Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation

Abstract

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, we construct solutions of the classical Leibniz Yang-Baxter equation using relative Rota-Baxter operators and Leibniz-dendriform algebras.

28 pages, comments are welcome. J. Noncommutative Geom. 2022

References in corpus (1)

• The Embedding Tensor, Leibniz-Loday Algebras, and Their Higher Gauge Theories

Cited by in corpus (6)

• Deformations of relative Rota-Baxter operators on Leibniz algebras
• Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras
• Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators
• Review of deformation theory I: Concrete formulas for deformations of algebraic structures
• Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
• $\mathcal{O}$-operators and related structures on Leibniz algebras