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arXiv · 2606.08627

Bialgebra theory, the Yang-Baxter equation and relative Rota-Baxter operators for diassociative algebras

Hui Hu·Yizhen Li·Guilai Liu·Shanghua Zheng
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Abstract

In this paper, we develop a bialgebra theory for diassociative algebras. Inspired by the notion of a quadratic diassociative algebra, we introduce the concept of a Manin triple of diassociative algebras. We then define a diassociative bialgebra, which is shown to be equivalent to a Manin triple of diassociative algebras through a specific matched pair of diassociative algebras. We further formulate the diassociative Yang-Baxter equation (DYBE) in a diassociative algebra, and prove that symmetric solutions of the DYBE give rise to diassociative bialgebras. To construct such solutions, we also introduce relative Rota-Baxter operators and pre-diassociative algebras. As a key application, we lift the known relationships between diassociative algebras and other algebraic structures to the bialgebra level. In particular, we show that every diassociative bialgebra naturally induces a Leibniz bialgebra, thereby extending Loday's classical result that a diassociative algebra gives rise to a Leibniz algebra. Moreover, we provide explicit constructions of Lie bialgebras via tensor products of diassociative bialgebras and quadratic dendriform algebras.

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Hui Hu, Yizhen Li, Guilai Liu, Shanghua Zheng. 2026-06-07. Bialgebra theory, the Yang-Baxter equation and relative Rota-Baxter operators for diassociative algebras. https://arxiv.org/abs/2606.08627

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