Date of Award

2026

Abstract

Tensor categories are algebraic objects used by physicists to describe symmetry, by computer scientists to describe programs, and by logicians to describe theorem and proof. Mathematicians invented them to capture the similarities of the representation theories of common algebraic objects including groups, algebras, and Lie algebras. In this dissertation we give presentations in terms of graphical generators and relations for two tensor categories arising from the two quantum subgroups of quantum G2.

We accomplish this by embedding a tensor-generating object's planar algebra into the graph planar algebra on that object's module fusion graph. Such an embedding allows one to explore the new category using concrete computations, and uncover new relations. To this end, we provide a modular collection of software allowing one to perform calculations inside of a graph planar algebra.

Document Type

Dissertation

First Advisor

Cain Edie-Michell

Second Advisor

Dmitri Nikshych

Third Advisor

Junhao Shen

Department or Program

Mathematics

Degree Name

Doctor of Philosophy

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