Mirror Hom-Lie algebras in semi-Euclidean spaces

In this paper, first we introduce the notion of a mirror Hom-Lie algebra and give some examples. Then, we endow the space of general linear mappings with a mirror Hom-Lie algebraic structure $(\mathfrak{g}\mathfrak{l}(V),[\cdot ,\cdot {]}_{\alpha },A{d}_{\alpha })$. Next, we study representations of mirror Hom-Lie algebras and that the pseudo-adjoint representation $a{d}^{\ast }:\mathfrak{g}⟶\mathfrak{g}\mathfrak{l}(\mathfrak{g})$, which is defined by $a{d}_x^{\ast }(y)=-[x,y]$, is a morphism from the mirror Hom-Lie algebra $(\mathfrak{g},[\cdot ,\cdot ],\beta )$ to the mirror Hom-Lie algebra $(\mathfrak{g}\mathfrak{l}(\mathfrak{g}),[\cdot ,\cdot {]}_{\beta },A{d}_{\beta })$. Then, we provide the coboundary operator of mirror Hom-Lie algebras. As an application, there exists a mirror Hom-Lie algebra $(R_2^4,[\cdot ,\cdot {]}_{\theta },P)$ in semi-Euclidean spaces. For the null space of semi-Euclidean spaces, there is a subset $V^{\ast }$ of the null space, and $V^{\ast }$ is invariant under the actions of $[\cdot ,\cdot {]}_{\theta }$ and $P$.