Primitive Deformations of Quantum $p$-groups

TL;DR

This paper introduces Primitive Deformations as a new method to classify certain finite-dimensional connected Hopf algebras in positive characteristic, achieving a complete classification for 8-dimensional cases.

Contribution

It develops the concept of Primitive Deformation to classify almost primitively generated Hopf algebras in positive characteristic, extending the classification to 8-dimensional cases.

Findings

Introduced Primitive Deformation technique for classification

Classified all 8-dimensional connected Hopf algebras in characteristic p>2

Extended previous results to new algebraic structures

Abstract

For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which are almost primitively generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional, whose primitive spaces are abelian restricted Lie algebras of dimension $n$. We illustrate this technique for the case $n=2$. Together with our preceding results in arXiv:1309.0286, we provide a complete classification of $p^3$-dimensional connected Hopf algebras over $\mathbf{k}$ of characteristic $p>2$.