Cohomology and Extensions of Relative Rota-Baxter groups
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| Format: | Preprint |
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2023
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| _version_ | 1866909663456395264 |
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| author | Belwal, Pragya Rathee, Nishant Singh, Mahender |
| author_facet | Belwal, Pragya Rathee, Nishant Singh, Mahender |
| contents | Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang-Baxter equation. In this paper, we develop an extension theory of relative Rota-Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota-Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota-Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension two. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_00692 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cohomology and Extensions of Relative Rota-Baxter groups Belwal, Pragya Rathee, Nishant Singh, Mahender Quantum Algebra Group Theory Rings and Algebras 17B38, 16T25, 81R50 Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang-Baxter equation. In this paper, we develop an extension theory of relative Rota-Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota-Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota-Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension two. |
| title | Cohomology and Extensions of Relative Rota-Baxter groups |
| topic | Quantum Algebra Group Theory Rings and Algebras 17B38, 16T25, 81R50 |
| url | https://arxiv.org/abs/2309.00692 |