---
tag: 2VQ771ENN0
kind: definition
name: Nichols algebra
msc: [16T05, 16T20, 17B37]
defines: KHMT0E4BRK
uses: [05EGHEQZAW, P4J5JZDW0B, SRS8WT4H8Y, 0RJMNEN3BJ, 663BP8ZHS5]
sources:
  - url: https://en.wikipedia.org/wiki/Nichols_algebra
---

## Definition

Let $k$ be a field and $V$ a [[vector space|05EGHEQZAW]] over $k$.
Let $c \colon V \otimes V \to V \otimes V$ 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]].
The pair $(V, c)$ is a *braided vector space*.
For $n \ge 2$ the [[braid group|SRS8WT4H8Y]] $B_n$ acts on $V^{\otimes n}$, the generator $\sigma_i$ acting as $c$ on the factors $i$ and $i+1$.
Let $s$ be a permutation in the [[symmetric group|0RJMNEN3BJ]] $S_n$, and take a reduced expression of $s$ in the simple transpositions.
Replace each transposition $(i\ i{+}1)$ by $\sigma_i$, and let $c_s$ be the action of the resulting element of $B_n$.
The *quantum symmetrizer* is $\Omega_n = \sum_{s \in S_n} c_s$.
The *Nichols algebra* of $(V, c)$ is
$$\mathfrak{B}(V) = T(V) \big/ \bigoplus_{n \ge 2} \ker \Omega_n,$$
where $T(V)$ is the [[tensor algebra|P4J5JZDW0B]] of $V$.

## Remarks

The 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 $T(V)$.
Example: 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 $N \ge 2$.
Each $c_s$ 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 $(m)_q = 1 + q + \cdots + q^{m-1}$.
This is zero exactly when $n \ge N$, so $\mathfrak{B}(V) = k[x]/(x^N)$.
Non-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 $\mathfrak{B}(V)$.
If $\operatorname{char} k = 0$, 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.
Equivalently, $\mathfrak{B}(V)$ 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 $V$.