Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation
Elisabete Barreiro, Saïd Benayadi, Carla Rizzo
TL;DR
The paper develops a cohesive bialgebra theory for nearly associative algebras and LR-algebras by extending Drinfeld's double to the non‑associative setting and showing the equivalence of nearly associative bialgebras with $L$-, $R$-, and $LR$-bialgebras. It provides a coproduct-based characterization and introduces nearly associative $L$-algebras to support the theory, including a coboundary framework that naturally leads to LR-Yang-Baxter equations and $r$-matrices. Through duality with nearly coassociative coalgebras and comprehensive representation theory, the work unifies several non‑associative structures and exposes algebraic mechanisms behind Yang-Baxter type equations in non‑associative contexts. The results yield explicit constructions and examples, expanding the toolkit for studying integrability and non‑associative algebraic systems in mathematical physics and related areas.
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.