On a q-Analogue of the McKay Correspondence and the ADE Classification of sl̂2 Conformal Field Theories

✍ Scribed by Alexander Kirillov Jr.; Viktor Ostrik

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 436 KB
• Volume: 171
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2002.2072

✦ Synopsis

The goal of this paper is to give a category theory based definition and classification of ''finite subgroups in U q ðsl 2 Þ'' where q ¼ e pi=l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U q ðsl 2 Þ; 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 b

sl sl 2 at level k ¼ l À 2: We show that ''finite subgroups in U q ðsl 2 Þ'' are classified by Dynkin diagrams of types A n ; D 2n ; E 6 ; E 8 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 ð b sl sl 2 Þ 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.