Skew-Hom-Lie algebras in Semi-Euclidean spaces

TL;DR

This paper introduces skew-Hom-Lie algebras, explores their representations and coboundary operators, and provides an example within semi-Euclidean spaces, highlighting invariance properties of null spaces.

Contribution

It defines skew-Hom-Lie algebras, studies their structure and representations, and constructs a specific example in semi-Euclidean spaces.

Findings

Defined skew-Hom-Lie algebras and their examples

Developed the coboundary operator for these algebras

Presented an invariant subset 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 $(R^4_2,[\cdot,\cdot]_\theta,P)$ in semi-Euclidean spaces. For the null space of Semi-Euclidean spaces, there have a subset $V^*$ of the null space, and $V^*$ is invariant under the actions of $[\cdot,\cdot]_\theta$ and $P$.