{pmatrix}40\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}$
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 4) ; the vectors are over the primal subalgebra.	$g_{4}+3/2g_{3}+3/2g_{2}+g_{1}$	$g_{7}+3/2g_{6}+g_{5}$	$g_{9}+g_{8}$	$g_{10}$
weight	$2\omega_{1}$	$4\omega_{1}$	$6\omega_{1}$	$8\omega_{1}$

Isotypic module decomposition over primal subalgebra (total 4 isotypic components).

Isotypical components + highest weight

$\displaystyle V_{2\omega_{1}}$ → (2)

$\displaystyle V_{4\omega_{1}}$ → (4)

$\displaystyle V_{6\omega_{1}}$ → (6)

$\displaystyle V_{8\omega_{1}}$ → (8)

Module label

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

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

Semisimple subalgebra component.

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

$h_{4}+3/2h_{3}+3/2h_{2}+h_{1}$

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

$g_{7}+3/2g_{6}+g_{5}$

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

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

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

$g_{-5}+g_{-6}+g_{-7}$

$g_{9}+g_{8}$

$g_{7}-g_{5}$

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

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

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

$-5g_{-5}+5g_{-7}$

$-5g_{-8}-5g_{-9}$

$g_{10}$

$g_{9}-g_{8}$

$g_{7}-2g_{6}+g_{5}$

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

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

$5g_{-1}-10g_{-2}+10g_{-3}-5g_{-4}$

$15g_{-5}-20g_{-6}+15g_{-7}$

$35g_{-8}-35g_{-9}$

$70g_{-10}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

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

$2\omega_{1}$

$0$

$-2\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$8\omega_{1}$

$6\omega_{1}$

$4\omega_{1}$

$2\omega_{1}$

$0$

$-2\omega_{1}$

$-4\omega_{1}$

$-6\omega_{1}$

$-8\omega_{1}$

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

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

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

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

$\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}} \oplus M_{-8\omega_{1}}$

Isotypic character

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

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

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

Semisimple subalgebra: W_{1}
Centralizer extension: 0

Made total 21786 arithmetic operations while solving the Serre relations polynomial system.