Deformations of modified $r$-matrices and cohomologies of related algebraic structures
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913817263341568 |
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| author | Jiang, Jun Sheng, Yunhe |
| author_facet | Jiang, Jun Sheng, Yunhe |
| contents | Modified $r$-matrices are solutions of the modified classical Yang-Baxter equation, introduced by Semenov-Tian-Shansky, and play important roles in mathematical physics. In this paper, first we introduce a cohomology theory for modified $r$-matrices. Then we study three kinds of deformations of modified $r$-matrices using the established cohomology theory, including algebraic deformations, geometric deformations and linear deformations. We give the differential graded Lie algebra that governs algebraic deformations of modified $r$-matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified $r$-matrix smooth in the space of all modified $r$-matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified $r$-matrix. Finally, applications are given to study deformations of complement of the diagonal Lie algebra and compatible Poisson structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_00411 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deformations of modified $r$-matrices and cohomologies of related algebraic structures Jiang, Jun Sheng, Yunhe Mathematical Physics Rings and Algebras Modified $r$-matrices are solutions of the modified classical Yang-Baxter equation, introduced by Semenov-Tian-Shansky, and play important roles in mathematical physics. In this paper, first we introduce a cohomology theory for modified $r$-matrices. Then we study three kinds of deformations of modified $r$-matrices using the established cohomology theory, including algebraic deformations, geometric deformations and linear deformations. We give the differential graded Lie algebra that governs algebraic deformations of modified $r$-matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified $r$-matrix smooth in the space of all modified $r$-matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified $r$-matrix. Finally, applications are given to study deformations of complement of the diagonal Lie algebra and compatible Poisson structures. |
| title | Deformations of modified $r$-matrices and cohomologies of related algebraic structures |
| topic | Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2206.00411 |