On a q-analogue of the McKay correspondence and the ADE classification of SI₂ conformal field theories

The goal of this paper is to give a category theory based definition and classification of "finite subgroups in U_q(sI₂)" 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(sI₂); 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 sI₂ at level k = l - 2. We show that "finite subgroups in U_q(sI₂)" are classified by Dynkin diagrams of types A_n, D_2n, 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 (sI₂)_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.