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

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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.

Original languageEnglish
Pages (from-to)30-42
Number of pages13
JournalJournal of Geometry and Physics
Volume84
DOIs
Publication statusPublished - 2014

Fingerprint

Nilpotent Orbits
W-algebras
Paul Adrien Maurice Dirac
equivalence
algebra
Equivalence
orbits
Poisson Structure
Simple Lie Algebra
Grading
Transverse
Subspace
polynomials
Polynomial

Keywords

  • Bihamiltonian reduction
  • Dirac reduction
  • Drinfeld-Sokolov reduction
  • Slodowy slice
  • Transverse poisson structure
  • W-algebras

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

Cite this

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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.",
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PY - 2014

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N2 - 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.

AB - 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.

KW - Bihamiltonian reduction

KW - Dirac reduction

KW - Drinfeld-Sokolov reduction

KW - Slodowy slice

KW - Transverse poisson structure

KW - W-algebras

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DO - 10.1016/j.geomphys.2014.06.003

M3 - Article

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SP - 30

EP - 42

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

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