{
  "id": "1502.07025",
  "version": "v1",
  "published": "2015-02-25T01:24:41.000Z",
  "updated": "2015-02-25T01:24:41.000Z",
  "title": "Quantum Hamiltonian reduction of W-algebras and category O",
  "authors": [
    "Stephen Morgan"
  ],
  "comment": "University of Toronto PhD thesis, defended July 2014, 57 pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "W-algebras are a class of non-commutative algebras related to the classical universal enveloping algebras. They can be defined as a subquotient of U(g) related to a choice of nilpotent element e and compatible nilpotent subalgebra m. The definition is a quantum analogue of the classical construction of Hamiltonian reduction. We define a quantum version of Hamiltonian reduction by stages and use it to construct intermediate reductions between different W-algebras U(g,e) in type A.This allows us to express the W-algebra U(g,e') as a subquotient of U(g,e) for nilpotent elements e' covering e. It also produces a collection of (U(g,e),U(g,e'))-bimodules analogous to the generalised Gel'fand-Graev modules used in the classical definition of the W-algebra; these can be used to obtain adjoint functors between the corresponding module categories. The category of modules over a W-algebra has a full subcategory defined in a parallel fashion to that of the Bernstein-Gel'fand-Gel'fand (BGG) category O; this version of category O(e) for W-algebras is equivalent to an infinitesimal block of O by an argument of Mili\\v{c}i\\'{c} and Soergel. We therefore construct analogues of the translation functors between the different blocks of O, in this case being functors between the categories O(e) for different W-algebras U(g,e). This follows an argument of Losev, and realises the category O(e') as equivalent to a full subcategory of the category O(e) where e' is greater than e in the refinement ordering.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-25T01:24:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum hamiltonian reduction",
      "nilpotent element",
      "full subcategory",
      "construct intermediate reductions",
      "definition"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}