Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
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| Format: | Preprint |
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2024
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| _version_ | 1866909335639031808 |
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| author | Bai, Chengming Liu, Guilai Sheng, Yunhe Tang, Rong |
| author_facet | Bai, Chengming Liu, Guilai Sheng, Yunhe Tang, Rong |
| contents | We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang-Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota-Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota-Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota-Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight $0$, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_03089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators Bai, Chengming Liu, Guilai Sheng, Yunhe Tang, Rong Quantum Algebra Mathematical Physics Rings and Algebras Representation Theory 17A32, 17A36, 17B10, 17B38, 17B40, 17B60, 17B62 We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang-Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota-Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota-Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota-Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight $0$, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights. |
| title | Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators |
| topic | Quantum Algebra Mathematical Physics Rings and Algebras Representation Theory 17A32, 17A36, 17B10, 17B38, 17B40, 17B60, 17B62 |
| url | https://arxiv.org/abs/2410.03089 |