Structure of classical (finite and affine) $\mathcal W$-algebras

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First, we derive an explicit formula for the Poisson bracket of the classical finite \mathcal W -algebra \mathcal W^{\mathrm {fin}}(\mathfrak g,f) , the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra \mathfrak 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 \mathcal W -algebra \mathcal W(\mathfrak g,f) . As an immediate consequence, we obtain a Poisson algebra isomorphism between \mathcal W^{\mathrm {fin}}(\mathfrak g,f) and the Zhu algebra of \mathcal W(\mathfrak g,f) . We also study the generalized Miura map for classical \mathcal W -algebras.