On the capablility of Hom-Lie algebras

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.