Hom-Lie structures on the algebra $\mathfrak{L}_{\lambda,\mu}$

Abstract

For parameters $\lambda,\mu\in\mathbb{C}$, the algebra $\mathfrak{L}_{\lambda,\mu}$ is the semi-direct product of the Witt algebra and its tensor density module. In this paper, we determine all (multiplicative) Hom-Lie structures on $\mathfrak{L}_{\lambda,\mu}$. As a result, we prove that any Hom-Lie structure on $\mathfrak{L}_{\lambda,\mu}$ is the direct sum of some special Hom-Lie structures, and there exist non-trivial multiplicative Hom-Lie structures on $\mathfrak{L}_{\lambda,\mu}$ if and only if $\lambda=0$ or $1$.