Lie Theory for Fusion Categories: a Research Primer

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.