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

The goal of this paper is to give a category theory based definition and classification of “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. The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory.