The Automorphisms group of a Current Lie algebra

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 $\mathrm{Der}({\mathfrak{g}}_A)$. Furthermore, we obtain the Levi decomposition of $\mathrm{Der}({\mathfrak{g}}_A)$. As a consequence of the last result, if ${\mathfrak{h}}_m$ is the Heisenberg Lie algebra of dimension $2m+1$, we obtain a faithful representation of $\mathrm{Der}({\mathfrak{h}}_{m,k})$ of the current truncated Heisenberg Lie algebra $\mathfrak{h}_{m,k}= \mathfrak{h}_m \otimes $\mathbb{C}$[t]/ (t^{k + 1})$forallpositiveinteger$k$.