Classification of Pointed Fusion Categories of dimension $p^3$ up to weak Morita Equivalence

TL;DR

This paper classifies all pointed fusion categories over the complex numbers with dimension p^3 for odd primes, detailing their equivalence classes and module categories, advancing understanding in fusion category theory.

Contribution

It provides a complete classification of pointed fusion categories of dimension p^3 and analyzes their module category equivalences, a novel comprehensive result.

Findings

Complete classification of pointed fusion categories of dimension p^3

Identification of equivalence classes and module category relations

Clarification of structure for categories of prime power dimension

Abstract

We give a complete classification of pointed fusion categories over $\mathbb{C}$ of global dimension $p^3$ for $p$ any odd prime. We proceed to classify the equivalence classes of pointed fusion categories of dimension $p^3$ and we determine which of these equivalence classes have equivalent categories of modules.