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Skew-Hom-Lie algebras in Semi-Euclidean spaces

Zhen Xiong

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 $(R^4_2,[\cdot,\cdot]_θ,P)$ 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 $[\cdot,\cdot]_θ$ and $P$.

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 in semi-Euclidean spaces. For the null space of Semi-Euclidean spaces, there have a subset of the null space, and is invariant under the actions of and .

Paper Structure

This paper contains 4 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Skew-Hom-Lie algebras and their representations
  3. Coboundary operators of skew-Hom-Lie algebras
  4. Skew-Hom-Lie algebras in semi-Euclidean 4-spaces

Key Result

Proposition 2.5

Let $\alpha\in \mathfrak g\mathfrak l(V)$ and $\alpha^2=-\rm{id}$, we define a linear map by $Ad_\alpha(B)=\alpha B\alpha$, and a bilinear map (bracket) then $(\mathfrak g\mathfrak l(V),[\cdot,\cdot]_\alpha,Ad_\alpha)$ is a skew-Hom-Lie algebra.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 4 more