Dissertation
Bimodules in tensor categories with applications to quantum symmetry
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2022
DOI: 10.17077/etd.006403
Abstract
In the first part of this thesis, we investigate examples of quantum symmetry by attempting to classify Hopf actions of $U_q(\mathfrak{b})$, $U_q(\mathfrak{sl}_2)$, generalized Taft algebras, and the small quantum group on path algebras. We begin by parametrizing these actions using linear algebraic data. Then, if $H$ is one of the quantum groups named above, we view quantum symmetries of path algebras as instances of tensor algebras in $\mathsf{rep}(H)$, as in the framework developed by Etingof, Kinser, and Walton. To understand the "building blocks" of these actions, we attempt to classify the minimal, faithful tensor algebras in $\mathsf{rep}(H)$ by constructing an equivalence between categories of bimodules in $\mathsf{rep}(H)$ and a subcategory of certain finite-dimensional representations of associative algebras, explicitly given in terms of quivers with relations. This allows us to determine whether classification of indecomposable bimodules in these categories is feasible based on the representation types of the quivers.
In general, we can understand quantum symmetries of path algebras by classifying tensor algebras in tensor categories. Motivated by this idea, we turn to bimodules in pointed fusion categories in the second part of this thesis. These bimodules have been classified by Ostrik and Natale. Using this classification, we decompose tensor products of bimodules in pointed fusion categories. This result also has applications in determining fusion rules of group-theoretical fusion categories.
Details
- Title: Subtitle
- Bimodules in tensor categories with applications to quantum symmetry
- Creators
- Amrei Oswald
- Contributors
- Ryan Kinser (Advisor)Frauke Bleher (Committee Member)Victor Camillo (Committee Member)Charles Frohman (Committee Member)Miodrag Iovanov (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2022
- DOI
- 10.17077/etd.006403
- Publisher
- University of Iowa
- Number of pages
- vi, 77 pages
- Copyright
- Copyright 2022 Amrei Oswald
- Language
- English
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 74-77).
- Public Abstract (ETD)
- In mathematics, symmetries are invertible, property-preserving transformations from an object to itself. In the classical setting, the symmetries of an object are given by actions of a group on this object. Quantum symmetry is a generalization of the notion of symmetry to the quantum setting, where actions of groups no longer give a complete description of symmetry. In this setting, quantum symmetries are given by Hopf actions of quantum groups on algebras. In the first part of this thesis, we investigate quantum symmetries of path al- gebras. Path algebras can be described in terms of directed graphs and play an important role in the representation theory of finite-dimensional algebras. Quantum symmetries can also be understood as algebras in the tensor category of represen- tations of the quantum group. As such, the study of algebraic structures in tensor categories is important for understanding quantum symmetry in general. Motivated by this idea, we investigate tensor products of bimodules in pointed fusion categories in the second part of this thesis.
- Academic Unit
- Mathematics
- Record Identifier
- 9984270953102771
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