On Cohomology group of current Lie algebras

Abstract

In this work we state a result that relates the cohomology groups of a Lie algebra $\mathfrak{g}$ and a current Lie algebra $\mathfrak{g} \otimes \mathcal{S}$, by means of a short exact sequence -- similar to the universal coefficients theorem for modules -- where $\mathcal{S}$ is a finite dimensional, commutative and associative algebra with unit over a field $\mathbb{F}$. Although this result can be applied to any Lie algebra, we determine the cohomology group of $\mathfrak{g} \otimes \mathcal{S}$, where $\mathfrak{g}$ is a semisimple Lie algebra.