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.
近关联代数和 $LR$ 代数的代数理论:等价、表征和 $LR$ 扬-巴克斯特方程
我们发展了两类非联立代数:近联立代数和 $LR$-代数的双代数理论。特别是,在揭示这些代数结构之间联系的最新研究的基础上,我们确立了近关联双代数和 $LR$ 双代数实际上是等价的概念。我们还提供了基于协积的双代数类的特征。此外,由于近关联双桥--以及推而广之的 $LR$ 双桥--的发展需要近关联 $L$-gebras 的框架,我们介绍了这一类非关联代数并探讨了它们的基本性质。此外,我们还发现并描述了近关联双桥的一个特殊类别--共界近关联双桥,这为在此背景下研究杨-巴克斯特方程(YBE)提供了一个自然框架。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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