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

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The goal of this paper is to classify ``finite subgroups in Uq sl(2)'' where q=eπı/ˡ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq 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 Uq sl(2)'' 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 (sl(2))ₖ conformal field theory.