{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a field and $V$ a [[vector space|05EGHEQZAW]] over $k$.\nLet $c \\colon V \\otimes V \\to V \\otimes V$ be an invertible linear map that satisfies the braid relation $c_{12} c_{23} c_{12} = c_{23} c_{12} c_{23}$ on $V^{\\otimes 3}$, the braid form of the [[Yang–Baxter equation|663BP8ZHS5]].\nThe pair $(V, c)$ is a *braided vector space*.\nFor $n \\ge 2$ the [[braid group|SRS8WT4H8Y]] $B_n$ acts on $V^{\\otimes n}$, the generator $\\sigma_i$ acting as $c$ on the factors $i$ and $i+1$.\nLet $s$ be a permutation in the [[symmetric group|0RJMNEN3BJ]] $S_n$, and take a reduced expression of $s$ in the simple transpositions.\nReplace each transposition $(i\\ i{+}1)$ by $\\sigma_i$, and let $c_s$ be the action of the resulting element of $B_n$.\nThe *quantum symmetrizer* is $\\Omega_n = \\sum_{s \\in S_n} c_s$.\nThe *Nichols algebra* of $(V, c)$ is\n$$\\mathfrak{B}(V) = T(V) \\big/ \\bigoplus_{n \\ge 2} \\ker \\Omega_n,$$\nwhere $T(V)$ is the [[tensor algebra|P4J5JZDW0B]] of $V$.\n\n## Remarks\n\nThe lift $c_s$ does not depend on the reduced expression, by Matsumoto's theorem, and $\\bigoplus_{n \\ge 2} \\ker \\Omega_n$ is a graded two-sided ideal of $T(V)$.\nExample: let $V = kx$ be one-dimensional with $c(x \\otimes x) = q\\, x \\otimes x$, where $q \\ne 1$ is a root of unity of order $N \\ge 2$.\nEach $c_s$ multiplies $x^{\\otimes n}$ by $q^{\\ell(s)}$, with $\\ell(s)$ the length of $s$, so $\\Omega_n(x^{\\otimes n}) = (n)_q!\\, x^{\\otimes n}$, where $(m)_q = 1 + q + \\cdots + q^{m-1}$.\nThis is zero exactly when $n \\ge N$, so $\\mathfrak{B}(V) = k[x]/(x^N)$.\nNon-example: for $c$ the flip and $\\dim V \\ge 2$, the tensor algebra $T(V)$ itself is not the Nichols algebra, because $\\Omega_2(u \\otimes v - v \\otimes u) = 0$ for $u, v \\in V$, so $uv - vu$ is zero in $\\mathfrak{B}(V)$.\nIf $\\operatorname{char} k = 0$, then $\\mathfrak{B}(V)$ is the symmetric algebra of $V$ for $c$ the flip, and the exterior algebra of $V$ for $c$ minus the flip.\nEquivalently, $\\mathfrak{B}(V)$ is the graded braided Hopf algebra $\\bigoplus_{n \\ge 0} R(n)$ with $R(0) = k$ and $R(1) = V$ that is generated by $V$ and whose primitive elements are exactly $V$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "KHMT0E4BRK",
    "kind": "definition",
    "msc": [
      "16T05",
      "16T20",
      "17B37"
    ],
    "name": "Nichols algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nichols_algebra"
      }
    ],
    "tag": "2VQ771ENN0",
    "uses": [
      "05EGHEQZAW",
      "P4J5JZDW0B",
      "SRS8WT4H8Y",
      "0RJMNEN3BJ",
      "663BP8ZHS5"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/2VQ771ENN0/",
  "links": [
    {
      "field": "defines",
      "target": "KHMT0E4BRK"
    },
    {
      "field": "uses",
      "target": "05EGHEQZAW"
    },
    {
      "field": "uses",
      "target": "P4J5JZDW0B"
    },
    {
      "field": "uses",
      "target": "SRS8WT4H8Y"
    },
    {
      "field": "uses",
      "target": "0RJMNEN3BJ"
    },
    {
      "field": "uses",
      "target": "663BP8ZHS5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2VQ771ENN0.md",
  "msc": [
    "16T05",
    "16T20",
    "17B37"
  ],
  "name": "Nichols algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2VQ/2VQ771ENN0.md",
  "status": null,
  "tag": "2VQ771ENN0",
  "uri": "m2://2VQ771ENN0",
  "url": "https://tamnd.github.io/math2/wiki/nichols-algebra-definition/",
  "writing": {
    "tag": "2VQ771ENN0",
    "kind": "definition",
    "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
      }
    ]
  }
}