Connected Hopf Algebras that are Not Hopf Ore Extensions of Enveloping Algebras

TL;DR

This paper constructs specific connected Hopf algebras with finite Gelfand-Kirillov dimension that are not Hopf Ore extensions of enveloping algebras, answering a previously posed question and showing they can be formed as iterated crossed products.

Contribution

It introduces a new family of connected Hopf algebras that are not Hopf Ore extensions, expanding understanding of their structure and relationships.

Findings

Constructed connected Hopf algebras with finite Gelfand-Kirillov dimension

Showed these algebras are not Hopf Ore extensions of enveloping algebras

Demonstrated they can be expressed as iterated crossed products

Abstract

We construct a family of connected Hopf algebras with finite Gelfand-Kirillov dimension, none of which is an iterated Hopf Ore extension of the universal enveloping algebra of its primitive part. This provides a negative answer to a question posed by Li and Zhou. It is also demonstrated that these connected Hopf algebras can be formulated as an iterated crossed product of enveloping algebras.