Classifying fusion categories $$\otimes $$-generated by an object of small Frobenius–Perron dimension

Abstract

The goal of this paper is to classify fusion categories ⊗-generated by a Knormal object (defined in this paper) of Frobenius-Perron dimension less than 2. This classification has recently become accessible due to a result of Morrison and Snyder, showing that any such category must be a cyclic extension of a category of adjoint ADE type. Our main tools in this classification are the results of [11], classifying cyclic extensions of a given category in terms of data computed from the Brauer-Picard group, and Drinfeld centre of that category, and the results of [8] which compute the Brauer-Picard group and Drinfeld centres of the categories of adjoint ADE type.Our classification includes the expected categories, constructed from cyclic groups and the categories of ADE type. More interestingly we have categories in our classification that are non-trivial de-equivariantizations of these expected categories. Most interesting of all, our classification includes three infinite families constructed from the exceptional quantum subgroups E 4 of C(sl 4 , 4), and E 16,6 of C(sl 2 , 16) ⊠ C(sl 3 , 6).

Classification of Super-Modular Categories by Rank

“…Theorem 3.1. Suppose B is a rank 8 self-dual super-modular category and G is its Galois group as in Table 1 then: 9 • If G = (23) , (01), (23) or (123) , then B does not exist.…”

A class of prime fusion categories of dimension $2^N$

“…Observe that, as shown [7], when N ≥ 2 there is another family of nonpointed Z 2 N -extensions of Vec Z 2 , not equivalent to any N -Ising fusion category. However, the fusion categories in this family do not admit any braiding (Theorem 5.3).…”

“…We call them N -Ising fusion categories. They are special instances of the cyclic extensions of adjoint categories of ADE type classified in [7] and are defined as follows: Let I be the semisimplification of the representation category of U −q (sl 2 ), with q = exp(iπ/4). Then I is an Ising fusion category.…”

A Complete Classification of Unitary Fusion Categories Tensor Generated by an Object of Dimension

“…While the results [30] and [11] do provide a broad classification, it is somewhat unsatisfying having conditions on the ⊗-generating object of small dimension. However very little is known about the classification of categories ⊗-generated by an arbitrary object of small dimension.…”

“…One of the earliest results in the field is the classification of unitary fusion categories generated by a self-dual object of dimension less than 2 [30,3,20,21,23,25,34], which contained the exceptional E 6 and E 8 examples. This result was generalised in [11], replacing the self-dual condition with a mild commutativity condition. Here again exceptional examples were found, namely the quantum subgroups E 4 of sl 4 and E 16,6 of sl 2 ⊕ sl 3 .…”