Post-groupoids and quiver-theoretical solutions of the Yang-Baxter equation
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| Format: | Preprint |
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2024
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| _version_ | 1866909338631667712 |
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| author | Sheng, Yunhe Tang, Rong Zhu, Chenchang |
| author_facet | Sheng, Yunhe Tang, Rong Zhu, Chenchang |
| contents | The notion of post-groups was introduced by Bai, Guo and the first two authors recently, which are the global objects corresponding to post-Lie algebras, equivalent to skew-left braces, and can be used to construct set-theoretical solutions of the Yang-Baxter equation. In this paper, first we introduce the notion of post-groupoids, which consists of a group bundle and some other structures satisfying some compatibility conditions. Post-groupoids reduce to post-groups if the underlying base is one point. An action of a group on a set gives rise to the natural example of post-groupoids. We show that a post-groupoid gives rise to a groupoid (called the Grossman-Larson groupoid), and an action on the original group bundle. Then we introduce the notion of relative Rota-Baxter operators on a groupoid with respect to an action on a group bundle. A relative Rota-Baxter operator naturally gives rise to a post-groupoid and a matched pair of groupoids. One important application of post-groupoids is that they give rise to quiver-theoretical solutions of the Yang-Baxter equation on the quiver underlying the Grossman-Larson groupoid. We also introduce the notion of a skew-left bracoid, which consists of a group bundle and a groupoid satisfying some compatibility conditions. A skew-left bracoid reduces to a skew-left brace if the underlying base is one point. We give the one-to-one correspondence between post-groupoids and skew-left bracoids. Finally, we show that post-Lie groupoids give rise to post-Lie algebroids via differentiation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_04973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Post-groupoids and quiver-theoretical solutions of the Yang-Baxter equation Sheng, Yunhe Tang, Rong Zhu, Chenchang Mathematical Physics Differential Geometry Rings and Algebras The notion of post-groups was introduced by Bai, Guo and the first two authors recently, which are the global objects corresponding to post-Lie algebras, equivalent to skew-left braces, and can be used to construct set-theoretical solutions of the Yang-Baxter equation. In this paper, first we introduce the notion of post-groupoids, which consists of a group bundle and some other structures satisfying some compatibility conditions. Post-groupoids reduce to post-groups if the underlying base is one point. An action of a group on a set gives rise to the natural example of post-groupoids. We show that a post-groupoid gives rise to a groupoid (called the Grossman-Larson groupoid), and an action on the original group bundle. Then we introduce the notion of relative Rota-Baxter operators on a groupoid with respect to an action on a group bundle. A relative Rota-Baxter operator naturally gives rise to a post-groupoid and a matched pair of groupoids. One important application of post-groupoids is that they give rise to quiver-theoretical solutions of the Yang-Baxter equation on the quiver underlying the Grossman-Larson groupoid. We also introduce the notion of a skew-left bracoid, which consists of a group bundle and a groupoid satisfying some compatibility conditions. A skew-left bracoid reduces to a skew-left brace if the underlying base is one point. We give the one-to-one correspondence between post-groupoids and skew-left bracoids. Finally, we show that post-Lie groupoids give rise to post-Lie algebroids via differentiation. |
| title | Post-groupoids and quiver-theoretical solutions of the Yang-Baxter equation |
| topic | Mathematical Physics Differential Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2410.04973 |