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 Cp of finite-dimensional representations of the restricted quantum group Uqs (2) at q = e iπ/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 W (p)-and Uqs (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.

Some remarks on the unrolled quantum group of sl(2)

“…Typical modules are projective and there exists a unique trace on Proj up to a scalar. Its associated modified dimension is given by formula (18).…”

“…These two blocks form a category similar to the category U q sl(2)-mod of representations of the standard small quantum group U q sl (2). The category U q sl(2)-mod, equivalent to that of modules over the triplet vertex operator algebra W(p) (see [26,28]), has been intensively studied in logarithmic conformal field theories (CFT) associated to the (1, p) triplet algebras (see [5,6,12,17,18]). In particular, some results of Section 6 are similar to the analysis of projective modules in [18].…”

Toward logarithmic extensions of $$\widehat{s\ell }(2)_k $$ conformal field models

“…Logarithmic conformal field theory models have nice quantum group counterparts, which capture at least part of the structure of logarithmic models and are therefore quite useful in investigating them [11], [31], [32]. On the quantum group side, the central role is played by two objects, the center and the Grothendieck ring.…”

“…Mathematically, various aspects of logarithmic conformal models and related structures in vertex-operator algebras were considered in [15]- [18]; relations to statistical mechanics models were studied in [19]- [21]; various aspects of logarithmic models were developed in [8], [10], [22]- [29]; relations to quantum groups and a "nonsemisimple" extension of the Kazhdan-Lusztig correspondence [30] were studied in [11], [31], [32]. Logarithmic conformal field theories can be viewed as an extension of rational conformal field theories [33]- [36] to the case involving indecomposable representations of the chiral algebra.…”

On the structure of ${\rm End}_{u_k(2)}(\Omega _k^{\otimes r})$ End uk(2)(Ωk⊗r)

“…For example, certain important relation between infinitesimal quantum groups and logarithmic conformal field theories has been found in Ref. 12.…”