On cohomology groups of current Lie algebras

Abstract

In this work we state a result that relates the cohomology groups of a Lie algebra $\frak g$ and a current Lie algebra $\frak g \otimes \mathcal{S}$, 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 $\Bbb F$. Using this result we determine the cohomology group of $\frak g \otimes \mathcal{S}$ where $\frak g$ is a semisimple Lie algebra.

Keywords

• Lie algebra cohomology
• current Lie algebra
• tensor product
• associative and commutative algebra
• semisimple Lie algebra

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