@article{Xiong2026, 
author = {Zhen Xiong},
title = {Mirror Hom-Lie algebras in semi-Euclidean spaces},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {15817-15830},
keywords = {deformations of Lie algebras, Hom-Lie algebras, representations, semi-Euclidean spaces, null space},
url = {https://www.sciopen.com/article/10.3934/math.2026651},
doi = {10.3934/math.2026651},
abstract = {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    (      g        l    (  V  )  ,  [  ⋅  ,  ⋅      ]    α    ,  A      d    α    ). Next, we study representations of mirror Hom-Lie algebras and that the pseudo-adjoint representation    a      d          ∗        :      g    ⟶      g        l    (      g    ), which is defined by    a      d    x          ∗        (  y  )  =  −  [  x  ,  y  ], is a morphism from the mirror Hom-Lie algebra    (      g    ,  [  ⋅  ,  ⋅  ]  ,  β  ) to the mirror Hom-Lie algebra    (      g        l    (      g    )  ,  [  ⋅  ,  ⋅      ]    β    ,  A      d    β    ). Then, we provide the coboundary operator of mirror Hom-Lie algebras. As an application, there exists a mirror Hom-Lie algebra    (      R    2    4    ,  [  ⋅  ,  ⋅      ]    θ    ,  P  ) in semi-Euclidean spaces. For the null space of semi-Euclidean spaces, there is a subset        V    ∗   of the null space, and        V    ∗   is invariant under the actions of    [  ⋅  ,  ⋅      ]    θ   and    P.}
}