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

@article{DeSole2012Classical, title={Classical \$\$\{\mathcal\{W\}\}\$\$W -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras}, author={Alberto De Sole and Victor G. Kac and Daniele Valeri}, journal={Communications in Mathematical Physics}, year={2012}, volume={323}, pages={663-711} }

We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical $${\mathcal{W}}$$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…

## 45 Citations

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The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel’d–Sokolov (D–S) reduction…

### Classical $${\mathcal{W}}$$W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

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We derive explicit formulas for λ-brackets of the affine classical $${\mathcal{W}}$$W -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra $${\mathfrak{g}}$$g .…

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We introduce a new family of Poisson vertex algebras
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### Supersymmetric Bi-Hamiltonian Systems

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### MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $${\mathcal {W}}$$W-algebras

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We give an introduction to the Mathematica packages MasterPVA and MasterPVAmulti used to compute $$\lambda $$λ-brackets in Poisson vertex algebras, which play an important role in the theory of…

### Classical $$\mathcal {W}$$-algebras for Centralizers

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