Existence of Hopf subalgebras of GK-dimension two

TL;DR

This paper proves that any pointed Hopf algebra over an algebraically closed field of characteristic zero, with finite Gelfand-Kirillov dimension at least two and being a domain, necessarily contains a Hopf subalgebra of dimension two.

Contribution

It establishes the existence of a Hopf subalgebra of GK-dimension two within a broad class of pointed Hopf algebras under specific conditions.

Findings

Any such Hopf algebra contains a GK-dimension two subalgebra.

The result applies to pointed Hopf algebras over algebraically closed fields of characteristic zero.

It advances understanding of the substructure of Hopf algebras with finite GK-dimension.

Abstract

Let $H$ be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If $H$ is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then $H$ contains a Hopf subalgebra of Gelfand-Kirillov dimension two.