Detecting and classifying algebra objects in fusion categories

Hannah, Samuel 2024. Detecting and classifying algebra objects in fusion categories. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

Algebra objects are categorical structures that appear in a wide range constructions and classification problems throughout category theory and mathematical physics. However, they can be difficult to find and describe explicitly so it is of great importance to develop methods of detecting and classifying them. The aim of this work is to develop such methods in the setting of fusion categories, which can be viewed as a categorical generalisation of a ring. We explore two approaches to this problem. We shall describe how different types of monoidal functors can be used to preserve algebraic structures, and shall construct a specific functor with the aim to classify algebra objects in the monoidal center of the category of group-graded vector spaces, ZpVectω Gq. We shall use an explicit description of this category in terms of Yetter-Drinfeld modules over the group Hopf algebra, and explore which algebraic structures can be preserved using the constructed functor. We shall classify a class of Frobenius algebras in terms of a choice of cohomological data. We also look at an alternate approach, which works not with an explicit category but with a fusion ring that can be extracted from the data of a fusion category. The representation theory of these rings will be studied using Non-negative Integer Matrix representations (NIM-reps), and we will describe how NIM-reps can be constructed from algebra objects. We shall look at this relationship in detail, providing a method of using NIM-reps to detect potential algebra structures. We will demonstrate how this technique works by classifying the NIM-reps of 3 families of fusion rings, providing a list of potential algebra objects and a platform to develop this technique further.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Funders:	EPSRC
Date of First Compliant Deposit:	20 November 2024
Last Modified:	20 Nov 2024 10:40
URI:	https://orca.cardiff.ac.uk/id/eprint/174153

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