On rationality of \(W\)-algebras
For a triple (\( \mathfrak{g},f,k\)), where \(\mathfrak g\) is a simple Lie algebra, \(f\) its nilpotent element and \(k \in \mathbb{C}\), let \(W_k( \mathfrak{g},f)\) be the associated simple \(W\)-algebra constructed in the papers [\textit{V. G. Kac, S.-S. Roan} and \textit{M. Wakimoto}, Commun. Math. Phys. 241, 307--342 (2003; Zbl 1106.17026) and \textit{V. G. Kac} and \textit{M. Wakimoto}, Adv. Math. 185, No. 2, 400--458 (2004; Zbl 1049.17025)]. In the paper under review, the authors study the problem of rationality of \(W\)-algebra \(W_k( \mathfrak{g},f)\). They define the notion of exceptional pair \((k, f)\) and conjecture that \(W_k( \mathfrak{g},f)\) is rational if and only if \((k, f)\) is exceptional. In this case irreducible \(W_k( \mathfrak{g},f)\)-modules can be obtained by the quantum Hamiltonian reduction from admissible modules of level \(k\) over the affine Lie algebra \(\hat{\mathfrak{g}}\). The authors support their conjectures by calculation of corresponding characters. They also study the problem of determining the exceptional pairs. The present paper contains important new concepts and results which will be very useful in the theory of vertex algebras.
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