Bialgebras, Frobenius algebras and associative Yang-Baxter equations for Rota-Baxter algebras

TL;DR

This paper introduces Rota-Baxter antisymmetric infinitesimal bialgebras, generalizing algebraic structures and equations like the associative Yang-Baxter equation within Rota-Baxter algebras, revealing new connections and constructions.

Contribution

It establishes a new bialgebra structure compatible with Rota-Baxter operators and extends the associative Yang-Baxter equation to this context, introducing novel algebraic concepts.

Findings

Defined Rota-Baxter ASI bialgebras and characterized them via matched pairs and Frobenius algebra constructions.

Extended the AYBE to Rota-Baxter algebras and constructed solutions using $ ext{O}$-operators and dendriform algebras.

Discovered that weight-zero Rota-Baxter ASI bialgebras lead to quadri-bialgebras instead of traditional bialgebras.

Abstract

Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-Baxter equation (AYBE) to Rota-Baxter algebras. Antisymmetric solutions of the equation are used to construct Rota-Baxter ASI bialgebras. The notions of an $\mathcal{O}$-operator on a Rota-Baxter algebra and a Rota-Baxter dendriform algebra are also introduced to produce solutions of the AYBE in Rota-Baxter algebras and thus to provide Rota-Baxter ASI bialgebras. An unexpected byproduct is that a Rota-Baxter ASI bialgebra of weight zero gives rise to a quadri-bialgebra instead of bialgebra constructions for the dendriform algebra.