Invariant Metrics on Nilpotent Lie algebras

TL;DR

This paper establishes criteria for nilpotent Lie algebras to admit invariant metrics, explores their structure via canonical ideals, and connects invariant bilinear forms to algebra reconstruction and simplicity.

Contribution

It introduces new criteria for invariant metrics on nilpotent Lie algebras and links invariant bilinear forms to algebraic properties and reconstruction.

Findings

Existence of invariant metrics characterized by ideal properties.

Invariant bilinear forms can reconstruct the algebra.

Invariant bilinear form's simplicity relates to algebra's simplicity.

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 $\g$.