Quasi-triangular, factorizable anti-dendriform bialgebras and relative Rota-Baxter operators

TL;DR

This paper introduces quasi-triangular anti-dendriform bialgebras, explores their connection to solutions of the AD-YBE, and relates factorizable structures to quadratic Rota-Baxter anti-dendriform algebras.

Contribution

It defines new algebraic structures, links them to solutions of the AD-YBE, and provides a characterization via relative Rota-Baxter operators.

Findings

Introduction of quasi-triangular anti-dendriform bialgebras

Connection between factorizable bialgebras and algebra factorizations

Interpretation of structures in terms of quadratic Rota-Baxter algebras

Abstract

We introduce the notion of quasi-triangular anti-dendriform bialgebras, which can be induced by the solutions of the AD-YBE whose symmetric parts are invariant. A factorizable anti-dendriform bialgebra leads to a factorization of the underlying anti-dendriform algebra. Moreover, relative Rota-Baxter operators with weights are introduced to characterize the solutions of the AD-YBE whose symmetric parts are invariant. Finally, we interpret factorizable anti-dendriform bialgebras in terms of quadratic Rota-Baxter anti-dendriform algebras.