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

TL;DR

This paper demonstrates that classical W-algebras associated with nilpotent orbits can be constructed via multiple reduction methods, showing their independence from specific choices and clarifying their relation to finite W-algebras.

Contribution

It establishes the equivalence of bihamiltonian, Drinfeld-Sokolov, and Dirac reductions in constructing classical W-algebras, highlighting their dependence solely on nilpotent orbits.

Findings

Classical W-algebras are independent of grading or isotropic subspace choices.

The transverse Poisson structure to a nilpotent orbit is polynomial.

Clarifies the relation between classical and finite W-algebras.

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.