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>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=⨁_{x∈ X}K e_x, X'subsetneq X is a nonempty subquandle, V'=⨁_{x∈ X'}K e_x, and s∈ X∖ X', then ℬ(V)≃K[e_s]/(e_s^m_s)⊗ C_s,X'⊗ℬ(V') for some graded vector space C_s,X', where m_s is the nilpotency order of e_s; in particular, (m_s)_z ℋ_ℬ(V')(z)|ℋ_ℬ(V)(z). In the non-group case, this yields a p²-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.

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BibTeX RIS

BibTeX

@misc{xiong2026qweyl,
  title = {{A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension}},
  author = {Rongchuan Xiong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29853},
  eprint = {2609.29853},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29853},
}

RIS

TY  - UNPB
TI  - A q-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension
AU  - Rongchuan Xiong
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29853
UR  - https://arxiv.org/abs/2609.29853
AB  - 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/|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=⨁_x∈ XK e_x, X'subsetneq X is a nonempty subquandle, V'=⨁_x∈ X'K e_x, and s∈ X∖ X', then ℬ(V)≃K[e_s]/(e_s^m_s)⊗ C_s,X'⊗ℬ(V') for some graded vector space C_s,X', where m_s is the nilpotency order of e_s; in particular, (m_s)_z ℋ_ℬ(V')(z)|ℋ_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.
ER  -

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