{
  "id": "2401.15751",
  "version": "v1",
  "published": "2024-01-28T20:29:35.000Z",
  "updated": "2024-01-28T20:29:35.000Z",
  "title": "Abstract Group Automorphisms of Heisenberg Groups and Partial Automatic Continuity",
  "authors": [
    "Tomoya Tatsuno"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GR",
    "math.DG",
    "math.RT"
  ],
  "abstract": "Any abstract (not necessarily continuous) group automorphism of a simple, compact Lie group must be continuous due to Cartan (1930) and van der Waerden (1933). The purpose of this paper is to study a similar question in nilpotent Lie groups. Let $N$ be the 2-step Iwasawa N-group of a simple Lie group of rank 1. The group $N$ is precisely one of the $(2n+1)$-dimensional Heisenberg group, $(4n+3)$-dimensional quaternionic Heisenberg group, and 15-dimensional octonionic Heisenberg group. We show that any abstract (not necessarily continuous) group automorphism of $N$ is a product of a (possibly discontinuous) central automorphism and a Lie group automorphism. Thus, the discontinuity only occurs at the center for these 2-step nilpotent Lie groups. Moreover, we gain a similar result about abstract homomorphisms under certain conditions. We give a uniform proof from a geometric perspective. We also present the first example of a nilpotent Lie group whose automorphism group is not of the type described above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-28T20:29:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E25",
      "22E46",
      "20F28"
    ],
    "keywords": [
      "partial automatic continuity",
      "abstract group automorphisms",
      "nilpotent lie group",
      "dimensional quaternionic heisenberg group",
      "van der waerden"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}