On modules over a Hopf brace

TL;DR

This paper establishes an isomorphism between the category of modules over a Hopf brace and modules over a smash product algebra, providing a new perspective on module categories in symmetric monoidal categories.

Contribution

It proves the equivalence of module categories over a Hopf brace and a smash product algebra, and characterizes modules in Zhu's sense via cocommutativity conditions.

Findings

Category of modules over a Hopf brace is isomorphic to modules over a smash product algebra.

Modules over a Hopf brace in Zhu's sense are characterized by cocommutativity conditions.

Provides a structural understanding of modules over Hopf braces in symmetric monoidal categories.

Abstract

Let $\mathbb{H}=(H_{1},H_{2})$ be a Hopf brace in a symmetric monoidal category ${\sf C}$. In this article it is proved that the category of modules over $\mathbb{H}$ is isomorphic to the category of modules over the smash product algebra $H_{1}\sharp H_{2}$. Furthermore, the category of modules over $\mathbb{H}$ in the sense of Zhu is characterized by the condition that a certain action lies in the cocommutativity class of $H_{2}$.