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Conformal Field Theories, Graphs and Quantum Algebras

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Book cover MathPhys Odyssey 2001

Part of the book series: Progress in Mathematical Physics ((PMP,volume 23))

Abstract

This article reviews some recent progress in our understanding of the structure of rational conformal field theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions—open, twisted—are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract “quantum” algebra, whose 6j-and 3j-symbols contain essential information on the operator product algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of “Weak C*- Hopf algebras”.

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Author information

Authors and Affiliations

  1. School of Computing and Mathematics, University of Northumbria, NEI 8ST, Newcastle upon Tyne, UK

    Valentina Petkova

  2. Institute for Nuclear Research and Nuclear Energy, 72 Tzarigrasko Chaussee, 1784, Sofia, Bulgaria

    Valentina Petkova

  3. Service de Physique Théorique, CEA Saclay, 91191, Gif-sur-Yvette cedex, France

    Jean-Bernard Zuber

Authors
  1. Valentina Petkova
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  2. Jean-Bernard Zuber
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Editor information

Editors and Affiliations

  1. RIMS, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Kashiwara

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Tetsuji Miwa

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Petkova, V., Zuber, JB. (2002). Conformal Field Theories, Graphs and Quantum Algebras. In: Kashiwara, M., Miwa, T. (eds) MathPhys Odyssey 2001. Progress in Mathematical Physics, vol 23. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0087-1_15

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  • DOI: https://doi.org/10.1007/978-1-4612-0087-1_15

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6605-1

  • Online ISBN: 978-1-4612-0087-1

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