Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators

TL;DR

This paper introduces quasi-triangular Leibniz bialgebras derived from solutions to the classical Leibniz Yang-Baxter equation, linking them to Rota-Baxter operators and expanding the understanding of Leibniz algebra factorizations.

Contribution

It defines quasi-triangular Leibniz bialgebras, connects them with Rota-Baxter operators, and characterizes factorizable Leibniz bialgebras through these operators.

Findings

Quasi-triangular Leibniz bialgebras constructed from solutions to the CLYBE.

Characterization of solutions via relative Rota-Baxter operators.

Introduction of skew-symmetric quadratic Rota-Baxter Leibniz algebras.

Abstract

We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang-Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota-Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota-Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota-Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight $0$, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.