Communications in Mathematical Physics

, Volume 323, Issue 2, pp 663–711 | Cite as

Classical \({\mathcal{W}}\) -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras

  • Alberto De SoleEmail author
  • Victor G. Kac
  • Daniele Valeri


We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical \({\mathcal{W}}\) -algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.


Poisson Structure Hamiltonian Equation Conformal Weight Differential Algebra Isotropic Subspace 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alberto De Sole
    • 1
    Email author
  • Victor G. Kac
    • 2
  • Daniele Valeri
    • 3
  1. 1.Dept. of Math.Univ. of Rome 1RomaItaly
  2. 2.Dept. of Math.MITCambridgeUSA
  3. 3.SISSA-ISASTriesteItaly

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