Schur multiplier and Schur covers of relative Rota-Baxter groups
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| Format: | Preprint |
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2023
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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 generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota-Baxter groups. We prove an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier $M_{RRB}(\mathcal{A})$ of a relative Rota-Baxter group $\mathcal{A} =(A,B,β,T)$, and prove that the exponent of $M_{RRB}(\mathcal{A})$ divides $|A||B|$ when $\mathcal{A}$ is finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12384 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Schur multiplier and Schur covers of relative Rota-Baxter groups Belwal, Pragya Rathee, Nishant Singh, Mahender Quantum Algebra Group Theory 17B38, 16T25, 81R50 Relative Rota-Baxter groups are generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota-Baxter groups. We prove an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier $M_{RRB}(\mathcal{A})$ of a relative Rota-Baxter group $\mathcal{A} =(A,B,β,T)$, and prove that the exponent of $M_{RRB}(\mathcal{A})$ divides $|A||B|$ when $\mathcal{A}$ is finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces. |
| title | Schur multiplier and Schur covers of relative Rota-Baxter groups |
| topic | Quantum Algebra Group Theory 17B38, 16T25, 81R50 |
| url | https://arxiv.org/abs/2311.12384 |