Hom-Lie algebra structures on quadratic Lie algebras and twisted invariant Killing-like forms defined on them

TL;DR

This paper explores Hom-Lie algebra structures on quadratic Lie algebras with invariant metrics, introducing twisted invariant forms that resemble classical Cartan-Killing forms, thus extending key Lie algebra properties to the Hom-Lie setting.

Contribution

It demonstrates how to define and analyze twisted invariant bilinear forms on Hom-Lie algebras derived from quadratic Lie algebras, extending classical Lie algebra invariants.

Findings

Hom-Lie algebras can be equipped with twisted invariant forms similar to Cartan-Killing forms.

These forms satisfy invariance conditions analogous to those in classical Lie theory.

Results analogous to classical Lie algebra properties are recoverable in the Hom-Lie context.

Abstract

Hom-Lie algebras defined on central extensions of a given quadratic Lie algebra that in turn admit an invariant metric, are studied. It is shown how some of these algebras are naturally equipped with other symmetric, bilinear forms that satisfy an invariant condition for their twisted multiplication maps. The twisted invariant bilinear forms so obtained resemble the Cartan-Killing forms defined on ordinary Lie algebras. This fact allows one to reproduce on the Hom-Lie algebras hereby studied, some results that are classically associated to the ordinary Cartan-Killing form.