Rota-Baxter paired modules and their constructions from Hopf algebras

TL;DR

This paper introduces Rota-Baxter paired modules, explores their properties, and constructs numerous examples from Hopf algebra related structures, expanding the understanding of Rota-Baxter algebraic frameworks.

Contribution

It defines Rota-Baxter paired modules, provides characterizations, and constructs many examples from various Hopf algebra related modules, broadening the scope of Rota-Baxter algebra applications.

Findings

Characterizations of Rota-Baxter paired modules

Construction of examples from Hopf algebra structures

New instances of Rota-Baxter algebras

Abstract

In this paper, we introduce the concept of a Rota-Baxter paired module to study Rota-Baxter modules without necessarily a Rota-Baxter operator. We obtain two characterizations of Rota-Baxter paired modules, and give some basic properties of Rota-Baxter paired modules. Beginning with the connection between the notion of integrals in the representations of Hopf algebra and of the notion of an integral algebra as a motivation and special case of Rota-Baxter algebra of weight zero, we obtain a large number of Rota-Baxter paired modules from Hopf related algebras and modules, including semisimple Hopf algebras, weak Hopf algebras, Long bialgebras, quasitriangular Hopf algebras, weak Hopf modules, dimodules and Doi-Hopf modules. Some of them give new examples of Rota-Baxter algebras.