Some properties of relative Rota--Baxter operators on groups
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910415118663680 |
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| author | Bardakov, V. G. Kozlovskaya, T. A. Sokolov, P. P. Zimireva, K. V. Zonov, M. N. |
| author_facet | Bardakov, V. G. Kozlovskaya, T. A. Sokolov, P. P. Zimireva, K. V. Zonov, M. N. |
| contents | We find connection between relative Rota--Baxter operators and usual Rota--Baxter operators. We prove that any relative Rota--Baxter operator on a group $H$ with respect to $(G, Ψ)$ defines a Rota--Baxter operator on the semi-direct product $H\rtimes_Ψ G$. On the other side, we give condition under which a Rota--Baxter operator on the semi-direct product $H\rtimes_Ψ G$ defines a relative Rota--Baxter operator on $H$ with respect to $(G, Ψ)$. We introduce homomorphic post-groups and find their connection with $λ$-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_12632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some properties of relative Rota--Baxter operators on groups Bardakov, V. G. Kozlovskaya, T. A. Sokolov, P. P. Zimireva, K. V. Zonov, M. N. Group Theory 17B38, 16T25 We find connection between relative Rota--Baxter operators and usual Rota--Baxter operators. We prove that any relative Rota--Baxter operator on a group $H$ with respect to $(G, Ψ)$ defines a Rota--Baxter operator on the semi-direct product $H\rtimes_Ψ G$. On the other side, we give condition under which a Rota--Baxter operator on the semi-direct product $H\rtimes_Ψ G$ defines a relative Rota--Baxter operator on $H$ with respect to $(G, Ψ)$. We introduce homomorphic post-groups and find their connection with $λ$-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group. |
| title | Some properties of relative Rota--Baxter operators on groups |
| topic | Group Theory 17B38, 16T25 |
| url | https://arxiv.org/abs/2404.12632 |