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

In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post -Lie algebra on the space of primitive elements and the fact that there is naturally a postHopf 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-Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang- Baxter equation. Then, we introduce the notion of relative Rota-Baxter operator 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. Finally, 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 to the Yang- Baxter equation in certain cocommutative Hopf algebras.