标题： Zinbiel双代数、相对Rota-Baxter算子及相关Yang-Baxter方程 显示英文标题

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

摘要： 本文首先引入了Zinbiel双代数的概念，并证明了Zinbiel双代数、Zinbiel代数的匹配对和Zinbiel代数的Manin三元组是等价的。 然后我们研究了余边界Zinbiel双代数，这导致了一个经典的Yang-Baxter方程的类比。 此外，我们引入了准三角和可分解的Zinbiel双代数作为特殊情况。一个准三角的Zinbiel双代数可以产生一个权值为$-1$的相对Rota-Baxter算子。一个可分解的Zinbiel双代数可以给出基础Zinbiel代数的一个分解。 作为一个例子，我们定义了Zinbiel双代数的Zinbiel双，它具有一个自然的可分解的Zinbiel双代数结构。 最后，我们引入了二次Rota-Baxter Zinbiel代数的概念，这是可分解Zinbiel双代数的Rota-Baxter特征化。我们证明了二次Rota-Baxter Zinbiel代数与可分解的Zinbiel双代数之间存在一一对应关系。 显示英文摘要

摘要： 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.