01002nas a2200133   4500008004100000245005800041210005500099260001000154520060700164100002100771700002000792700002000812856003600832       2014                        en  d00aStructure of classical (finite and affine) W-algebras0 aStructure of classical finite and affine Walgebras  bSISSA3 aFirst, 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.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele  uhttp://hdl.handle.net/1963/7314