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Quantum Groups: (English)

Quantum Groups: (English)

          
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About the Book

The category of representations of a quantum group has tensor products, duals, and a braiding making it a ribbon tensor category. The universal ribbon category has a topological description as the category of ribbon tangles, thus any ribbon category yields an invariant of knots and links. This process was developed by Reshetikhin and Turaev, based on earlier work of Drinfel'd, Jones, Witten, and others. In this thesis we will develop several generalizations of this process. In 2-dimensions, lattice topological quantum field theories relate semisimple rings to invariants of surfaces. This correspondence has an application to the dimensions of group representations called Mednykh's formula. We give a rapid proof of this formula with a special emphasis on the nonorientable case. For a quantum group you can interpret the 360-degree full-twist of a ribbon as a map between representations. We extend this picture to allow for 180-degree half-twists. One benefit is that the braiding itself can be recovered from the half-twist, but not from the full-twist. We formalize the notion of a half-ribbon Hopf algebra, and prove that every quantum group Uqg is an example. There is a quotient of the semisimplified category of representations of Uqsl2 at q a root of unity, called D 2n in the subfactor literature. We give a skein theoretic description of this tensor category. As applications we prove several new identities between knot polynomials and prove a conjecture of Rowell's on the unitarity of G2 13 .


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Product Details
  • ISBN-13: 9781243667014
  • Publisher: Proquest, Umi Dissertation Publishing
  • Publisher Imprint: Proquest, Umi Dissertation Publishing
  • Height: 246 mm
  • No of Pages: 202
  • Series Title: English
  • Weight: 372 gr
  • ISBN-10: 124366701X
  • Publisher Date: 01 Sep 2011
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Spine Width: 11 mm
  • Width: 189 mm


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