Leibniz-dendriform bialgebras and relative Rota-Baxter operators
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| Format: | Preprint |
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2025
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| _version_ | 1866909894773309440 |
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| author | Sun, Qinxiu Guo, Shuangjian |
| author_facet | Sun, Qinxiu Guo, Shuangjian |
| contents | In this paper, we introduce the notion of Leibniz-dendriform bialgebras and establish their equivalence with phase spaces and matched pairs of Leibniz algebras. The study of the coboundary case leads naturally to the Leibniz-dendriform Yang-Baxter equation (LD-YBE). We prove that skew-symmetric solutions of the LD-YBE give rise to coboundary Leibniz-dendriform bialgebras.
Furthermore, we demonstrate that solutions not necessarily skew-symmetric can also induce such bialgebras. This observation motivates the introduction of quasi-triangular and factorizable Leibniz-dendriform bialgebras. In particular, we show that solutions of the LD-YBE with invariant symmetric parts yield quasi-triangular Leibniz-dendriform bialgebras. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras
and factorizable Leibniz-dendriform bialgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16826 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Leibniz-dendriform bialgebras and relative Rota-Baxter operators Sun, Qinxiu Guo, Shuangjian Rings and Algebras In this paper, we introduce the notion of Leibniz-dendriform bialgebras and establish their equivalence with phase spaces and matched pairs of Leibniz algebras. The study of the coboundary case leads naturally to the Leibniz-dendriform Yang-Baxter equation (LD-YBE). We prove that skew-symmetric solutions of the LD-YBE give rise to coboundary Leibniz-dendriform bialgebras. Furthermore, we demonstrate that solutions not necessarily skew-symmetric can also induce such bialgebras. This observation motivates the introduction of quasi-triangular and factorizable Leibniz-dendriform bialgebras. In particular, we show that solutions of the LD-YBE with invariant symmetric parts yield quasi-triangular Leibniz-dendriform bialgebras. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras and factorizable Leibniz-dendriform bialgebras. |
| title | Leibniz-dendriform bialgebras and relative Rota-Baxter operators |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2510.16826 |