On q-analog of McKay correspondence and ADE classification of sl^(2) conformal field theories

TL;DR

This paper classifies finite subgroups of quantum sl(2) at roots of unity using ADE Dynkin diagrams, extending classical McKay correspondence to quantum groups and relating to conformal field theory invariants.

Contribution

It introduces a definition of finite subgroups in U_q sl(2) at roots of unity and establishes their classification via ADE Dynkin diagrams, linking quantum groups with conformal field theory.

Findings

Finite subgroups correspond to ADE Dynkin diagrams with Coxeter number l.

Classification generalizes classical McKay correspondence to quantum groups.

Relation established between subgroups and modular invariants in conformal field theory.

Abstract

The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\pi\i/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl^(2) at level k=l-2. We show that ``finite subgroups in U_q sl(2)'' are classified by Dynkin diagrams of types A_n, D_{2n}, E_6, E_8 with Coxeter number equal to $l$, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in (sl(2))_k conformal field theory.