Deformed Virasoro algebras and \({\mathcal W}\)-algebras from elliptic algebras
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.
scientific article; zbMATH DE number 1551592
scientific article