Lie bialgebras constructed from Zinbiel bialgebras and Leibniz bialgebras

Abstract

There is a Lie algebra structure on the tensor product of a Leibniz algebra and a Zinbiel algebra for the operads of Leibniz algebras and Zinbiel algebras are Koszul dual. In this paper, we extend such conclusion to the context of bialgebras. We show that there is a Lie bialgebra structure on the tensor product of a Leibniz bialgebra and a quadratic Zinbiel algebra; there is an infinite-dimensional Lie bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic $\mathbb{Z}$-graded Leibniz algebra. For special quadratic $\mathbb{Z}$-graded Leibniz algebra, the tensor product with a Zinbiel bialgebra being a Lie bialgebra characterizes the Zinbiel bialgebra. By analyzing the relationship between solutions of the classical Yang-Baxter equation in a Zinbiel algebra (resp. a Leibniz algebra) and solutions of the classical Yang-Baxter equation in the induced Lie algebra, we prove that the induced Lie bialgebra is quasi-triangular (resp. triangular, factorizable) if the original Zinbiel bialgebra (resp. Leibniz bialgebra) is quasi-triangular (resp. triangular, factorizable). Finally, we provide a construction of a quasi-Frobenius Lie algebra on the tensor product of a quasi-Frobenius Zinbiel algebra and a quadratic Leibniz algebra.