On a \(q\)-analogue of the McKay correspondence and the ADE classification of \(\mathfrak{sl}_{2}\) conformal field theories
type: Item
instance of: scholarly article
title: On a \(q\)-analogue of the McKay correspondence and the ADE classification of \(\mathfrak{sl}_{2}\) conformal field theories
zbMATH Open document ID: 1024.17013
published in: Advances in Mathematics
publication date: 2003-03-26T00:00:00Z
review text: A famous result of McKay establishes a bijective correspondence between finite subgroups \(\Gamma \subseteq SU(2)\) and affine simply laced Dynkin diagrams, such that the vertices of the Dynkin diagram correspond to the irreducible representations of \(\Gamma\) and tensoring with the fundamental 2-dimensional representation of \(SU(2)\) has matrix \(2 - A\) in this basis, where \(A\) is the Cartan matrix of the Dynkin diagram. The paper under review proposes an analogue of this correspondence for `finite subgroups' of \(U_q(\mathfrak{sl}_2)\), where \(q = e^{i\pi / l}\) is a root of unity: these are commutative associative algebras in the category \(\mathcal C\), where \(\mathcal C\) is the semisimple part of the category of representations of \(U_q(\mathfrak{sl}_2)\) introduced by \textit{H. H. Andersen} and \textit{J. Paradowski} [Commun. Math. Phys. 169, 563-588 (1995; Zbl 0827.17010)]. Such an algebra \(A\) gives rise naturally to a module category \(\operatorname {Rep} A\) over \(\mathcal C\), endowed with functors \(F: \mathcal C \to \operatorname {Rep} A\) and \(G: \operatorname {Rep} A \to \mathcal C\). The main result of the paper says that such commutative associative algebras \(A\) are in bijective correspondence with Dynkin diagrams of types \(A_n\), \(D_{2n}\), \(E_6\), \(E_8\) with Coxeter number \(l\), the vertices of the Dynkin diagram corresponding to irreducible objects of \(\operatorname {Rep} A\) and the matrix of tensor product with \(F(V)\) in this basis is \(2 - A\), where \(A\) is the associated Cartan matrix and \(V\) is the fundamental 2-dimensional representation of \(U_q(\mathfrak{sl}_2)\). The proof of this result is based on the classification of the integer modules over the Grothendieck ring of \(\mathcal C\) due to \textit{P. Etingof} and \textit{M. Khovanov} [Int. Math. Res. Not. 1995, 235-247 (1995; Zbl 0854.17016)]. A different approach appears in the work of \textit{A. Ocneanu} [Lond. Math. Soc. Lect. Note Ser. 136, 119-172 (1988;…
reviewed by: Sonia Natale
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 1888323
zbMATH Keywords: McKay correspondence
zbMATH Keywords: modular category
zbMATH Keywords: ADE classification
author: Alexander jun. Kirillov
author: Victor Ostrik
MaRDI profile type: Publication
full work available at URL: aima.2002.2072
OpenAlex ID: W2065247876
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DOI: 10.1006/AIMA.2002.2072
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description: scientific article; zbMATH DE number 1888323