Leibniz-dendriform bialgebras and relative Rota-Baxter operators
Qinxiu Sun, Shuangjian Guo
TL;DR
The paper develops a bialgebra theory for Leibniz-dendriform algebras by introducing Leibniz-dendriform bialgebras and establishing their equivalence with phase spaces and matched pairs. It introduces the Leibniz-dendriform Yang-Baxter equation (LD-YBE) and shows skew-symmetric solutions yield coboundary bialgebras, while non-skew-symmetric solutions can also generate such structures, including quasi-triangular and factorizable examples. The authors connect LD-YBE data with relative Rota-Baxter operators of various weights and show a tight correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras and factorizable Leibniz-dendriform bialgebras. They also develop O-operators and Leibniz-quadri-algebras as tools for constructing LD-YBE solutions, and establish a comprehensive framework linking bialgebras, phase spaces, and operator theories in the Leibniz-dendriform setting.
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.