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

### Yunnan Li

Guangzhou University, China### Yunhe Sheng

Jilin University, Changchun, China### Rong Tang

Jilin University, Changchun, China

## Abstract

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 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–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.

## Cite this article

Yunnan Li, Yunhe Sheng, Rong Tang, Post-Hopf algebras, relative Rota–Baxter operators and solutions to the Yang–Baxter equation. J. Noncommut. Geom. 18 (2024), no. 2, pp. 605–630

DOI 10.4171/JNCG/537