Lie theory for fusion categories: A research primer

Abstract

A diverse collection of fusion categories, in the language of [22], 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. Our 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. A multitude of figures and worked examples are included to elucidate the material, and additional references are abundant for those readers looking to delve deeper. Note that in general our chosen references are intended to provide useable resources for the reader and do not always indicate provenance. Lastly we have included several open and approachable questions of general interest throughout the final sections.The organization of this paper is as follows: Sections 1 and 2 summarize the classical representation theory of semisimple Lie algebras in the spirit of [37] to introduce the chosen language and notation used extensively in what follows. Those unfamiliar with Lie algebras are encouraged to work through the provided examples themselves, while readers who possesses this prerequisite knowledge can safely begin reading in Section 3 referring back to earlier sections as needed. Terminology most relevant to future explanation is italicized for this purpose. Section 3 explains computationally relevant subtleties of the modern generalization of the representation theory of quantum groups including quantum dimensions and the affine Weyl group, followed by Section 4 which defines our primary objects of study: the fusion categories C(g, ℓ, q) where g is a finite-dimensional simple complex Lie algebra and q is a root of unity such that q 2 has order ℓ ∈ Z ≥1 . Section 5 discusses the fusion rules of C(g, ℓ, q), the classification of fusion subcategories, and simple factorizations. Modular data and the Galois symmetry thereof is covered in Section 6, while tensor autoequivalences, module categories, and commutative algebras are contained in Section 7.

Modular tensor categories, subcategories, and Galois orbits

“…One major source of examples of modular tensor categories comes from the semisimple representation theory of quantum groups at roots of unity; these are indexed by a complex finitedimensional simple Lie algebra g and positive integer k, and denoted C(g, k). The details of this construction are but ancillary to our general theory, so we refer the reader to [27] for further reading.…”

Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

“…The main object of study in this paper will be the modular tensor categories C(sl r+1 , k), the category of level k integrable representations of ŝ l n . For an overview us these categories see [50]. For our purposes we will only require some basic combinatorics of these categories.…”

Simple current auto-equivalences of modular tensor categories

“…Our example modular tensor categories will all be categories of level k integrable representations of an affine Lie algebraĝ, which we denote C(g, k). For details on these categories we direct the reader to [9]. sl 4 at level 2.…”