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

Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$. We prove that either $H$ is a group algebra or $\dim H\mathrm{/}\mathrm{\mid }\text{G}(H)\mathrm{\mid }=p$, 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 $V={⨁}_{x\in X}\text{K}{e}_x$, $X^{\mathrm{\prime }}⊊X$ is a nonempty subquandle, $V^{\mathrm{\prime }}={⨁}_{x\in X^{\mathrm{\prime }}}\text{K}{e}_x$, and $s\in X\setminus X^{\mathrm{\prime }}$, then $\mathcal{B}(V)\simeq \text{K}[e_s]\mathrm{/}(e_s^{m_s})\otimes C_{s,X^{\mathrm{\prime }}}\otimes \mathcal{B}(V^{\mathrm{\prime }})$ for some graded vector space $C_{s,X^{\mathrm{\prime }}}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s{)}_z{\mathcal{H}}_{\mathcal{B}(V^{\mathrm{\prime }})}(z)\mid {\mathcal{H}}_{\mathcal{B}(V)}(z)$. In the non-group case, this yields a $p^2$-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 $p$.