{
  "id": "2107.03095",
  "version": "v1",
  "published": "2021-07-07T09:26:43.000Z",
  "updated": "2021-07-07T09:26:43.000Z",
  "title": "Tame quivers and affine bases I: a Hall algebra approach to the canonical bases",
  "authors": [
    "Jie Xiao",
    "Han Xu",
    "Minghui Zhao"
  ],
  "categories": [
    "math.RT",
    "math.QA",
    "math.RA"
  ],
  "abstract": "For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-07T09:26:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hall algebra approach",
      "tame quiver",
      "affine bases",
      "simple perverse sheaves",
      "bar-invariant basis coincides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}