Hopf group braces, post-Hopf group algebras and Rota–Baxter operators on Hopf group algebras

In this paper, we introduce the notions of Hopf group braces, post-Hopf group algebras, and Rota–Baxter Hopf group algebras as important generalizations of Hopf braces, post-Hopf algebras, and Rota–Baxter Hopf algebras, respectively. We show that a Hopf group brace could give a set-theoretical solution of the Yang–Baxter equation. We introduce the notion of matched pairs of Hopf group algebras and demonstrate the one-to-one correspondence between matched pairs of Hopf group algebras and Hopf group braces under the condition of cocomutativity. Then we prove that a cocommutative post-Hopf group algebra and its subadjacent Hopf group algebra form a Hopf group brace. Finally, we prove that Rota–Baxter Hopf group algebras could also lead to Hopf group braces.