Reduction by stages for finite W-algebras

Naoki Genra, Thibault Juillard*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let g be a simple Lie algebra: its dual space g is a Poisson variety. It is well known that for each nilpotent element f in g, it is possible to construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of g, the Slodowy slice Sf. Given two nilpotent elements f1 and f2 with some compatibility assumptions, we prove Hamiltonian reduction by stages: the slice Sf2 is the Hamiltonian reduction of the slice Sf1. We also state an analogous result in the setting of finite W-algebras, which are quantizations of Slodowy slices. These results were conjectured by Morgan in his Ph.D. thesis. As corollary in type A, we prove that any hook-type W-algebra can be obtained as Hamiltonian reduction from any other hook-type one. As an application, we establish a generalization of the Skryabin equivalence. Finally, we make some conjectures in the context of affine W-algebras.

Original languageEnglish
Article number15
JournalMathematische Zeitschrift
Volume308
Issue number1
DOIs
StatePublished - 2024/09

Keywords

  • Hamiltonian reduction
  • Nilpotent orbits
  • Slodowy slices
  • W-algebras

ASJC Scopus subject areas

  • General Mathematics

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