A field guide to categories with $A_n$ fusion rules

TL;DR

This paper provides a comprehensive overview and classification of fusion categories with $A_n$ fusion rules, including their realizations, automorphism groups, subcategories, and Drinfeld centres, enriching the understanding of their structure.

Contribution

It offers a detailed classification and explicit descriptions of $A_n$ fusion rule categories, including realizations and automorphism groups, which were previously not systematically documented.

Findings

Classification of $A_n$ fusion rule categories as monoidal, dagger, pivotal, and braided.

Realizations via Temperley-Lieb categories.

Descriptions of automorphism groups and Drinfeld centres.

Abstract

We collate information about the fusion categories with $A_n$ fusion rules. This note includes the classification of these categories, a realisation via the Temperley-Lieb categories, the auto-equivalence groups (both braided and tensor), identifications of the subcategories of invertible objects, and explicit descriptions of the Drinfeld centres. The first section describes the classification of these categories (as monoidal, dagger, pivotal, and braided categories). The second section describes the properties of these categories.