Year: 2026
Type: preprint
Abstract: We study when a finite-dimensional pre-Nichols algebra $R$ of $V$ over an algebraically closed field of characteristic $p>0$, with $p\nmid\dim R$, 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 $G(H)\simeq G$ 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.
Source: arXiv (Cornell University)
Author: Xiong, Rongchuan
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223034189