On a q-Analogue of the McKay Correspondence and the ADE Classification of 2 Conformal Field Theories

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The goal of this paper is to give a category theory based definition and classification of “finite subgroups in Uq(2)” where q=ei/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(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 2 at level k=l-2. We show that “finite subgroups in Uq(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 (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. © 2002 Elsevier Science (USA).

Document Type: Research Article

DOI: http://dx.doi.org/10.1006/aima.2002.2072

Affiliations: 1: Department of Mathematics, 3-116 Mathematics Building, SUNY at Stony Brook, Stony Brook, New York, 11794 2: Department of Mathematics, MIT, Cambridge, Massachusetts, 02139

Publication date: November 1, 2002

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