On a \(q\)-analogue of the McKay correspondence and the ADE classification of \(\mathfrak{sl}_{2}\) conformal field theories
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; Zbl 0696.46048)] in connection with the classification of subfactors. Attached to \(A\) one also has a modular category \(\operatorname {Rep}^0 A\): this is the full subcategory of \(\operatorname {Rep} A\) consisting of objects \(\mu_V: A \otimes V \to V\) such that \(\mu_V R_{VA} R_{AV}= \mu_V\), where \(R\) is the braiding. The fact that this modular category gives rise to a modular invariant relates the present paper with the results of \textit{A. Cappelli, C. Itzykson} and \textit{J. B. Zuber} [Commun. Math. Phys. 113, 1-26 (1987; Zbl 0639.17008)] on conformal field theory.
scientific article; zbMATH DE number 1888323
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