Rota-Baxter operators on cocommutative Hopf algebras and Hopf braces

Abstract: This paper studies the relationship of Rota-Baxter operators on cocommutative Hopf algebras with Hopf braces and the Yang-Baxter equation, with emphasis on the embedding of cocommutative Hopf braces into Rota-Baxter Hopf algebras. Through Hopf braces, we establish a connection between relative Rota-Baxter operators on cocommutative Hopf algebras and bijective 1-cocycles. Finally, we introduce the notion of symmetric Hopf braces, and establish the relationship between symmetric Hopf braces and Rota-Baxter Hopf algebras.