Invariant Metrics on Nilpotent Lie algebras

Abstract

We state criteria for a nilpotent Lie algebra $\g$ to admit an invariant metric. We use that $\g$ possesses two canonical abelian ideals $\ide(\g) \subset \mathfrak{J}(\g)$ to decompose the underlying vector space of $\g$ and then we state sufficient conditions for $\g$ to admit an invariant metric. The properties of the ideal $\mathfrak{J}(\g)$ allows to prove that if a current Lie algebra $\g \otimes \Sa$ admits an invariant metric, then there must be an invariant and non-degenerate bilinear map from $\Sa \times \Sa$ into the space of centroids of $\g/\mathfrak{J}(\g)$. We also prove that in any nilpotent Lie algebra $\g$ there exists a non-zero, symmetric and invariant bilinear form. This bilinear form allows to reconstruct $\g$ by means of an algebra with unit. We prove that this algebra is simple if and only if the bilinear form is an invariant metric on $\g$.