{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$ with identity.\nThe algebra $A$ is a *Frobenius algebra* if there is a [[bilinear form|0SQZMXB6GP]] $\\sigma \\colon A \\times A \\to k$ such that:\n\n1. $\\sigma$ is [[nondegenerate|01NHFZF2SK]];\n2. $\\sigma(ab, c) = \\sigma(a, bc)$ for all $a, b, c \\in A$.\n\n## Remarks\n\nExample: the matrix algebra $M_n(k)$ with $\\sigma(a, b) = \\tr(ab)$.\nClause 2 holds because both sides equal $\\tr(abc)$.\nClause 1 holds because $\\tr(a E_{ji}) = a_{ij}$ for the matrix unit $E_{ji}$, so $\\sigma(a, \\cdot) = 0$ forces $a = 0$.\nAnother example is the group algebra $k[G]$ of a finite group $G$, with $\\sigma(a, b)$ the coefficient of the identity element of $G$ in $ab$.\nNon-example: $A = k[x, y]/(x^2, xy, y^2)$, with basis $1, x, y$, is not Frobenius.\nIf $\\sigma$ satisfies clause 2, then $\\sigma(a, b) = \\lambda(ab)$ for the linear form $\\lambda = \\sigma(1, \\cdot)$.\nThe plane spanned by $x$ and $y$ meets $\\ker \\lambda$ in some nonzero $u = \\alpha x + \\beta y$.\nSince $ub \\in k u$ for every $b \\in A$, $\\sigma(u, b) = 0$ for all $b$, and clause 1 fails.\nEquivalent forms: $A$ is Frobenius if and only if some linear form $\\lambda \\colon A \\to k$ has a kernel that contains no nonzero left ideal.\nThis holds if and only if $A \\cong \\Hom_k(A, k)$ as left $A$-modules.\nIf moreover $\\sigma$ is symmetric, $A$ is called a symmetric algebra.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "8EVP1K2CZH",
    "kind": "definition",
    "msc": [
      "16L60",
      "16D50"
    ],
    "name": "Frobenius algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Frobenius_algebra"
      }
    ],
    "tag": "9MXPNHHMT1",
    "uses": [
      "04C5ZBG1WT",
      "3CTW1P4HBP",
      "0SQZMXB6GP",
      "01NHFZF2SK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/9MXPNHHMT1/",
  "links": [
    {
      "field": "defines",
      "target": "8EVP1K2CZH"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    },
    {
      "field": "uses",
      "target": "3CTW1P4HBP"
    },
    {
      "field": "uses",
      "target": "0SQZMXB6GP"
    },
    {
      "field": "uses",
      "target": "01NHFZF2SK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9MXPNHHMT1.md",
  "msc": [
    "16L60",
    "16D50"
  ],
  "name": "Frobenius algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9MX/9MXPNHHMT1.md",
  "status": null,
  "tag": "9MXPNHHMT1",
  "uri": "m2://9MXPNHHMT1",
  "url": "https://tamnd.github.io/math2/wiki/frobenius-algebra-definition/",
  "writing": {
    "tag": "9MXPNHHMT1",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}