Classification of connected Hopf algebras of dimension $p^3$ I

Abstract

Let $p$ be a prime, $k$ be an algebraically closed field of characteristic $p$. In this paper, we provide the classification of connected Hopf algebras of dimension $p^3$, except the case when the primitive space of the Hopf algebra is two dimensional and abelian. Each isomorphism class is presented by generators $x,y,z$ with relations and Hopf algebra structures. Let $\mu$ be the multiplicative group of $(p^2+p-1)$-th roots of unity. When the primitive space is one-dimensional and $p$ is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by $A_k^1/\mu$.

32 pages, no figures

References in corpus (7)

• Connected Hopf Algebras of Dimension $p^2$
• On Hopf algebras of dimension 4p
• Classification of pointed Hopf algebras of dimension $p^2$ over any algebraically closed field
• Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic
• Techniques for classifying Hopf algebras and applications to dimension p^3
• Local criteria for cocommutative Hopf algebras
• Connected Hopf algebras of Gelfand-Kirillov dimension four

Cited by in corpus (8)

• Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic
• Finite symmetric integral tensor categories with the Chevalley property
• Torsors over the Rational Double Points in Characteristic $\mathbf{p}$
• Universal enveloping algebras of Poisson Hopf algebras
• On non-connected pointed Hopf algebras of dimension 16 in characteristic 2
• Pointed Hopf algebras of dimension $p^{2}q$ in characteristic $p$
• Frobenius-Perron dimensions of integral ${\mathbb{Z}}_{+}$-rings and applications
• Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra