Subalgebra \(A^{1}_1\) ↪ \(A^{1}_1\)
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Computations done by the calculator project.

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

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{1}_1\): (1): 2
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}\)
Positive simple generators: \(\displaystyle g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}2\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}2\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{2\omega_{1}}\)
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 1) ; the vectors are over the primal subalgebra.\(g_{1}\)
weight\(2\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 1 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)
Module label \(W_{1}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{1}\)
\(h_{1}\)
\(2g_{-1}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 272 arithmetic operations while solving the Serre relations polynomial system.