Nichols algebra

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 $V$.