Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


MainDocument.pdf (1.18 MB)

ETD Abstract Container

Abstract Header

Quantum Symmetries for Quantum Spaces

Hernandez Palomares, Roberto
http://orcid.org/0000-0002-7647-8444
http://rave.ohiolink.edu/etdc/view?acc_num=osu1618524562050801

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis consists of three self-contained papers from my graduate work at The Ohio State University. In Chapter 2, we introduce most of the necessary results, notation, techniques, and examples used in the reminder of this document, involving unitary tensor categories (UTC) and C*-2-categories. In Chapter 3 we introduce Q-Systems in a $\dag$-2-category and their bimodules, and define the Q-System completion of a $\dag$-2-category. We later prove $\rCorr,$ the C*-2-category of right C*-correspondences, is Q-System complete by constructing an inverse to the canonical embedding of $\rCorr$ into its Q-System completion $\QSys(\rCorr)$. In Chapter 4 we introduce generalized Temperley-Lieb-Jones ($\TLJ$) $\dag$-2-categories associated to weighted bidirected graphs, meant to represent lattices of inclusions of finite-index subfactors. We formally define unitary modules for these generalized $\TLJ$ $\dag$-2-categories as strong $\dag$-2-functors into $\BigHilb,$ the $\dag$-2-category of row-finite separable bigraded Hilbert spaces. We then classify these modules up to $\dag$-2-equivalence in terms of weighted bi-directed $\Gamma$-fair and balanced graphs, in the spirit of Yamagami's classification of fiber functors on $\TLJ(\delta)$ categories [Yam04], and De Commer and Yamashita's classification of unitary modules for ${\rm Rep(SU}_q(2))$ [DCY15]. In Chapter 5 we show that every UTC acts on some separable simple and monotracial C*-algebra, using diagrammatic, functional analytic and free probabilistic techniques. We construct a fully-faithful bi-involutive strong tensor functor onto a full subcategory of fgp bimodules over a simple exact separable unital monotracial C*-algebra. This C*-algebra is built solely from the category using the Guionnet-Jones-Shlyakhtenko construction introduced in [GJS11]. Proving the aforementioned properties of this functor involves a certain Hilbertification tensor functor, whose source is the UTC of finitely generated projective Hilbert C*-bimodules with certain tracial compatibilities, into the category of bi-finite spherical bimodules over an interpolated free group factor. The composite of these two functors recovers the UTC-action constructed in {BHP12].
David Penneys (Advisor)
Thomas Kerler (Committee Member)
Sachin Gautam (Committee Member)
170 p.
Mathematics
Operator Algebras, Quantum Algebra, Category Theory

Recommended Citations

Citations

  • Hernandez Palomares, R. (2021). Quantum Symmetries for Quantum Spaces [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618524562050801

    APA Style (7th edition)

  • Hernandez Palomares, Roberto. Quantum Symmetries for Quantum Spaces. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1618524562050801.

    MLA Style (8th edition)

  • Hernandez Palomares, Roberto. "Quantum Symmetries for Quantum Spaces." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618524562050801

    Chicago Manual of Style (17th edition)

osu1618524562050801
104
© 2021, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.