The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras

Shanshan Liu; Abdenacer Makhlouf; Lina Song; Shanshan Liu; Abdenacer Makhlouf; Lina Song

doi:10.3934/era.2022141

Electronic Research Archive

2022, Volume 30, Issue 8: 2748-2773. doi: 10.3934/era.2022141

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Research article

The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras

1.
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China
2.
University of Haute Alsace, IRIMAS-Mathematics department, Mulhouse, France
3.
Department of Mathematics, Jilin University, Changchun 130012, Jilin, China

Received: 18 September 2021 Revised: 10 March 2022 Accepted: 12 March 2022 Published: 23 May 2022

The main purpose of this paper is to provide a full cohomology of a Hom-pre-Lie algebra with coefficients in a given representation. This new type of cohomology exploits strongly the Hom-type structure and fits perfectly with simultaneous deformations of the multiplication and the homomorphism defining a Hom-pre-Lie algebra. Moreover, we show that its second cohomology group classifies abelian extensions of a Hom-pre-Lie algebra by a representation.
- Hom-pre-Lie algebra,
- cohomology,
- abelian extension,
- formal deformation
Citation: Shanshan Liu, Abdenacer Makhlouf, Lina Song. The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras[J]. Electronic Research Archive, 2022, 30(8): 2748-2773. doi: 10.3934/era.2022141

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Abstract

The main purpose of this paper is to provide a full cohomology of a Hom-pre-Lie algebra with coefficients in a given representation. This new type of cohomology exploits strongly the Hom-type structure and fits perfectly with simultaneous deformations of the multiplication and the homomorphism defining a Hom-pre-Lie algebra. Moreover, we show that its second cohomology group classifies abelian extensions of a Hom-pre-Lie algebra by a representation.

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