Matched pairs of Hopf algebras and Rota-Baxter Hopf algebras

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 $(C,\text{wtd}C)$ 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 $(C,\text{wtd}C)$. Furthermore, we study the Rota-Baxter Hopf algebra structure on $\text{im}C$ that is Rota-Baxter isomorphic to $(H,B)$, and investigate the relationship between the Rota-Baxter Hopf algebra structure on $\text{im}\text{wtd}C$ and the descendent Rota-Baxter Hopf algebra $(H_B,B)$.