Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras

Abstract

In this paper, we mainly provide some methods for constructing antisymmetric infinitesimal (ASI) bialgebras and Lie bialgebras using Zinbiel bialgebras and diassociative bialgebras. We first show that there is an ASI bialgebra structure on the tensor product of a diassociative bialgebra and a quadratic Zinbiel algebra, while there is an infinite-dimensional ASI bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic $\bz$-graded diassociative algebra. For a special quadratic $\bz$-graded Leibniz algebra, the property that its tensor product with a Zinbiel bialgebra forms an ASI bialgebra characterizes the Zinbiel bialgebra. By examining the relationship between solutions of the (classical) Yang-Baxter equation in a Zinbiel algebra and the induced associative algebra, we prove that the induced ASI bialgebra is quasi-triangular (resp. triangular, factorizable) whenever the original Zinbiel bialgebra is quasi-triangular (resp. triangular, factorizable). These conclusions enable us to provide a method for constructing Lie bialgebras from Zinbiel bialgebras and two approaches for constructing Lie bialgebras from diassociative bialgebras. We also provide specific descriptions of the connections between the solutions of the Yang-Baxter equations and the connections between the $O$-operators corresponding to these constructions.