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TQFTs from Quasi-Hopf Algebras and Group Cocycles
George, Jennifer Lynn

2013, Doctor of Philosophy, Ohio State University, Mathematics.
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.
Thomas Kerler (Advisor)
Henri Moscovici (Committee Member)
Sergei Chmutov (Committee Member)
177 p.

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George, J. (2013). TQFTs from Quasi-Hopf Algebras and Group Cocycles. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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George, Jennifer. "TQFTs from Quasi-Hopf Algebras and Group Cocycles." Electronic Thesis or Dissertation. Ohio State University, 2013. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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George, Jennifer "TQFTs from Quasi-Hopf Algebras and Group Cocycles." Electronic Thesis or Dissertation. Ohio State University, 2013. https://etd.ohiolink.edu/

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