Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929269851029504 |
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| author | Baishya, Anusuiya Das, Apurba |
| author_facet | Baishya, Anusuiya Das, Apurba |
| contents | The notion of a matched pair of Lie algebras was introduced in the study of Lie bialgebras and Poisson-Lie groups. In this paper, we introduce representations and cohomology of a matched pair of Lie algebras. We show that there is a morphism from the cohomology of a Lie bialgebra to the cohomology of the corresponding matched pair of Lie algebras. Our cohomology is also useful to study infinitesimal deformations and abelian extensions. In the last part, we define the notion of a matched pair of $L_\infty$-algebras and construct the corresponding bicrossed product. Finally, we show that a matched pair of skeletal $L_\infty$-algebras is closely related to the cohomology introduced in the paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras Baishya, Anusuiya Das, Apurba Representation Theory K-Theory and Homology Rings and Algebras 17B10, 17B56, 17B62, 17B70, 18N40 The notion of a matched pair of Lie algebras was introduced in the study of Lie bialgebras and Poisson-Lie groups. In this paper, we introduce representations and cohomology of a matched pair of Lie algebras. We show that there is a morphism from the cohomology of a Lie bialgebra to the cohomology of the corresponding matched pair of Lie algebras. Our cohomology is also useful to study infinitesimal deformations and abelian extensions. In the last part, we define the notion of a matched pair of $L_\infty$-algebras and construct the corresponding bicrossed product. Finally, we show that a matched pair of skeletal $L_\infty$-algebras is closely related to the cohomology introduced in the paper. |
| title | Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras |
| topic | Representation Theory K-Theory and Homology Rings and Algebras 17B10, 17B56, 17B62, 17B70, 18N40 |
| url | https://arxiv.org/abs/2402.16495 |