On modules over a Hopf brace

Abstract

Let $\mathbb{H}=(H_1,H_2)$ be a Hopf brace in a symmetric monoidal category $\mathsf{C}$. 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 $H_1♯H_2$. 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 $H_2$.