Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid
R. García-Delgado
TL;DR
The paper addresses the problem of constructing and classifying indecomposable quadratic Hom-Lie algebras with twist maps in the centroid that are not Lie algebras. It develops an inductive double-extension framework, using a Fitting decomposition to show that the twist $T$ is nilpotent and identifying a maximal ideal $\mathcal I$ with $\ker T+\operatorname{Im} T\subset \mathcal I$, such that the quotient $\mathfrak g/\mathcal I$ is a simple Lie algebra. It then proves that every indecomposable quadratic Hom-Lie algebra not satisfying the Jacobi identity arises from data (A)–(G) via a generalized double-extension on $\mathfrak g=\mathfrak s\oplus\mathfrak h\oplus\mathfrak s^{\ast}$, where $\mathfrak s$ is simple and $\mathfrak h$ carries a quadratic Hom-Lie structure. An explicit example with $\mathfrak s=\mathfrak{sl}_3$ demonstrates the construction and shows that $\mathfrak s$ need not be a subalgebra of the ambient Hom-Lie algebra, highlighting the novelty of the approach compared to classical Lie theory.
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.