Mirror Hom-Lie algebras in semi-Euclidean spaces

Zhen Xiong; Zhen Xiong

doi:10.3934/math.2026651

AIMS Mathematics

2026, Volume 11, Issue 6: 15817-15830. doi: 10.3934/math.2026651

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Research article

Mirror Hom-Lie algebras in semi-Euclidean spaces

Zhen Xiong ^,

Department of Basic Education and Research, Jiangxi Police College, Nanchang 330100, China

Received: 26 December 2025 Revised: 12 February 2026 Accepted: 25 February 2026 Published: 04 June 2026
MSC : 17B99, 51P05

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, Ad_\alpha) $. Next, we study representations of mirror Hom-Lie algebras and that the pseudo-adjoint representation $ ad^{*}: \mathfrak g\longrightarrow \mathfrak g \mathfrak l(\mathfrak g) $, which is defined by $ ad^{*}_x(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, Ad_\beta) $. Then, we provide the coboundary operator of mirror Hom-Lie algebras. As an application, there exists a mirror Hom-Lie algebra $ (R^4_2, [\cdot, \cdot]_\theta, 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 $ [\cdot, \cdot]_\theta $ and $ P $.
- deformations of Lie algebras,
- Hom-Lie algebras,
- representations,
- semi-Euclidean spaces,
- null space
Citation: Zhen Xiong. Mirror Hom-Lie algebras in semi-Euclidean spaces[J]. AIMS Mathematics, 2026, 11(6): 15817-15830. doi: 10.3934/math.2026651

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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 $ (\mathfrak g \mathfrak l(V), [\cdot, \cdot]_\alpha, Ad_\alpha) $. Next, we study representations of mirror Hom-Lie algebras and that the pseudo-adjoint representation $ ad^{*}: \mathfrak g\longrightarrow \mathfrak g \mathfrak l(\mathfrak g) $, which is defined by $ ad^{*}_x(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, Ad_\beta) $. Then, we provide the coboundary operator of mirror Hom-Lie algebras. As an application, there exists a mirror Hom-Lie algebra $ (R^4_2, [\cdot, \cdot]_\theta, 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 $ [\cdot, \cdot]_\theta $ and $ P $.

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