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JennysThesis.pdf (13.2 MB)
ETD Abstract Container
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TQFTs from Quasi-Hopf Algebras and Group Cocycles
Author Info
George, Jennifer Lynn
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1369834588
Abstract Details
Year and Degree
2013, Doctor of Philosophy, Ohio State University, Mathematics.
Abstract
In three dimensions, a topological quantum field theory, or TQFT, is a functor from the category of 3-dimensional framed cobordisms to the category of vector spaces. Two well-known TQFTs are the Hennings TQFT and the Dijkgraaf-Witten TQFT. The Hennings TQFT is built from a link invariant, by applying elements of a Hopf algebra in a systematic way to tangle diagrams. The Dijkgraaf-Witten TQFT is built by counting principal bundles on a 3-manifold which have been weighted by a 3-cocycle. We prove that the Hennings TQFT applied on the double of the group algebra is equivalent to the Dijkgraaf-Witten TQFT applied on a trivial cocycle. In order to extend this result to the more general case of a non-trivial cocycle, we discuss the notion of a quasi-Hopf algebra, which is an almost-cocommutative Hopf algebra. We then extend the definition of the Hennings TQFT so that instead of applying elements of a Hopf algebra to the tangle, we instead apply elements of a quasi-Hopf algebra. The specific quasi-Hopf algebra in which we are interested is the twisted double of the group algebra, where the twisting occurs via a 3-cocycle. Finally, we conjecture that the Hennings TQFT applied on the twisted double of the group algebra is equivalent to the Dijkgraaf-Witten TQFT applied on the same cocycle.
Committee
Thomas Kerler (Advisor)
Henri Moscovici (Committee Member)
Sergei Chmutov (Committee Member)
Pages
177 p.
Subject Headings
Mathematics
Keywords
TQFTs
;
Hennings TQFT
;
Dijkgraaf Witten TQFT
;
Quasi-Hopf Algebras
;
Quantum Invariants
;
3-manifolds
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Citations
George, J. L. (2013).
TQFTs from Quasi-Hopf Algebras and Group Cocycles
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1369834588
APA Style (7th edition)
George, Jennifer.
TQFTs from Quasi-Hopf Algebras and Group Cocycles.
2013. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1369834588.
MLA Style (8th edition)
George, Jennifer. "TQFTs from Quasi-Hopf Algebras and Group Cocycles." Doctoral dissertation, Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1369834588
Chicago Manual of Style (17th edition)
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Document number:
osu1369834588
Download Count:
1,177
Copyright Info
© 2013, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.