Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911151876472832 |
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| author | Jiang, Jun |
| author_facet | Jiang, Jun |
| contents | In this paper, we construct a categorical solution $(\huaC, R)$ of the Yang-Baxter equation, i.e. $\huaC$ is a small category and $R: \huaC\times\huaC\lon\huaC\times\huaC$ is an invertible functor satisfying $$ (R\times\Id_\huaC)(\Id_\huaC\times R)(R\times\Id_\huaC)=(\Id_\huaC\times R)(R\times\Id_\huaC)(\Id_\huaC\times R),
$$ where $\huaC\times\huaC$ is the product category. First, the notion of Rota-Baxter operators on crossed modules of Lie groups is defined and its various properties are established. Then, we use Rota-Baxter operators on crossed modules of Lie groups to construct categorical solutions of the Yang-Baxter equation. We also study the Rota-Baxter operators on crossed modules of Lie algebras which are infinitesimals of Rota-Baxter operators on crossed modules of Lie groups, they can give connections on manifolds. Finally, we study the integration of Rota-Baxter operators on crossed modules of Lie algebras and the differentials of Rota-Baxter operators on crossed modules of Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation Jiang, Jun Mathematical Physics Group Theory Rings and Algebras In this paper, we construct a categorical solution $(\huaC, R)$ of the Yang-Baxter equation, i.e. $\huaC$ is a small category and $R: \huaC\times\huaC\lon\huaC\times\huaC$ is an invertible functor satisfying $$ (R\times\Id_\huaC)(\Id_\huaC\times R)(R\times\Id_\huaC)=(\Id_\huaC\times R)(R\times\Id_\huaC)(\Id_\huaC\times R), $$ where $\huaC\times\huaC$ is the product category. First, the notion of Rota-Baxter operators on crossed modules of Lie groups is defined and its various properties are established. Then, we use Rota-Baxter operators on crossed modules of Lie groups to construct categorical solutions of the Yang-Baxter equation. We also study the Rota-Baxter operators on crossed modules of Lie algebras which are infinitesimals of Rota-Baxter operators on crossed modules of Lie groups, they can give connections on manifolds. Finally, we study the integration of Rota-Baxter operators on crossed modules of Lie algebras and the differentials of Rota-Baxter operators on crossed modules of Lie groups. |
| title | Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation |
| topic | Mathematical Physics Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2411.16040 |