Structure of classical (finite and affine) W-algebras

Abstract

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