The definitions of the n th Gauss sum and the associated n th central charge are introduced for premodular categories C and n ∈ Z. We first derive an expression of the n th Gauss sum of a modular category C, for any integer n coprime to the order of the Tmatrix of C, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these n, the higher Gauss sums are d-numbers and the associated central charges are roots of unity. In particular, if C is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.

The Witt group of nondegenerate braided fusion categories W contains a subgroup Wun consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra g and positive integer k there exists a pseudo-unitary category C(g, k) consisting of highest weight integerableĝ-modules of level k whereĝ is the corresponding affine Lie algebra. Relations between the classes [C(sl2, k)], k ≥ 1 have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes [C(sl3, k)], k ≥ 1 with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras g and positive integer levels k.

There is a long-standing belief that the modular tensor categories C(g, k), for k ∈ Z ≥1 and finite-dimensional simple complex Lie algebras g, contain exceptional connectedétale algebras at only finitely many levels k. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here we confirm this conjecture when g has rank 2, contributing proofs and explicit bounds when g is of type B2 or G2, adding to the previously known positive results for types A1 and A2.fusion categories whose braidings are entirely non-degenerate, or furthest from symmetric as possible.Definition 2.1.1. A pre-modular category C is a modular tensor category if 1 ∈ C is the only simple transparent object.Lastly we recall that each pre-modular category has a family of natural isomorphisms θ(X) : X ∼ → X for all X ∈ C known as twists (or ribbon structure), compatible with the inherent braiding isomorphisms [12, Definition 8.10.1]. We will abuse this notation when it suits our purposes by denoting the complex number α such that θ(X) = α · id X simply as θ(X) for any simple X ∈ C.

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.

No abstract