{
  "id": "2012.11089",
  "version": "v1",
  "published": "2020-12-21T02:44:56.000Z",
  "updated": "2020-12-21T02:44:56.000Z",
  "title": "Structure of centralizer algebras",
  "authors": [
    "Changchang Xi",
    "Jinbi Zhang"
  ],
  "comment": "25",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Given an $n\\times n$ matrix $c$ over a unitary ring $R$, the centralizer of $c$ in the full $n\\times n$ matrix ring $M_n(R)$ is called a principal centralizer matrix ring, denoted by $S_n(c,R)$. We investigate its structure and prove: $(1)$ If $c$ is an invertible matrix with a $c$-free point, or if $R$ has no zero-divisors and $c$ is a Jordan-similar matrix with all eigenvalues in the center of $R$, then $M_n(R)$ is a separable Frobenius extension of $S_{n}(c,R)$ in the sense of Kasch. $(2)$ If $R$ is an integral domain and $c$ is a Jordan-similar matrix, then $S_n(c,R)$ is a cellular $R$-algebra in the sense of Graham and Lehrer. In particular, if $R$ is an algebraically closed field and $c$ is an arbitrary matrix in $M_n(R)$, then $S_n(c,R)$ is always a cellular algebra, and the extension $S_n(c,R)\\subseteq M_n(R)$ is always a separable Frobenius extension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-21T02:44:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S50",
      "15B33",
      "16U70",
      "15A27",
      "20C05",
      "16W22",
      "11C20"
    ],
    "keywords": [
      "centralizer algebras",
      "separable frobenius extension",
      "jordan-similar matrix",
      "principal centralizer matrix ring",
      "cellular algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}