The classification of categories generated by an object of small dimension

Abstract

The goal of this thesis is to attempt the classification of unitary fusion categories generated by a normal object (\refi{an object comuting with its dual}{1}) of dimension less than 2. This classification has recently become accessible due to a result of Morrison and Snyder, which shows that any such category must be a cyclic extension of an adjoint subcategory of one of the $ADE$ fusion categories. Our main tool is the classification of graded categories from \cite{MR2677836}, which classifies graded extensions of a fusion category in terms of the Brauer-Picard group, and Drinfeld centre of that category.

We compute the Drinfeld centres, and Brauer-Picard groups of the adjoint subcategories of the $ADE$ fusion categories. Using this information we apply the machinery of graded extensions to classify the cyclic extensions that are generated by a normal object of dimension less than 2, of the adjoint subcategories of the $ADE$ fusion categories. Unfortunately, our classification has a gap when the dimension of the object is $\sqrt{2+\sqrt{2}}$ corresponding to the possible existence of an interesting new fusion category. Interestingly we prove the existence of a new category, generated by a normal object of dimension $2\cos(\frac{\pi}{18})$, which we call the DEE fusion category. We include the fusion rules for the DEE fusion categories in an appendix to this thesis.