Lie theory and cohomology of relative Rota-Baxter operators
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| Format: | Preprint |
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2021
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| _version_ | 1866910378082959360 |
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| author | Jiang, Jun Sheng, Yunhe Zhu, Chenchang |
| author_facet | Jiang, Jun Sheng, Yunhe Zhu, Chenchang |
| contents | In this paper, we establish a local Lie theory for relative Rota-Baxter operators of weight $1$. First we recall the category of relative Rota-Baxter operators of weight $1$ on Lie algebras and construct a cohomology theory for them. We use the second cohomology group to study infinitesimal deformations of relative Rota-Baxter operators and modified $r$-matrices. Then we introduce a cohomology theory of relative Rota-Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota-Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving the Van Est theorem between the two cohomology theories. We integrate a relative Rota-Baxter operator of weight 1 on a Lie algebra to a local relative Rota-Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other. Finally, we give two applications of our integration of Rota-Baxter operators: one is to give an explicit formula for the factorization problem, and the other is to provide an integration for matched pairs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_02627 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lie theory and cohomology of relative Rota-Baxter operators Jiang, Jun Sheng, Yunhe Zhu, Chenchang Rings and Algebras Mathematical Physics Differential Geometry In this paper, we establish a local Lie theory for relative Rota-Baxter operators of weight $1$. First we recall the category of relative Rota-Baxter operators of weight $1$ on Lie algebras and construct a cohomology theory for them. We use the second cohomology group to study infinitesimal deformations of relative Rota-Baxter operators and modified $r$-matrices. Then we introduce a cohomology theory of relative Rota-Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota-Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving the Van Est theorem between the two cohomology theories. We integrate a relative Rota-Baxter operator of weight 1 on a Lie algebra to a local relative Rota-Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other. Finally, we give two applications of our integration of Rota-Baxter operators: one is to give an explicit formula for the factorization problem, and the other is to provide an integration for matched pairs. |
| title | Lie theory and cohomology of relative Rota-Baxter operators |
| topic | Rings and Algebras Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2108.02627 |