Reynolds Leibniz bialgebras of any weight

TL;DR

This paper explores the structure and classification of Reynolds Leibniz bialgebras of any weight, linking them with solutions to the Leibniz Yang-Baxter equation and providing a classification for two-dimensional cases.

Contribution

It introduces equivalent characterizations of Reynolds Leibniz bialgebras, examines their relation to the Leibniz Yang-Baxter equation, and classifies two-dimensional triangular Reynolds Leibniz bialgebras.

Findings

Characterizations via matched pairs and Manin triples.

Compatibility conditions with Leibniz Yang-Baxter solutions.

Classification of 2D triangular Reynolds Leibniz bialgebras.

Abstract

This paper studies bialgebraic structures associated with a Reynolds Leibniz algebra of weight $\lambda$, that is, a Leibniz algebra equipped with a Reynolds operator of weight $\lambda$. We first present equivalent characterizations of Reynolds Leibniz bialgebras of weight $\lambda$, using matched pairs and Manin triples. Next, we examine compatibility conditions between solutions of the classical Leibniz Yang-Baxter equation and Reynolds operators of weight $\lambda$, framed in terms of triangular Reynolds Leibniz bialgebras. Finally, building on results of Ayupov {\em et al.}, we classify two-dimensional triangular Reynolds Leibniz bialgebras of weight $\lambda$.