{
  "id": "2212.06022",
  "version": "v1",
  "published": "2022-12-12T16:41:41.000Z",
  "updated": "2022-12-12T16:41:41.000Z",
  "title": "Reduction by stages for finite W-algebras",
  "authors": [
    "Naoki Genra",
    "Thibault Juillard"
  ],
  "comment": "30 pages, 4 figures",
  "categories": [
    "math.RT",
    "math.AG",
    "math.QA"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a semisimple Lie algebra: its dual space $\\mathfrak{g}^*$ is a Poisson variety. It is well known that for each nilpotent element $f$ in $\\mathfrak{g}^*$, it is possible to construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of $\\mathfrak{g}^*$, the Slodowy slice $S_f$. We prove that, given two nilpotent elements $f_1$ and $f_2$, with some assumptions, it is possible to perform a Hamiltonian reduction by stages: the slice $S_{f_2}$ is a Hamiltonian reduction of the slice $S_{f_1}$. We also state an analogue result in the setting of finite W-algebras, which are quantizations of Slodowy slices. As corollary in type A, we prove the conjecture of Morgan stating 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 about a possible analogue of our results in the context of affine W-algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T16:41:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite w-algebras",
      "hamiltonian reduction",
      "slodowy slice",
      "nilpotent element",
      "hook type w-algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}