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 Author Edie-Michell, Cain Title The classification of categories generated by an object of small dimension URL http://hdl.handle.net/1885/146634 Publication Date 2018 Date Accessioned 2018-08-24 01:26:09 University/Publisher Australian National University 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. Subjects/Keywords Unitary fusion categories; classification; ADE Language en Country of Publication au Record ID handle:1885/146634 Repository anu Date Retrieved 2019-12-26 Date Indexed 2019-12-30 Issued Date 2018-01-01 00:00:00

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…fusion categories generated by a normal object X of dimension less than 2. There is of course significant work to be done to complete such a classification. Namely we need to compute the Brauer-Picard groups of the adjoint subcategories of the ADE fusion…

…of the ADE fusion categories. Again this is no easy task, in fact entire papers [23] have dealt with constructing cyclic extensions of certain fusion categories. The purpose of this thesis is to attempt the classification of unitary fusion…

…Picard group, the classification of graded categories, and planar algebras. In particular we define the ADE planar algebras, and associated unitary fusion categories. These categories will play a key role in this thesis, due to Theorem 1.0.1. While the…

…play a large role in this Thesis, as we plan to classify all cyclic 14 extensions, generated by a normal object of dimension less than 2, of the adjoint subcategories of the ADE fusion categories. An important piece of data in the classification of…

…fusion categories With Theorem 1.0.1 in mind, we want to classify certain cyclic extensions of the adjoint subcategories of the ADE fusion categories. The recipe for such a classification, as laid out in [17] (and described in Chapter 2…

…Contents Acknowledgements v Abstract vii 1 Introduction 1 2 Preliminaries 9 3 The Brauer-Picard groups of the adjoint subcategories of the ADE fusion categories 3.1 The Drinfeld Centres of the ADE fusion categories . . . . . . . . . 3.2…

…Planar algebra automorphisms and auto-equivalences of tensor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Brauer-Picard groups of the ADE fusion categories . . . . . . 3.4 Explicit constructions of invertible…

…bimodules from braided autoequivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 27 36 46 48 4 Classifying cyclic extensions of the adjoint subcategories of the ADE fusion categories generated by an object of dimension less than…