Generalized representations, deformations and extensions of 3-Lie superalgebras

Junxia Zhu; Rongsheng Ma; Junxia Zhu; Rongsheng Ma

doi:10.3934/math.2025994

AIMS Mathematics

2025, Volume 10, Issue 9: 22314-22335. doi: 10.3934/math.2025994

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Research article

Generalized representations, deformations and extensions of 3-Lie superalgebras

Junxia Zhu ,
Rongsheng Ma ^,

School of Science, Yanshan University, Qinhuangdao 066004, China

Received: 09 June 2025 Revised: 09 September 2025 Accepted: 11 September 2025 Published: 26 September 2025
MSC : 17B10, 17B56, 17A42

In this paper, we introduced generalized representations of 3-Lie superalgebras, which are associated to generalized semidirect product 3-Lie superalgebras. We also developed the corresponding cohomology theory and generalized one-parameter formal deformations. Furthermore, we proved that the infinitesimals and the extensibility of finite-order deformations of generalized one-parameter formal deformations are controlled by the new first and second cohomology groups, respectively. At last, we described split and non-split Abelian extensions by generalized semidirect products and Maurer-Cartan elements, respectively.
- 3-Lie superalgebra,
- generalized representation,
- cohomology,
- deformation,
- Abelian extension,
- Maurer-Cartan element
Citation: Junxia Zhu, Rongsheng Ma. Generalized representations, deformations and extensions of 3-Lie superalgebras[J]. AIMS Mathematics, 2025, 10(9): 22314-22335. doi: 10.3934/math.2025994

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Abstract

In this paper, we introduced generalized representations of 3-Lie superalgebras, which are associated to generalized semidirect product 3-Lie superalgebras. We also developed the corresponding cohomology theory and generalized one-parameter formal deformations. Furthermore, we proved that the infinitesimals and the extensibility of finite-order deformations of generalized one-parameter formal deformations are controlled by the new first and second cohomology groups, respectively. At last, we described split and non-split Abelian extensions by generalized semidirect products and Maurer-Cartan elements, respectively.

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