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 language | English |
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Article number | 15 |
Journal | Mathematische Zeitschrift |
Volume | 308 |
Issue number | 1 |
DOIs | |
State | Published - 2024/09 |
Keywords
- Hamiltonian reduction
- Nilpotent orbits
- Slodowy slices
- W-algebras
ASJC Scopus subject areas
- General Mathematics