ON q-ANALOG OF MCKAY CORRESPONDENCE AND ADE CLASSIFICATION OF ŝl2 CONFORMAL FIELD THEORIES

The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = eπi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); 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 ŝl2 at level k = l − 2. We show that “finite subgroups in Uq(sl2)” are classified by Dynkin diagrams of types An,D2n, E6, E8 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 (ŝl2)k conformal field theory.