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 $\mathrm{Der}({\mathfrak{g}}_A)$. 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 ${\mathfrak{h}}_{m,k}={\mathfrak{h}}_m\otimes \mathbb{C}[t]/(t^{k+1})$ for all positive integer $k$.

14 pages

References in corpus (1)

• Six-dimensional nilpotent Lie algebras

Cited by in corpus (1)

• A new invariant for finite dimensional Leibniz/Lie algebras