q-generalized (anti -) flexible algebras and bialgebras

TL;DR

This paper introduces a q-generalization of flexible and antiflexible algebras, exploring their properties, connections with existing structures, and establishing a comprehensive framework including bialgebras and Manin triples.

Contribution

It provides the first systematic development of q-generalized flexible algebras and their bialgebraic structures, extending classical results and establishing new equivalences.

Findings

Derived basic properties of q-generalized flexible algebras.

Established connections with known algebraic structures.

Proved the equivalence between q-generalized flexible algebras, Manin triples, and bialgebras.

Abstract

In this work, we provide a q-generalization of flexible algebras and related bialgebraic structures, including center-symmetric (also called antiflexible) algebras, and their bialgebras. Their basic properties are derived and discussed. Their connection with known algebraic structures, previously developed in the literature, is established. A q-generalization of Myung theorem is given. Main properties related to bimodules, matched pairs and dual bimodules as well as their algebraic consequences are investigated and analyzed. Finally, the equivalence between q-generalized flexible algebras, their Manin triple and bialgebras is established.