Communications in Mathematical Physics

, Volume 265, Issue 1, pp 47–93 | Cite as

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

  • B.L. FeiginEmail author
  • A.M. Gainutdinov
  • A.M. Semikhatov
  • I.Yu. Tipunin


The SL(2, ℤ)-representation π on the center of the restricted quantum group Open image in new window at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the Open image in new window ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of Open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of Open image in new window .


Neural Network Complex System Nonlinear Dynamics Group Representation Quantum Computing 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • B.L. Feigin
    • 1
    Email author
  • A.M. Gainutdinov
    • 2
  • A.M. Semikhatov
    • 3
  • I.Yu. Tipunin
    • 3
  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.Physics DepartmentMoscow State UniversityMoscowRussia
  3. 3.Lebedev Physics InstituteMoscowRussia

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