Pre-Nichols algebras and generation of pointed Hopf algebras in positive characteristic

@misc{xiong2026prenichols,
  title = {{Pre-Nichols algebras and generation of pointed Hopf algebras in positive characteristic}},
  author = {Rongchuan Xiong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25242},
  eprint = {2610.25242},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25242},
}

TY  - UNPB
TI  - Pre-Nichols algebras and generation of pointed Hopf algebras in positive characteristic
AU  - Rongchuan Xiong
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25242
UR  - https://arxiv.org/abs/2610.25242
AB  - We study when a finite-dimensional pre-Nichols algebra R of V over an algebraically closed field of characteristic p>0, with pnmiddim R, is necessarily isomorphic to the Nichols algebra B(V). A trace identity gives a rigidity criterion involving the twist on minimal defining relations. Using integral forms and Hochschild cohomology, we establish uniform rigidity for Yetter--Drinfeld modules of bounded dimension over fixed finite abelian groups, and obtain related results for certain modules over nonabelian groups. For every finite group G of odd order, outside finitely many characteristics depending only on G, every finite-dimensional pointed Hopf algebra H with G(H)≃ G and dimension prime to the characteristic is generated by group-like and skew-primitive elements. If G is abelian, then, outside a possibly larger finite set of characteristics, every such H is a Hopf $2$-cocycle deformation of its associated graded Hopf algebra.
ER  -