Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On modules over a Hopf brace

Ramón González Rodríguez, Brais Ramos Pérez, Ana Belén Rodríguez Raposo

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 $H_{2}$.

On modules over a Hopf brace

Abstract

Let be a Hopf brace in a symmetric monoidal category . In this article it is proved that the category of modules over is isomorphic to the category of modules over the smash product algebra . Furthermore, the category of modules over in the sense of Zhu is characterized by the condition that a certain action lies in the cocommutativity class of .
Paper Structure (4 sections, 15 theorems, 85 equations)

This paper contains 4 sections, 15 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries about Hopf algebras and Hopf braces
  3. From modules over a Hopf brace to modules over an algebra
  4. On the notion of modules over a Hopf brace in Zhu's sense

Key Result

Theorem 2.9

Let $H$ be a Hopf algebra and $(A, \varphi_A)$ a left $H$-module algebra, and define where $\Psi^{H}_{A}\coloneqq (\varphi_{A}\otimes H)\circ(H\otimes c_{H,A})\circ(\delta_{H}\otimes A)$. Then, $A\sharp H$ is an algebra in ${\sf C}$, known as the smash product algebra.

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 43 more