Deformed Virasoro algebras and \({\mathcal W}\)-algebras from elliptic algebras
type: Item
instance of: scholarly article
title: Deformed Virasoro algebras and \({\mathcal W}\)-algebras from elliptic algebras
zbMATH Open document ID: 1032.17019
DOI: 10.1007/BF02551401
published in: Theoretical and Mathematical Physics
publication date: 2004-03-11T00:00:00Z
review text: The results of Avan, Frappat, Rossi and Sorba concerning the construction of the \(q\)-deformed Virasoro and \(W\)-algebras from the elliptic algebras \(A_{p,q} (\widehat {sl}(N)_c)\) [\textit{O. Foda} et al., Lett. Math. Phys. 32, 259-268 (1994; Zbl 0821.17011); \textit{M. Jimbo}, \textit{M. Lashkevich}, \textit{T. Miwa} and \textit{Ya. Pugai} Phys. Lett. A 229, 285-292 (1997)] in the classical (Poisson structures) and in the quantum (exchange algebras) cases are summarized. In the beginning, the existence of an extended center in the algebra \(A_{p,q}(\widehat{sl}(N)_c)\) is proved, and the Poisson structures in this center are computed, which are identified as the \(q\)-deformed Virasoro algebras \(Vir_q(sl(N))\). Then the existence of the closed (quadratic) exchange algebras is shown, which admit a classical limit (commuting algebras) and, at last, realize the new quantizations of the Poisson structures. In the conclusion, the essential features of the extension of the presented computations to the higher-spin fields are briefly described and the possible future developments of the approach under consideration are indicated. The paper is dedicated to the memory of Prof. M. Saveliev.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81Q50
Mathematics Subject Classification ID: 17B68
zbMATH DE Number: 1551592
zbMATH Keywords: \(q\)-deformed Virasoro algebras
zbMATH Keywords: \(W\)-algebras
zbMATH Keywords: elliptic algebras
zbMATH Keywords: Poisson structures
zbMATH Keywords: exchange algebras
author: P. Sorba
reviewed by: A. A. Bogush
MaRDI profile type: Publication
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cites work: Q3324796
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full work available at URL: https://doi.org/10.1007/bf02551401
OpenAlex ID: W2052139989
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description: scientific article; zbMATH DE number 1551592