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Quantum Groups, Quantum Categories and Quantum Field Theory

  • Book
  • © 1993

Overview

Authors:
  1. Jürg Fröhlich
  2. Thomas Kerler

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1542)

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About this book

This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.

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Table of contents (8 chapters)

  1. Front Matter

    Pages I-VII
    Download chapter PDF

  2. Introduction and survey of results

    • Jürg Fröhlich, Thomas Kerler
    Pages 1-16

  3. Local quantum theory with braid group statistics

    • Jürg Fröhlich, Thomas Kerler
    Pages 17-44

  4. Superselection sectors and the structure of fusion rule algebras

    • Jürg Fröhlich, Thomas Kerler
    Pages 45-101

  5. Hopf algebras and quantum groups at roots of unity

    • Jürg Fröhlich, Thomas Kerler
    Pages 102-118

  6. Representation theory of U redq (sℓ 2)

    • Jürg Fröhlich, Thomas Kerler
    Pages 119-140

  7. Path representations of the braid groups for quantum groups at roots of unity

    • Jürg Fröhlich, Thomas Kerler
    Pages 141-175

  8. Duality theory for local quantum theories, dimensions and balancing in quantum categories

    • Jürg Fröhlich, Thomas Kerler
    Pages 176-283

  9. The quantum categories with a generator of dimension less than two

    • Jürg Fröhlich, Thomas Kerler
    Pages 284-411

  10. Back Matter

    Pages 412-431
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