Mathematics

First, we derive an explicit formula for the Poisson bracket of the classical
finite W-algebra W^{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^{fin}(g,f) and the Zhu algebra of W(g,f). We also
study the generalized Miura map for classical W-algebras.