Lie Theory for Fusion Categories: a Research Primer

Equivalences of graded fusion categories

“…An important class of fusion categories are the categories of level k integrable representations of an affine Lie algebra g, which we denote as C(g, k). We direct the reader to [15] for more details. Using the above Theorem we are able to show that if g is a Lie algebra of type A n , B n , C n , D 2n+1 , or G 2 , then the category C(g, k) has no non-trivial gauge auto-equivalences, and if g is a Lie algebra of type D 2n , then the gauge auto-equivalence group of C(g, k) is isomorphic to Z 2 .…”

“…Example 9.4. Let k ∈ N and consider C(sl 2 , k), the modular tensor category of level k integrable representations of sl 2 (see [15]). This modular category always has a non-trivial invertible object (k), which is a boson if k ≡ 0 (mod 4), and is a fermion if k ≡ 2 (mod 4).…”

Auto-equivalences of the modular tensor categories of type $A$, $B$, $C$ and $G$

“…Surprising little knowledge of these categories is required for this paper, so we keep the details to a minimum. For additional details we point the reader to [45] for an expository explanation of the modular categories C(g, k), and to [1,35] for a detailed exposition. We mainly restrict our attention to the case when g is one of sl r+1 , so 2r+1 , so 2r+1 , or g 2 , as these are the Lie algebras we deal with in this paper.…”

“…Here a ∨ j are the co-labels of g, given in Table 1 for the relevant Lie algebras in this paper. The fusion rules for the categories C(g, k) can be computed using the Weyl chamber of g. The details of this procedure are complicated, so we direct the reader to [45,Section 5] for the details. We will recall relevant fusion rules of the categories C(g, k) as needed throughout the paper.…”

Quadratic $d$-numbers

“…To each rank n ∈ Z ≥1 complex finite-dimensional simple Lie algebra of Dynkin type X n , and level k ∈ Z ≥1 one associates a modular tensor category which we abbreviate X n,k . One can refer to [46] for a technical outline of these examples and further references. Weakly quadratic categories X n,k are entirely described and the only fields appearing are Q( √ N ) for N = 2, 3, 5, 6, 21.…”