%0 Journal Article
%T $W$-algebras and the equivalence of bihamiltonian, Drinfeld-Sokolov and Dirac reductions
%A Yassir Dinar
%J Mathematics
%D 2009
%I arXiv
%X We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld-Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.
%U http://arxiv.org/abs/0911.2116v4