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