Some Lie algebra structures on symmetric powers
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917902845739008 |
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| author | Chen, Yin |
| author_facet | Chen, Yin |
| contents | Let $k$ be a field of any characteristic, $V$ a finite-dimensional vector space over $k$, and $S^d(V^*)$ be the $d$-th symmetric power of the dual space $V^*$. Given a linear map $φ$ on $V$ and an eigenvector $w$ of $φ$, we prove that the pair $(φ, w)$ can be used to construct a new Lie algebra structure on $S^d(V^*)$. We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if $φ$ is a nilpotent map. We also classify the Lie algebras for all possible pairs $(φ, w)$, when $k=\mathbb{C}$ and $V$ is two-dimensional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14934 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Lie algebra structures on symmetric powers Chen, Yin Rings and Algebras Mathematical Physics 17B05, 17B30 Let $k$ be a field of any characteristic, $V$ a finite-dimensional vector space over $k$, and $S^d(V^*)$ be the $d$-th symmetric power of the dual space $V^*$. Given a linear map $φ$ on $V$ and an eigenvector $w$ of $φ$, we prove that the pair $(φ, w)$ can be used to construct a new Lie algebra structure on $S^d(V^*)$. We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if $φ$ is a nilpotent map. We also classify the Lie algebras for all possible pairs $(φ, w)$, when $k=\mathbb{C}$ and $V$ is two-dimensional. |
| title | Some Lie algebra structures on symmetric powers |
| topic | Rings and Algebras Mathematical Physics 17B05, 17B30 |
| url | https://arxiv.org/abs/2402.14934 |