Pointed finite tensor categories over abelian groups

TL;DR

This paper characterizes finite pointed tensor categories derived from abelian group-based pointed Hopf algebras using associator cohomology, and shows all such braided tensor categories are de-equivariantizations of these Hopf algebras.

Contribution

It provides a cohomological criterion for classifying pointed tensor categories over abelian groups and proves their braided versions are all de-equivariantizations of finite-dimensional pointed Hopf algebras.

Findings

Characterization of pointed tensor categories via associator cohomology

Every coradically graded pointed finite braided tensor category is a de-equivariantization

Connection between braided categories and pointed Hopf algebras over abelian groups

Abstract

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of a finite dimensional pointed Hopf algebras over an abelian group.