On a q-Analogue of the McKay Correspondence and the ADE Classification of [formula omitted]2 Conformal Field Theories

Abstract

The goal of this paper is to give a category theory based definition and classification of “finite subgroups in U q ( s l 2)” where q= e πi/ l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U q ( s l 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 s l ̂ 2 at level k= l−2. We show that “finite subgroups in U q ( s l 2)” are classified by Dynkin diagrams of types A n , D 2 n , 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 ( s l ̂ 2) k conformal field theory. The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory.