Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension

TL;DR

This paper investigates the properties of pointed and connected Hopf algebras with finite Gelfand-Kirillov dimension, establishing their dimensional relations and classifying certain low-dimensional cases.

Contribution

It proves that pointed Hopf algebras and their associated graded algebras share the same Gelfand-Kirillov dimension under mild conditions and classifies connected Hopf algebras of GK-dimension three.

Findings

GK-dimension of connected Hopf algebras is either infinite or a positive integer

Pointed Hopf algebras and their graded counterparts have equal GK-dimension under mild assumptions

Classification of connected Hopf algebras of GK-dimension three over algebraically closed fields

Abstract

Let $H$ be a pointed Hopf algebra. We show that under some mild assumptions $H$ and its associated graded Hopf algebra $\gr H$ have the same Gelfand-Kirillov dimension. As an application, we prove that the Gelfand-Kirillov dimension of a connected Hopf algebra is either infinity or a positive integer. We also classify connected Hopf algebras of GK-dimension three over an algebraically closed field of characteristic zero.