On the capablility of Hom-Lie algebras

TL;DR

This paper investigates the concept of capability in Hom-Lie algebras, providing characterizations and exploring their homological properties, including exact sequences and Hopf-type formulas for their second homology.

Contribution

It introduces a characterization of capable Hom-Lie algebras using their epicentre and applies this to homology theories and formulas.

Findings

Characterization of capable Hom-Lie algebras via epicentre

Analysis of the six-term exact sequence in homology

Derivation of Hopf-type formulas for second homology

Abstract

A Hom-Lie algebra $(L, \alpha_L)$ is said to be capable if there exists a Hom-Lie algebra $(H, \alpha_H)$ such that $L \cong H/Z(H)$. We obtain a characterisation of capable Hom-Lie algebras involving its epicentre and we use this theory to further study the six-term exact sequence in homology and to obtain a Hopf-type formulae of the second homology of perfect Hom-Lie algebras.