Post-Hopf group algebras, Hopf group braces and Rota-Baxter operators on Hopf group algebras

TL;DR

This paper introduces and explores the relationships between Hopf group braces, post-Hopf group algebras, and Rota-Baxter Hopf group algebras, generalizing existing algebraic structures and establishing their interconnections.

Contribution

It defines new algebraic structures—Hopf group braces, post-Hopf group algebras, Rota-Baxter Hopf group algebras—and analyzes their relationships, especially under cocommutativity.

Findings

Mutual derivation of Hopf group braces and post-Hopf group algebras under cocommutativity.

Rota-Baxter Hopf group algebras can generate Hopf group braces.

Established relationships among the introduced algebraic structures.

Abstract

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 brace, post Hopf algebra and Rota-Baxter Hopf algebras respectively. We also discuss their relationships. Explicitly under the condition of cocomutativity, Hopf group braces, post-Hopf group algebras could be mutually obtained, and Rota-Baxter Hopf group algebras could lead to Hopf group braces.