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arXiv · 2402.14934

Some Lie algebra structures on symmetric powers

Yin Chen
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Abstract

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 $\varphi$ on $V$ and an eigenvector $w$ of $\varphi$, we prove that the pair $(\varphi, w)$ 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 $\varphi$ is a nilpotent map. We also classify the Lie algebras for all possible pairs $(\varphi, w)$, when $k=\mathbb{C}$ and $V$ is two-dimensional.

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Yin Chen. 2024-02-22. Some Lie algebra structures on symmetric powers. https://doi.org/10.1080/00029890.2024.2416375

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