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arXiv · 2409.00390

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

Elisabete Barreiro·Saïd Benayadi·Carla Rizzo
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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.

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Elisabete Barreiro, Saïd Benayadi, Carla Rizzo. 2024-08-31. Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation. https://doi.org/10.1016/j.laa.2025.02.017

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