Theoretical and Mathematical Physics

, Volume 148, Issue 3, pp 1210–1235 | Cite as

Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT

  • A. M. Gainutdinov
  • A. M. Semikhatov
  • I. Yu. Tipunin
  • B. L. Feigin
Article

Abstract

To study the representation category of the triplet W-algebra \(\mathcal{W}\left( p \right)\) that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finite-dimensional representations of the restricted quantum group Ūqsℓ(2) at \(\mathfrak{q} = e^{{{i\pi } \mathord{\left/ {\vphantom {{i\pi } p}} \right. \kern-\nulldelimiterspace} p}} \). We fully describe the category Cp by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the \(\mathcal{W}\left( p \right)\)-and Ūqsℓ(2)-representation categories is conjectured for all p = 2 and proved for p = 2. The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal R-matrix with the braiding matrix.

Keywords

Kazhdan-Lusztig correspondence quantum groups logarithmic conformal field theories indecomposable representations 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

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

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