Lie bialgebras constructed from Zinbiel bialgebras and Leibniz bialgebras

TL;DR

This paper explores the construction of Lie bialgebras from Zinbiel and Leibniz bialgebras, extending known algebraic structures and analyzing solutions to the classical Yang-Baxter equation.

Contribution

It extends the tensor product construction of Lie bialgebras to Zinbiel and Leibniz bialgebras, and characterizes when these structures are quasi-triangular or factorizable.

Findings

Lie bialgebra structure on tensor product of Leibniz bialgebra and Zinbiel algebra

Infinite-dimensional Lie bialgebra on tensor product of Zinbiel bialgebra and graded Leibniz algebra

Characterization of Zinbiel bialgebra via tensor product with special Leibniz algebra

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.