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arXiv · 2510.16826

Leibniz-dendriform bialgebras and relative Rota-Baxter operators

Qinxiu Sun·Shuangjian Guo
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Abstract

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.

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Qinxiu Sun, Shuangjian Guo. 2025-10-19. Leibniz-dendriform bialgebras and relative Rota-Baxter operators. https://arxiv.org/abs/2510.16826

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