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

Abstract

To study the representation category of the triplet W-algebra W(p) that is the symmetry of the (1,p) logarithmic conformal field theory model, we propose the equivalent category C(p) of finite-dimensional representations of the restricted quantum group $U_{q}SL2$ at $q=e^{\frac{i\pi }{p}}$. We fully describe the category C(p) 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 W(p)- and $U_{q}SL2$-representation categories is conjectured for all $p\ge 2$ and proved for p=2, the implications including the identifications of the quantum-group center with the logarithmic conformal field theory center and of the universal R-matrix with the braiding matrix.

31 pages, AMSLaTeX, xy, graphicx. V2: minor changes, references added

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Cited by in corpus (7)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Boundary algebras and Kac modules for logarithmic minimal models
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)