Date of Award

1-1-2020

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (ix, 158 pages) : illustrations.

Dissertation/Thesis Chair

Antun Milas

Committee Members

Cristian Lenart, Alexandre Tchernev

Keywords

Quantum Groups, Topological Quantum Field Theory, Unrolled Quantum Groups, Vertex Operator Algebra, Quantum groups, Vertex operator algebras, Quantum field theory, Hopf algebras, Mathematical physics

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

This dissertation expands on the work of Constantino, Geer, and Patureau-Mirand \cite{CGP}, as well as Creutzig, Milas, and Rupert \cite{CMR} on exploring the unrolled quantum group of $\mathfrak{sl}(2)$ (denoted by $\overline{U_q}^H(\mathfrak{sl}_2)$) and the category of finite dimensional weight $\overline{U_q}^H(\mathfrak{sl}_2)$-modules (denoted by $\mathscr{C}$) for any $p$\textsuperscript{th} root of unity where $p>2$. This unrolled quantum group and category has connections to topological quantum field theory (TQFT) and to vertex operator algebras (VOAs). Significant results of this dissertation include proving that the $\mathscr{C}$ is a ribbon category, finding all simple and projective modules of $\mathscr{C}$ and proving that they form a full sub-tensor category, calculating the modified quantum dimension of all projective modules, as well as finding logarithmic typical modules and their tensor decomposition. Furthermore, we explored the center of the quantum group and showed it is not finitely generated when $p=4$, as well as defining typical modules for higher rank versions of the unrolled quantum group.

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