$W$algebras and the equivalence of bihamiltonian, DrinfeldSokolov and Dirac reductions
Abstract
We prove that the classical $W$algebra associated to a nilpotent orbit in a simple Liealgebra can be constructed by preforming bihamiltonian, DrinfeldSokolov 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2009
 arXiv:
 arXiv:0911.2116
 Bibcode:
 2009arXiv0911.2116D
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Representation Theory;
 Mathematics  Symplectic Geometry;
 37K10;
 35D45
 EPrint:
 revised for clarity