Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models

Abstract

The subject of our study is the Kazhdan-Lusztig (KL) equivalence in the context of a one-parameter family of logarithmic CFTs based on Virasoro symmetry with the (1,p) central charge. All finite-dimensional indecomposable modules of the KL-dual quantum group - the "full" Lusztig quantum sl(2) at the root of unity - are explicitly described. These are exhausted by projective modules and four series of modules that have a functorial correspondence with any quotient or a submodule of Feigin-Fuchs modules over the Virasoro algebra. Our main result includes calculation of tensor products of any pair of the indecomposable modules. Based on the Kazhdan-Lusztig equivalence between quantum groups and vertex-operator algebras, fusion rules of Kac modules over the Virasoro algebra in the (1,p) LCFT models are conjectured.

40pp. V2: a new introduction, corrected typos, some explanatory comments added, references added

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Cited by in corpus (23)

• Logarithmic Conformal Field Theory: a Lattice Approach
• Lattice fusion rules and logarithmic operator product expansions
• Some remarks on the unrolled quantum group of sl(2)
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• The Nichols algebra of screenings
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
• Logarithmic ^sl(2) CFT models from Nichols algebras. 1
• Lattice W-algebras and logarithmic CFTs
• On the extended W-algebra of type sl_2 at positive rational level
• Renormalized Hennings Invariants and 2+1-TQFTs
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
• Coset construction of logarithmic minimal models: branching rules and branching functions
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• Representations of ${\bar{U}}_{q}s\ell (2|1)$ at even roots of unity
• On the Representation theory of the Infinite Temperley-Lieb algebra
• Staggered and affine Kac modules over $A_1^{(1)}$
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra