Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms

TL;DR

This paper investigates the structural properties of Lie algebras that admit symmetric, invariant, and nondegenerate bilinear forms, highlighting their rarity and specific instances where they occur.

Contribution

It characterizes Lie algebras with these bilinear forms and identifies the limited cases where such properties are satisfied, including certain abelian and free nilpotent Lie algebras.

Findings

Most nilradicals of parabolic subalgebras do not satisfy these properties.

Only a few specific nilpotent Lie algebras admit such forms.

Rare cases include abelian and certain free nilpotent Lie algebras.

Abstract

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, with only few exceptions. These rare cases are, essentially, abelian Lie algebras, the free 3-step nilpotent Lie algebra on 2-generators and the free 2-step nilpotent Lie algebra on 3-generators.