On quadratic Hom-Lie algebras with twist maps in their centroids and their relationship with quadratic Lie algebras

TL;DR

This paper explores quadratic Hom-Lie algebras with non-invertible twist maps, their central extensions, and their relationship to quadratic Lie algebras, providing conditions for invariant metrics and examples of such structures.

Contribution

It introduces new conditions for quadratic Hom-Lie algebras with non-invertible twist maps and analyzes their central extensions and connections to quadratic Lie algebras.

Findings

Central extensions preserve key properties of Hom-Lie algebras.

Conditions for invariant metrics in central extensions are established.

Examples of non-trivial quadratic Hom-Lie algebras are provided.

Abstract

Hom-Lie algebras having non-invertible twist maps in their centroids are studied. Central extensions of Hom-Lie algebras having these properties are obtained and shown how the same properties are preserved. Conditions are given so that the produced central extension has an invariant metric with respect to its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie algebra has such a metric. This work is focused on algebras with these properties and following Benayadi and Makhloufwe call them quadratic Hom-Lie algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the Lie algebra associated to the given Hom-Lie central extension is a Lie algebra central extension of it. It is also shown that if the Hom-Lie product is not a Lie product, there exists a non-abelian algebra, which is in general non-associative too, the commutator of whose product is precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose commutator realizes this Hom-Lie product is shown to be simple if the associated Lie algebra is nilpotent. Non-trivial examples are provided.