Module categories over pointed Hopf algebras

TL;DR

This paper develops techniques to study exact module categories over pointed finite-dimensional Hopf algebras and classifies such categories over small quantum groups, advancing understanding of their representation theory.

Contribution

It introduces new methods for analyzing module categories over pointed Hopf algebras and provides a classification for the case of small quantum groups.

Findings

Classification of exact module categories over u_q(sl_2)

Development of techniques for pointed Hopf algebras

Enhanced understanding of representation categories

Abstract

We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups $u_q(\mathfrak{sl}_2)$.