Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916527251390464 |
|---|---|
| author | Hou, Shuai Sheng, Yunhe |
| author_facet | Hou, Shuai Sheng, Yunhe |
| contents | In this paper, first we give the notion of a compatible $3$-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible $3$-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible $3$-Lie algebra. Then we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible $3$-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible $3$-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible $3$-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible $3$-Lie algebra using the second cohomology group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_12647 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras Hou, Shuai Sheng, Yunhe Rings and Algebras Mathematical Physics In this paper, first we give the notion of a compatible $3$-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible $3$-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible $3$-Lie algebra. Then we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible $3$-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible $3$-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible $3$-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible $3$-Lie algebra using the second cohomology group. |
| title | Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras |
| topic | Rings and Algebras Mathematical Physics |
| url | https://arxiv.org/abs/2208.12647 |