Dissertation
Skein theory and its applications to tensor categories
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2022
DOI: 10.25820/etd.006546
Abstract
This thesis is composed of two main parts. The first part focuses on families of unitary fusion categories $\C$ that contain at least one selfdual object, $q$, satisfying a fusion rule of the form $q\otimes q \cong \bm{1}\oplus\bigoplus^k_{i=1}x_{i}$. Assuming the category is ribbon, we classify all associated ribbon structures when $k=1,2$. In particular, we also consider the cases when $q$ is antisymmetrically selfdual. Further, a comprehensive discussion on construction of a ``canonical diagrammatic basis'' for any unitary fusion category is provided.\\
In the second part, we focus on spider categories associated to the representation category of the quantum group $U_q(SL_n)$. We work with the full subcategory of the spider category, $\mathcal{S}p(SL_n)$ from \cite{ckm}, whose objects are monoidally generated by the standard representation and its dual, and show that this category is equivalent as a spherical braided (ribbon) category to the spider category which is a quotient category coming from Sikora's relations in \cite{sik}. This answers a question from Morrison's Ph.D. thesis \cite{mor}.We also show that the full subcategory (with standard representation and its dual as objects) of MOY's quotient category defined using their relations in \cite{moy} is equivalent as a spherical braided category to the HOMFLYPT category, which is a quotient category with standard representation and its dual as objects with relations coming from the HOMFLYPT relations.
Details
- Title: Subtitle
- Skein theory and its applications to tensor categories
- Creators
- Anup Poudel
- Contributors
- Charles Frohman (Advisor)Benjamin Cooper (Committee Member)Ryan Kinser (Committee Member)Mohammad Tehrani (Committee Member)Miodrag Iovanov (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2022
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.006546
- Number of pages
- viii, 95 pages
- Copyright
- Copyright 2022 Anup Poudel
- Language
- English
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 92-95).
- Public Abstract (ETD)
This thesis focuses is composed of two parts. The first part focuses on the study of mathematical knots using tools coming from abstract algebraic objects. There are many algebraic tools developed to distinguish two knotted surfaces sitting in an ambient space. We investigate general properties of one of these tools arising from representation theory of Lie algebras.
The second part involves the study of tensor categories which is an active area of research. We apply combinatorial (diagrammatic) and geometric tools to study tensor categories. This approach uses diagrammatic methods to obtain information regarding the knot invariants coming from these categories. Further, using combinatorial methods, we show equivalence between certain known tensor categories.
- Academic Unit
- Mathematics
- Record Identifier
- 9984284951802771
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