Factorizable ribbon quantum groups in logarithmic conformal field theories

TL;DR

This paper reviews the properties of factorizable ribbon quantum groups at roots of unity, highlighting their role in logarithmic conformal field theories and their modular properties.

Contribution

It introduces and discusses the structure of factorizable ribbon quantum groups that are dual to logarithmic conformal field theories, emphasizing their modular and algebraic features.

Findings

Quantum groups at roots of unity are factorizable and ribbon but not quasitriangular.

Modular group representation on the quantum group center matches that on logarithmic CFT characters.

Provides insights into the algebraic structure underlying logarithmic conformal field theories.

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.