Zinbiel bialgebras, relative Rota-Baxter operators and the related Yang-Baxter Equation

In this paper, we first introduce the notion of a Zinbiel bialgebra and show that Zinbiel bialgebras, matched pairs of Zinbiel algebras and Manin triples of Zinbiel algebras are equivalent. Then we study the coboundary Zinbiel bialgebras, which leads to an analogue of the classical Yang-Baxter equation. Moreover, we introduce the notions of quasi-triangular and factorizable Zinbiel bialgebras as special cases. A quasi-triangular Zinbiel bialgebra can give rise to a relative Rota-Baxter operator of weight -1. A factorizable Zinbiel bialgebra can give a factorization of the underlying Zinbiel algebra. As an example, we define the Zinbiel double of a Zinbiel bialgebra, which enjoys a natural factorizable Zinbiel bialgebra structure. Finally, we introduce the notion of quadratic Rota-Baxter Zinbiel algebras, as the Rota-Baxter characterization of factorizable Zinbiel bialgebras. We show that there is a one-to-one correspondence between quadratic Rota-Baxter Zinbiel algebras and factorizable Zinbiel bialgebras.