The Automorphisms group of a Current Lie algebra

TL;DR

This paper characterizes the derivation Lie algebra of current Lie algebras formed from a finite dimensional complex Lie algebra and a commutative algebra, including their Levi decomposition and applications to truncated Heisenberg algebras.

Contribution

It provides a detailed description of the derivation algebra structure and Levi decomposition for current Lie algebras, extending to representations of truncated Heisenberg algebras.

Findings

Derived the structure of derivation Lie algebras for current Lie algebras.

Obtained the Levi decomposition of these derivation algebras.

Constructed faithful representations for derivations of truncated Heisenberg Lie algebras.

Abstract

Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra and let $A$ be a finite dimensional complex, associative and commutative algebra with unit. We describe the structure of the derivation Lie algebra of the current Lie algebra $\mathfrak{g}_A= \mathfrak{g} \otimes A$, denoted by $\operatorname{Der}(\mathfrak{g}_A)$. Furthermore, we obtain the Levi decomposition of $\operatorname{Der}(\mathfrak{g}_A)$. As a consequence of the last result, if $\mathfrak{h}_m$ is the Heisenberg Lie algebra of dimension $2 m + 1$, we obtain a faithful representation of $\operatorname{Der}(\mathfrak{h}_{m,k})$ of the current truncated Heisenberg Lie algebra $\mathfrak{h}_{m,k}= \mathfrak{h}_m \otimes \mathbb{C}[t]/ (t^{k + 1})$ for all positive integer $k$.