Invariant metrics on current Lie algebras

In this work we state conditions for a current Lie algebra $\text{g}\otimes \mathcal{S}$ to admit an invariant metric, where $\text{g}$ is a quadratic Lie algebra and $\mathcal{S}$ is an associative and commutative algebra with unit. We also consider the reciprocal: if $\text{g}\otimes \mathcal{S}$ admits an invariant metric, we state necessary and sufficient conditions for $\text{g}$ to admit an invariant metric. In particular, we show that if $\text{g}$ is an indecomposable quadratic Lie algebra, then $\text{g}\otimes \mathcal{S}$ 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 $\text{g}\otimes \mathcal{S}$, where $\text{g}$ is an indecomposable, nilpotent and quadratic Lie algebra.