A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension

Abstract: Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p&gt;0$. We prove that either $H$ is a group algebra or $\dim H/|\G(H)|=p$, and that in the latter case $H$ belongs to exactly one of two explicit families of rank-one pointed Hopf algebras. We develop a truncated $q$-Weyl freeness principle for finite-dimensional Nichols algebras of quandle type. If $V=\bigoplus_{x\in X}\K e_x$, $X&#39;\subsetneq X$ is a nonempty subquandle, $V&#39;=\bigoplus_{x\in X&#39;}\K e_x$, and $s\in X\setminus X&#39;$, then $\mathcal B(V)\simeq\K[e_s]/(e_s^{{m_s})\otimes} C_{s,X&#39;}\otimes\mathcal B(V&#39;)$ for some graded vector space $C_{s,X&#39;}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s)<em>z\,\mathcal H</em>{\mathcal B(V&#39;)}(z)\mid\mathcal H_{\mathcal B(V)}(z)$. In the non-group case, this yields a $p<sup>2$-divisibility obstruction that rules out noncentral support for the infinitesimal braiding. Together with a graded-dual argument, the resulting rank-one reduction forces the diagram of $H$ to have dimension $p$.