The classification of categories generated by an object of small dimension
| dc.contributor.author | Edie-Michell, Cain | |
| dc.date.accessioned | 2018-08-24T01:26:09Z | |
| dc.date.available | 2018-08-24T01:26:09Z | |
| dc.date.issued | 2018 | |
| dc.description.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. | en_AU |
| dc.identifier.other | b53532107 | |
| dc.identifier.uri | http://hdl.handle.net/1885/146634 | |
| dc.language.iso | en_AU | en_AU |
| dc.subject | Unitary fusion categories | en_AU |
| dc.subject | classification | en_AU |
| dc.subject | ADE | en_AU |
| dc.title | The classification of categories generated by an object of small dimension | en_AU |
| dc.type | Thesis (PhD) | en_AU |
| dcterms.valid | 2018 | en_AU |
| local.contributor.affiliation | Mathematical Sciences Institute, The Australian National University | en_AU |
| local.contributor.authoremail | cain.edie-michell@anu.edu.au | en_AU |
| local.contributor.supervisor | Morrison, Scott | |
| local.contributor.supervisorcontact | scott.morrison@anu.edu.au | en_AU |
| local.description.notes | the author deposited 24/08/2018 | en_AU |
| local.identifier.doi | 10.25911/5d650ee43fe0c | |
| local.mintdoi | mint | |
| local.type.degree | Doctor of Philosophy (PhD) | en_AU |