Primitive Deformations of Quantum $p$-groups

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 $p>2$.

38 pages, 9 tables. We expanded and formalized our original version to a structured technique called Primitive Deformation, applicable in the classification of Hopf algebras in positive characteristic. We changed the title to fit in this new context. Comments and suggestions are welcome

References in corpus (3)

• Connected Hopf Algebras of Dimension $p^2$
• Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic
• A note on Masuoka's Theorem for semisimple irreducible Hopf algebras

Cited by in corpus (2)

• Pointed $p^3$-dimensional Hopf algebras in positive characteristic
• Indicators of Hopf algebras in positive characteristic