Invariant metrics on current Lie algebras

Abstract

In this work we state conditions for a current Lie algebra $\g \otimes \mathcal{S}$ to admit an invariant metric, where $\g$ is a quadratic Lie algebra and $\mathcal{S}$ is an associative and commutative algebra with unit. We also consider the reciprocal: if $\g \otimes \mathcal{S}$ admits an invariant metric, we state necessary and sufficient conditions for $\g$ to admit an invariant metric. In particular, we show that if $\g$ is an indecomposable quadratic Lie algebra, then $\g \otimes \mathcal{S}$ admits an invariant metric if and only if $\mathcal{S}$ also admits an invariant, symmetric and non-degenerate bilinear form. In addition, we prove a theorem similar to the double extension for $\g \otimes \mathcal{S}$, where $\g$ is an indecomposable, nilpotent and quadratic Lie algebra.