Skew-Hom-Lie algebras in Semi-Euclidean spaces

Abstract

In this paper, first we introduce the notion of a skew-Hom-Lie algebra and give some examples. Then we study their representations and give the coboundary operator of skew-Hom-Lie algebras. As an application, there have a skew-Hom-Lie algebra $(R_2^4,[\cdot ,\cdot {]}_{\theta },P)$ in semi-Euclidean spaces. For the null space of Semi-Euclidean spaces, there have a subset $V^{\ast }$ of the null space, and $V^{\ast }$ is invariant under the actions of $[\cdot ,\cdot {]}_{\theta }$ and $P$.