From convolution pi-algebras to Rota-Baxter Hopf pi-algebras

In this paper, we study Goncharov-type Rota-Baxter operators in the framework of group-graded Hopf pi-algebras. By constructing convolution pi-algebras as the core calculus tool, we systematically derive fundamental algebraic properties of Hopf pi-algebras, including antipode anti-homomorphism and generalized homomorphism compatibility. On this basis, we establish the axiomatic system of two classes of graded Rota-Baxter operators of weight 1 and weight -1, prove their bijective correspondence via the antipode, construct several non-trivial examples, and present two derived structures: descendant Hopf pi-algebras and image Hopf pi-subalgebras. This work enriches the operator theory of graded Hopf algebras and provides algebraic constructions for related research on graded Yang-Baxter equations.