The classification of categories generated by an object of small dimension
Date
2018
Authors
Edie-Michell, Cain
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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.
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Keywords
Unitary fusion categories, classification, ADE
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Thesis (PhD)
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