Dissertation
Classifying spherical categories by fusion rules
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2021
DOI: 10.17077/etd.006132
Abstract
Taking webs as fundamental and fusion rules as data, a pair of state functions is constructed. One function gives Kuperberg's $\mathfrak{sp}(4)$ quantum invariant while the other gives an $\mathfrak{osp}(1,4)$ invariant. The construction shows that any spider satisfying the $\mathfrak{sp}(4)$ fusion rules must be isomorphic to $Rep (\mathfrak{sp}(4))$ or $Rep (\mathfrak{osp}(1,4))$. The relationship between the two state functions is described by a quantum covering group.
Details
- Title: Subtitle
- Classifying spherical categories by fusion rules
- Creators
- Shawn Nevalainen
- Contributors
- Charles Frohman (Advisor)Frauke Bleher (Committee Member)Benjamin Cooper (Committee Member)Miodrag Iovanov (Committee Member)Ryan Kinser (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2021
- DOI
- 10.17077/etd.006132
- Publisher
- University of Iowa
- Number of pages
- ix, 87 pages
- Copyright
- Copyright 2021 Shawn Nevalainen
- Language
- English
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 84-87).
- Public Abstract (ETD)
In this work we present new discoveries at the intersection of three mathematical fields: category theory, knot theory and representation theory.
Category theory provides an overarching framework for mathematics and unifies different fields. It distills the essence of structures and processes that arise across disciplines and presents them in a universal way so that a statement in category theory implies statements in several different domains at once as special cases.
Knot theory is the study of mathematical knots. A physical model of a mathematical knot is a rope twisting around itself with its ends fused. Knot theory seeks to provide ways to distinguish one knot from another and to connect the properties of knots to those of other mathematical objects.
Representation theory is a means to study mathematical objects by associating to them objects from another context which are more easily understood. Information about structure or the results of computations can be formally translated from one context to another in a rigorously meaningful way.
We classify certain category theoretic structures called spiders by using representation theory to analyze the building blocks of a special type of knot. We develop a method of distinguishing between these knots, and information from the construction translates to information about categories.
- Academic Unit
- Mathematics
- Record Identifier
- 9984097278202771
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