Subalgebra \(A^{1}_2\) ↪ \(G^{1}_2\)
6 out of 7

Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{1}_2\) (click on type for detailed printout).
The subalgebra is regular (= the semisimple part of a root subalgebra).
Subalgebra is (parabolically) induced from \(\displaystyle A^{1}_1\) .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle G^{1}_2\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_2\): (3, 2): 2, (0, -1): 2
Dimension of subalgebra generated by predefined or computed generators: 8.
Negative simple generators: \(\displaystyle g_{-6}\), \(\displaystyle g_{2}\)
Positive simple generators: \(\displaystyle g_{6}\), \(\displaystyle g_{-2}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2 & -1\\ -1 & 2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2 & -1\\ -1 & 2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{\omega_{1}+\omega_{2}}\oplus V_{\omega_{2}}\oplus V_{\omega_{1}}\)
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 3) ; the vectors are over the primal subalgebra.	\(g_{4}\)	\(g_{1}\)	\(g_{5}\)
weight	\(\omega_{1}\)	\(\omega_{2}\)	\(\omega_{1}+\omega_{2}\)

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

Isotypical components + highest weight

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

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

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

Module label

\(W_{1}\)

\(W_{2}\)

\(W_{3}\)

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

\(g_{4}\)

\(g_{-3}\)

\(g_{-1}\)

\(g_{1}\)

\(-g_{3}\)

\(g_{-4}\)

Semisimple subalgebra component.

\(-g_{5}\)

\(g_{-2}\)

\(-g_{6}\)

\(h_{2}\)

\(2h_{2}+3h_{1}\)

\(g_{-6}\)

\(-2g_{2}\)

\(-g_{-5}\)

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

\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)

\(\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)

\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)

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

\(\omega_{1}\)
\(-\omega_{1}+\omega_{2}\)
\(-\omega_{2}\)

\(\omega_{2}\)
\(\omega_{1}-\omega_{2}\)
\(-\omega_{1}\)

\(\omega_{1}+\omega_{2}\)
\(-\omega_{1}+2\omega_{2}\)
\(2\omega_{1}-\omega_{2}\)
\(0\)
\(0\)
\(-2\omega_{1}+\omega_{2}\)
\(\omega_{1}-2\omega_{2}\)
\(-\omega_{1}-\omega_{2}\)

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

\(\displaystyle M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\)

\(\displaystyle M_{\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}\)

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

Isotypic character

\(\displaystyle M_{\omega_{1}}\oplus M_{-\omega_{1}+\omega_{2}}\oplus M_{-\omega_{2}}\)

\(\displaystyle M_{\omega_{2}}\oplus M_{\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}}\)

Semisimple subalgebra: W_{3}
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.

Made total 357 arithmetic operations while solving the Serre relations polynomial system.