Pre-anti-flexible bialgebra

TL;DR

This paper introduces pre-anti-flexible bialgebras, explores their structures via Rota-Baxter operators, and establishes their relation to Yang-Baxter equations, providing new insights into algebraic splitting and bialgebra theory.

Contribution

It defines pre-anti-flexible bialgebras, links them to matched pairs and Yang-Baxter equations, and connects $ ext{O}$-operators with bimodules, advancing the understanding of algebraic structures.

Findings

Pre-anti-flexible bialgebras are characterized via matched pairs.

Symmetric solutions of the pre-anti-flexible Yang-Baxter equation yield bialgebras.

The pre-anti-flexible Yang-Baxter equation coincides with the $ ext{D}$-equation.

Abstract

In this paper, we derive pre-anti-flexible algebras structures in term of zero weight's Rota-Baxter operators defined on anti-flexible algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce the notion of pre-anti-flexible bialgebras and establish equivalences among matched pair of anti-flexible algebras, matched pair of pre-anti-flexible algebras and pre-anti-flexible bialgebras. Investigation on special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation. Both dual bimodules of pre-anti-flexible algebras and dendriform algebras have the same shape and this induces that both pre-anti-flexible Yang-Baxter equation and $\mathcal{D}$-equation are identical. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link $\mathcal{O}$-operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation.