Factorizable ribbon quantum groups in logarithmic conformal field theories

Abstract

ABSTRACT. We review the properties of quantum groups occurring as Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.

Double affine Hecke algebra in logarithmic conformal field theory

“…The center of U q s (2) coincides with the symmetrization of Z, and C = −(X + X −1 ) coincides with the U q s (2) Casimir element. Probably, the entire representation Z can be realized as a subalgebra of U q s (2) such that X is realized as the multiplication by an element of U q s (2).…”

“…• Under this isomorphism, the eigenvectors of C correspond to the Radford images of the characters of irreducible representations of U q s (2), and the eigenvectors of H correspond to the Drinfeld images of these characters.…”

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

“…The interest in the GR structure of U q ≡ U q sℓ(2) has been justified by its relation to the fusion ring of the corresponding CFT model. This relation, noticed by a number of physicists at the outbreak of interest in quantum groups in the late 1980's was made precise by the KazhdanLusztig correspondence (of the 1990's) verified for the logarithmic c 1p Virasoro model in [8,9,13,18] and discussed for logarithmic extensions of minimal and sℓ(2) k conformal theories ( [10,11,35,36]). The present paper considers instead the infinite dimensional…”

Higher String Functions, Higher-Level Appell Functions, and the Logarithmic $${{{\widehat{s\ell}2_k/u(1)}}}$$ CFT Model

“…The interesting representation theory may be considered the basic reason underlying fascinating features of logarithmic conformal field models and their links with several related problems, e.g., in [7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In particular, modular group representations generated from characters in logarithmic models are of a different structure than the modular group representations occurring in rational models (cf.…”