Matched pairs of Hopf algebras and Rota-Baxter Hopf algebras

TL;DR

This paper explores the structure of Rota-Baxter Hopf algebras of weight -1, establishing a correspondence with matched pairs of Hopf algebras and projection homomorphism pairs, revealing new structural insights.

Contribution

It introduces the concept of projection homomorphism pairs on matched Hopf algebras and links them to Rota-Baxter Hopf algebra structures, providing a novel framework.

Findings

Constructed matched pairs of Hopf algebras from Rota-Baxter Hopf algebras

Established a correspondence between projection homomorphism pairs and Rota-Baxter Hopf algebras

Analyzed the Rota-Baxter structures on images of projection homomorphisms

Abstract

In this paper, we first study Rota-Baxter Hopf algebras of weight $-1$ and construct a matched pair of Hopf algebras on every Rota-Baxter Hopf algebra of weight $-1$. Then we propose the notion of projection homomorphism pairs on a matched pair of Hopf algebras, and show that every projection homomorphism pair $(C,\wtd{C})$ induces a Rota-Baxter Hopf algebra. Conversely, we prove that the matched pair of Hopf algebras on a Rota-Baxter Hopf algebra of weight $-1$ $(H,B)$ gives rise to a projection homomorphism pair $(C,\wtd{C})$. Furthermore, we study the Rota-Baxter Hopf algebra structure on $\im C$ that is Rota-Baxter isomorphic to $(H,B)$, and investigate the relationship between the Rota-Baxter Hopf algebra structure on $\im \wtd{C}$ and the descendent Rota-Baxter Hopf algebra $(H_{B},B)$.