Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation

TL;DR

This paper develops a unified bialgebra theory for nearly associative and LR-algebras, establishing their equivalence, characterizing them via coproducts, and exploring their connection to the Yang-Baxter equation.

Contribution

It introduces nearly associative L-algebras, proves the equivalence of nearly associative and LR-bialgebras, and links coboundary nearly associative bialgebras to the Yang-Baxter equation.

Findings

Nearly associative and LR-bialgebras are equivalent concepts.

Characterization of these bialgebras via coproducts.

Identification of coboundary nearly associative bialgebras related to the Yang-Baxter equation.

Abstract

We develop the bialgebra theory for two classes of non-associative algebras: nearly associative algebras and $LR$-algebras. In particular, building on recent studies that reveal connections between these algebraic structures, we establish that nearly associative bialgebras and $LR$-bialgebras are, in fact, equivalent concepts. We also provide a characterization of these bialgebra classes based on the coproduct. Moreover, since the development of nearly associative bialgebras - and by extension, $LR$-bialgebras - requires the framework of nearly associative $L$-algebras, we introduce this class of non-associative algebras and explore their fundamental properties. Furthermore, we identify and characterize a special class of nearly associative bialgebras, the coboundary nearly associative bialgebras, which provides a natural framework for studying the Yang-Baxter equation (YBE) within this context.