Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid

TL;DR

This paper introduces an inductive method to construct quadratic Hom-Lie algebras with nilpotent twist maps in the centroid, expanding the understanding of non-Lie Hom-Lie algebra structures.

Contribution

It develops a double extension procedure for constructing non-Lie quadratic Hom-Lie algebras with nilpotent twist maps in the centroid.

Findings

Hom-Lie algebras of this type have trivial center.

Their twist maps are nilpotent.

Any indecomposable quadratic Hom-Lie algebra with these properties can be constructed via the proposed method.

Abstract

In this work we give an inductive way to construct quadratic Hom-Lie algebras with twist maps in the centroid. We focus on those Hom-Lie algebras which are not Lie algebras. We prove that a Hom-Lie algebra of this type has trivial center and its twist map is nilpotent. We show that there exists a maximal ideal containing the kernel and the image of the twist map. Then we state an inductive way to construct this type of Hom-Lie algebras, similar to the double extension procedure for Lie algebras, and prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist map in the centroid, which is not a Lie algebra, can be constructed using this type of double extension.