The Automorphisms group of a Current Lie algebra

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 $k$.