Lie Theory for Fusion Categories: a Research Primer

TL;DR

This paper provides a self-contained, computationally accessible introduction to the Lie-theoretic aspects of fusion categories, focusing on their realization via quantum groups and aiming to bridge gaps for researchers from diverse backgrounds.

Contribution

It offers a comprehensive primer that simplifies the complex Lie theory connections in fusion categories, making the subject more approachable for computational study.

Findings

Clarifies the relationship between Lie theory and fusion categories.

Provides computational tools and frameworks for studying fusion categories.

Bridges gaps between abstract theory and practical research applications.

Abstract

A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the representation-theoretic properties these algebras possess. Here we will forego technical intricacy as a growing number of researchers study fusion categories disjoint from Lie theory, representation theory, and a laundry list of other obstacles to understanding the mostly combinatorial, geometric, and numerical descriptions of the examples of fusion categories arising from quantum groups. This expository piece aims to create a self-contained guide for researchers to study from a computational standpoint with only the prerequisite knowledge of fusion categories.