On rationality of \(W\)-algebras
type: Item
instance of: scholarly article
title: On rationality of \(W\)-algebras
zbMATH Open document ID: 1221.17033
author: Victor G. Kac
author: Minoru Wakimoto
published in: Transformation Groups
publication date: 2009-06-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0711.2296
review text: 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.
reviewed by: Dražen Adamović
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B68
zbMATH DE Number: 5565876
zbMATH Keywords: W-algebras
zbMATH Keywords: rationality
zbMATH Keywords: affine Lie algebras
zbMATH Keywords: admissible representations
MaRDI profile type: Publication
OpenAlex ID: W1972120167
arXiv ID: 0711.2296
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DOI: 10.1007/S00031-008-9028-7
description: scientific article