This talk is about the Rota-Baxter operators on Hopf algebras and paired Hopf modules. We obtain two characterizations of Rota-Baxter paired modules, and 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. We also introduce the concept of Rota-Baxter paired Hopf modules by combining the notion of Rota-Baxter paired module with Rota-Baxter paired comodule, and prove a structure theorem for generic Rota-Baxter paired Hopf modules.

(This work is a joint work with Zhongwei Wang and   Huihui Zheng)