$W$-algebras and the equivalence of bihamiltonian, Drinfeld-Sokolov and Dirac reductions

Abstract

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.

revised for clarity

References in corpus (4)

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies
• On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions
• Frobenius manifolds from regular classical $W$-algebras