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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.08213 |
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| _version_ | 1866909343607160832 |
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| author | Abdesselam, Abdelmalek Thomas, Alexander |
| author_facet | Abdesselam, Abdelmalek Thomas, Alexander |
| contents | For a simple complex Lie algebra $\mathfrak{g}$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak{g}$ as vector space. For $\mathfrak{sl}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08213 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structure constants for simple Lie algebras from principal $\mathfrak{sl}_2$-triple Abdesselam, Abdelmalek Thomas, Alexander Representation Theory Mathematical Physics Geometric Topology 17B05, 13A50 For a simple complex Lie algebra $\mathfrak{g}$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak{g}$ as vector space. For $\mathfrak{sl}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed. |
| title | Structure constants for simple Lie algebras from principal $\mathfrak{sl}_2$-triple |
| topic | Representation Theory Mathematical Physics Geometric Topology 17B05, 13A50 |
| url | https://arxiv.org/abs/2309.08213 |