{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $F$ be a [[field|0QA8CHH8MG]] of characteristic different from $2$.\nA *Jordan algebra* over $F$ is a [[vector space|05EGHEQZAW]] $J$ over $F$ with a bilinear product $J \\times J \\to J$, $(x, y) \\mapsto xy$, such that for all $x, y \\in J$:\n\n1. $xy = yx$;\n2. $(xy)(xx) = x(y(xx))$.\n\nClause 2 is the *Jordan identity*.\nThe product need not be associative, so $J$ is a [[nonassociative ring|YP5TQ6WFVN]] with an $F$-vector space structure.\n\n## Remarks\n\nExample: for an associative $F$-algebra $A$, put $x \\circ y = \\frac{1}{2}(xy + yx)$.\nClause 1 is clear, and clause 2 follows by expanding: with $x^2 = xx$ in $A$, one has $x \\circ x = x^2$, and both $(x \\circ y) \\circ x^2$ and $x \\circ (y \\circ x^2)$ equal $\\frac{1}{4}(xyx^2 + yx^3 + x^3y + x^2yx)$.\nThe resulting Jordan algebra is written $A^+$; the Hermitian $n \\times n$ complex matrices form a real Jordan subalgebra of $M_n(\\C)^+$.\nNon-example: $\\R^3$ with the cross product satisfies clause 2, since $x \\times x = 0$ makes both sides $0$, but fails clause 1, because $x \\times y = -y \\times x$.\nA Jordan algebra is power-associative: the powers of a single element do not depend on the bracketing.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6C2MCVB8C7",
    "kind": "definition",
    "msc": [
      "17C10",
      "17C50"
    ],
    "name": "Jordan algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Jordan_algebra"
      }
    ],
    "tag": "4DV7PEPZ7R",
    "uses": [
      "0QA8CHH8MG",
      "05EGHEQZAW",
      "YP5TQ6WFVN"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/4DV7PEPZ7R/",
  "links": [
    {
      "field": "defines",
      "target": "6C2MCVB8C7"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "05EGHEQZAW"
    },
    {
      "field": "uses",
      "target": "YP5TQ6WFVN"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4DV7PEPZ7R.md",
  "msc": [
    "17C10",
    "17C50"
  ],
  "name": "Jordan algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4DV/4DV7PEPZ7R.md",
  "status": null,
  "tag": "4DV7PEPZ7R",
  "uri": "m2://4DV7PEPZ7R",
  "url": "https://tamnd.github.io/math2/wiki/jordan-algebra-definition/",
  "writing": {
    "tag": "4DV7PEPZ7R",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}