Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

TL;DR

This paper introduces post-Hopf algebras and relative Rota-Baxter operators, establishing their roles in constructing solutions to the Yang-Baxter equation within cocommutative Hopf algebras.

Contribution

It defines post-Hopf algebras and relative Rota-Baxter operators, revealing their interrelations and applications in solving the Yang-Baxter equation.

Findings

Post-Hopf algebras induce post-Lie algebra structures.

Cocommutative post-Hopf algebras lead to solutions of the Yang-Baxter equation.

Relative Rota-Baxter operators characterize solutions and relate to module structures.

Abstract

In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.