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Edie-Michell, Cain.
The classification of *categories* generated by an object of small dimension
.

Degree: 2018, Australian National University

URL: http://hdl.handle.net/1885/146634

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 , 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 √{2+√{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{π}{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

…*Unitary* *fusion* *categories* are an important class of algebraic objects, providing a
unifying… …algebras, *unitary* *fusion* *categories* appear as the even part of
the standard invariant of a… …subfactor. Using the machinery available to *unitary*
*fusion* *categories*, many strong results… …*fusion*
*categories* [11]. For representation theory, *unitary* *fusion* *categories* arise as… …classification of *unitary* *fusion* *categories*
generated by a self-dual object of dimension less than 2…

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APA (6^{th} Edition):

Edie-Michell, C. (2018). The classification of categories generated by an object of small dimension . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/146634

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Edie-Michell, Cain. “The classification of categories generated by an object of small dimension .” 2018. Thesis, Australian National University. Accessed December 05, 2020. http://hdl.handle.net/1885/146634.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Edie-Michell, Cain. “The classification of categories generated by an object of small dimension .” 2018. Web. 05 Dec 2020.

Vancouver:

Edie-Michell C. The classification of categories generated by an object of small dimension . [Internet] [Thesis]. Australian National University; 2018. [cited 2020 Dec 05]. Available from: http://hdl.handle.net/1885/146634.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Edie-Michell C. The classification of categories generated by an object of small dimension . [Thesis]. Australian National University; 2018. Available from: http://hdl.handle.net/1885/146634

Not specified: Masters Thesis or Doctoral Dissertation