Prime dimension of the nontrivial diagram for general coradicals

Determine whether the nontrivial diagram of a finite-dimensional pointed Hopf algebra of square-free dimension over an algebraically closed field of positive characteristic necessarily has prime dimension when the group of group-like elements is not assumed to be abelian.

Background

For a pointed Hopf algebra H, the associated graded Hopf algebra has the form gr H ≃ R#K[G(H)], where R is the connected graded diagram and dim R = dim H/|G(H)|. In the square-free setting, the diagram dimension is constrained by divisibility properties of Nichols algebras.

The paper notes that prior results establish prime dimension for the nontrivial diagram when the coradical, equivalently the group of group-like elements, is abelian. The unresolved question is whether the same prime-dimension conclusion holds for arbitrary, possibly nonabelian, coradicals. The present paper addresses the broader square-free case in positive characteristic and proves a prime-dimensional conclusion under its stated hypotheses.