{
  "id": "math/9901130",
  "version": "v1",
  "published": "1999-01-28T00:01:55.000Z",
  "updated": "1999-01-28T00:01:55.000Z",
  "title": "Group Actions on Central Simple Algebras",
  "authors": [
    "Daniel S. Sage"
  ],
  "comment": "Latex2e, 20 pages",
  "categories": [
    "math.RT",
    "math.GR",
    "math.RA"
  ],
  "abstract": "Let $G$ be a group, $F$ a field, and $A$ a finite-dimensional central simple algebra over $F$ on which $G$ acts by $F$-algebra automorphisms. We study the ideals and subalgebras of $A$ which are preserved by the group action. Let $V$ be the unique simple module of $A$. We show that $V$ is a projective representation of $G$ and $A\\cong\\text{End}_D(V)$ makes $V$ into a projective representation. We then prove that there is a natural one-to-one correspondence between $G$-invariant $D$-submodules of $V$ and invariant left (and right) ideals of $A$. Under the assumption that $V$ is irreducible, we show that an invariant (unital) subalgebra must be a simply embedded semisimple subalgebra. We introduce induction of $G$-algebras. We show that each invariant subalgebras is induced from a simple $H$-algebra for some subgroup $H$ of finite index and obtain a parametrization of the set of invariant subalgebras in terms of induction data. We then describe invariant central simple subalgebras. For $F$ algebraically closed, we obtain an entirely explicit classification of the invariant subalgebras. Furthermore, we show that the set of invariant subalgebras is finite if $G$ is a finite group. Finally, we consider invariant subalgebras when $V$ is a continuous projective representation of a topological group $G$. We show that if the connected component of the identity acts irreducibly on $V$, then all invariant subalgebras are simple. We then apply our results to obtain a particularly nice solution to the classification problem when $G$ is a compact connected Lie group and $F=\\mathbf C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-01-28T00:01:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16W20",
      "20C99",
      "16K20"
    ],
    "keywords": [
      "invariant subalgebras",
      "group action",
      "projective representation",
      "invariant central simple subalgebras",
      "finite-dimensional central simple algebra"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......1130S"
    }
  }
}