Reduction by stages for finite W-algebras

2022/12/12 by Naoki Genra, Genra, Naoki, Thibault Juillard +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2212.06022

openalex publication_date 2022/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakg be a simple Lie algebra: its dual space \mathfrakg^* is a Poisson variety. It is well known that for each nilpotent element f in \mathfrakg, it is possible to construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of \mathfrakg^*, the Slodowy slice S_f. Given two nilpotent elements f₁ and f₂ with some compatibility assumptions, we prove Hamiltonian reduction by stages: the slice S_f2 is the Hamiltonian reduction of the slice S_f1. 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 PhD 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.

Cited by

• Inverse Hamiltonian reduction for affine W-algebras in type A

Related

• Reduction by stages for affine W-algebras
• Inverse Hamiltonian reduction in type A and generalized slices in the affine Grassmannian
• The Poisson geometry of the conjugation quotient map for simple algebraic groups and deformed Poisson W-algebras
• Equivariant deformation theory for nilpotent slices in symplectic Lie algebras
• Inverse Hamiltonian reduction for affine W-algebras in type A