{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nConsider symbols $\\operatorname{Sq}^i$ of degree $i$, for $i \\ge 0$.\nThe mod $2$ *Steenrod algebra* $\\mathcal{A}_2$ is the [[graded|8KB4SQGC0C]] associative algebra over $\\Z/2\\Z$ generated by them, subject to these relations.\n\n1. $\\operatorname{Sq}^0 = 1$.\n2. The Adem relations hold:\n$$\\operatorname{Sq}^a \\operatorname{Sq}^b = \\sum_{j=0}^{\\floor{a/2}} \\binom{b - 1 - j}{a - 2j} \\operatorname{Sq}^{a + b - j} \\operatorname{Sq}^j \\quad \\text{for all } 0 \u003c a \u003c 2b,$$\nwhere the [[binomial coefficients|092GEF38YD]] are taken modulo $2$.\n\n## Remarks\n\nThe algebra acts on the mod $2$ cohomology of every space, with $\\operatorname{Sq}^i$ acting as the [[Steenrod square|VR7RHP41M0]] of the same name.\nIt is isomorphic to the algebra of all stable cohomology operations on mod $2$ cohomology.\n\nExample: for $a = b = 1$, the sum has the single term $j = 0$ with coefficient $\\binom{0}{1} = 0$, so $\\operatorname{Sq}^1 \\operatorname{Sq}^1 = 0$.\nFor $a = 1$ and $b = 2$, the coefficient is $\\binom{1}{1} = 1$, so $\\operatorname{Sq}^1 \\operatorname{Sq}^2 = \\operatorname{Sq}^3$.\nThus $\\operatorname{Sq}^3$ is not needed as a generator; the algebra is generated by the $\\operatorname{Sq}^{2^k}$, $k \\ge 0$.\n\nNon-example: the free associative algebra over $\\Z/2\\Z$ on the symbols $\\operatorname{Sq}^i$ is not the Steenrod algebra, since in it $\\operatorname{Sq}^1 \\operatorname{Sq}^1 \\ne 0$.\n\nA basis of $\\mathcal{A}_2$ as a vector space is given by the admissible monomials $\\operatorname{Sq}^{i_1} \\cdots \\operatorname{Sq}^{i_r}$ with $i_s \\ge 2 i_{s+1}$ for all $s$ and $i_r \\ge 1$, together with $1$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "JWTTXSGD0E",
    "kind": "definition",
    "msc": [
      "55S10"
    ],
    "name": "Steenrod algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Steenrod_algebra"
      }
    ],
    "tag": "HEV7170SEK",
    "uses": [
      "VR7RHP41M0",
      "8KB4SQGC0C",
      "092GEF38YD"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/HEV7170SEK/",
  "links": [
    {
      "field": "defines",
      "target": "JWTTXSGD0E"
    },
    {
      "field": "uses",
      "target": "VR7RHP41M0"
    },
    {
      "field": "uses",
      "target": "8KB4SQGC0C"
    },
    {
      "field": "uses",
      "target": "092GEF38YD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HEV7170SEK.md",
  "msc": [
    "55S10"
  ],
  "name": "Steenrod algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HEV/HEV7170SEK.md",
  "status": null,
  "tag": "HEV7170SEK",
  "uri": "m2://HEV7170SEK",
  "url": "https://math2.tamnd.com/wiki/steenrod-algebra-definition/",
  "writing": {
    "tag": "HEV7170SEK",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}