## Structure of classical (finite and affine) W-algebras

##### Author(s)

De Sole, Alberto; Kac, Victor; Valeri, Daniele
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First, we derive an explicit formula for the Poisson bracket of the classical finite W- Algebra W[superscript fin] (g, f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f . On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W- Algebra W(g, f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W[superscript fin] (g, f) and the Zhu algebra of W(g, f).We also study the generalized Miura map for classical W- Algebras. Keywords: W-algebra, Poisson algebra, Poisson vertex algebra, Slodowy slice, Hamiltonian reduction, Zhu algebra, Miura map

##### Date issued

2016-07##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Journal of the European Mathematical Society

##### Publisher

European Mathematical Publishing House

##### Citation

De Sole, Alberto, et al. “Structure of Classical (Finite and Affine) W-Algebras.” Journal of the European Mathematical Society, vol. 18, no. 9, 2016, pp. 1873–908.

Version: Original manuscript

##### ISSN

1435-9855