Matched pairs of Hopf algebras and Rota-Baxter Hopf algebras

TL;DR

The paper develops a structural bridge between weight-$-1$ Rota-Baxter Hopf algebras and matched pairs of Hopf algebras. It shows that any RB Hopf algebra $(H,B)$ with weight $-1$ induces a matched pair $(H_+,H_-,\rhd,\vartriangleleft)$ via $H_+=\operatorname{Im}B$ and $H_- = \operatorname{Im}\widetilde{B}$, and forms the double cross product. It then introduces projection homomorphism pairs on matched pairs, proving they yield RB Hopf algebras of weight $-1$, and demonstrates that the RB structure on a given $(H,B)$ gives rise to a projection pair $(C,\widetilde{C})$ whose images carry RB Hopf algebra structures isomorphic to $(H,B)$ and related descriptions to the descendant $(H_B,B)$. The results unify RB Hopf algebras, matched pairs, and projection concepts, extending prior work and offering a cohesive framework for exploring descendant and image-based RB structures.

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 $(H_{B},B)$.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• proof
• Proposition 2.8
• proof