{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "49VHC2VWNF"
    },
    {
      "field": "defines",
      "target": "PWDQ80EKZ8"
    }
  ],
  "body": "\n## Summary\n\nThe antipode of a [[bialgebra|Q9HHVN606G]] is a linear map $S$ that is a two-sided inverse of the identity map for the convolution product.\nA bialgebra with an antipode is a [[Hopf algebra|7XFYP0N84H]].\nThe antipode generalizes the inverse map $g \\mapsto g^{-1}$ of a group, and it is used to define dual representations.\nWhen it exists it is unique, and it reverses the order of products and of coproducts.\n\n## Explanations\n\n### L1 human\n\nIn a group, multiplying an element by its inverse gives the identity.\nIn a bialgebra one first splits an element into pairs, applies $S$ to the left part of each pair, and multiplies the parts back together.\nThe antipode is the map for which this always gives the counit of the element times $1$.\nFor a group algebra, the antipode sends $g$ to $g^{-1}$.\nFor polynomials $K[x]$ with $x$ split as $x \\otimes 1 + 1 \\otimes x$, the antipode sends $x$ to $-x$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16T05"
    ],
    "name": "Antipode",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hopf_algebra"
      }
    ],
    "tag": "PG3ECG5D0G"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PG3ECG5D0G/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/PG3ECG5D0G.md",
  "msc": [
    "16T05"
  ],
  "name": "Antipode",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PG3/PG3ECG5D0G.md",
  "status": null,
  "tag": "PG3ECG5D0G",
  "uri": "m2://PG3ECG5D0G",
  "url": "https://tamnd.github.io/math2/wiki/antipode/",
  "writing": {
    "tag": "PG3ECG5D0G",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}