Invariant metrics on current Lie algebras

TL;DR

This paper investigates conditions under which current Lie algebras formed from a quadratic Lie algebra and an associative commutative algebra admit invariant metrics, providing necessary and sufficient conditions and extending classical theorems.

Contribution

It establishes criteria for invariant metrics on current Lie algebras and extends the double extension theorem to certain nilpotent cases.

Findings

Invariant metric existence depends on the algebra's properties

Characterization for indecomposable quadratic Lie algebras

Extension of double extension theorem to nilpotent cases

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.