Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras

TL;DR

This paper explores the relationships between noetherianity, Gelfand-Kirillov dimension, and the Dixmier-Moeglin equivalence in group algebras and pointed Hopf algebras, establishing key equivalences and conditions.

Contribution

It proves the equivalence of Gelfand-Kirillov dimension and Dixmier-Moeglin properties for certain group algebras and extends these results to specific cocommutative Hopf algebras.

Findings

Gelfand-Kirillov dimension and Dixmier-Moeglin equivalence are equivalent for polycyclic-by-finite groups

Conditions for group algebras to satisfy Goldie properties are characterized

Analysis of minimal counterexamples to the noetherian group algebra conjecture

Abstract

This paper addresses the interactions between three properties that a group algebra or more generally a pointed Hopf algebra may possess: being noetherian, having finite Gelfand-Kirillov dimension, and satisfying the Dixmier-Moeglin equivalence. First it is shown that the second and third of these properties are equivalent for group algebras $kG$ of polycyclic-by-finite groups, and are, in turn, equivalent to $G$ being nilpotent-by-finite. In characteristic $0$, this enables us to extend this equivalence to certain cocommutative Hopf algebras. In sections 3 and 4 of the paper finiteness conditions for group algebras are studied. Thus in $\S$3 we examine when a group algebra satisfies the Goldie conditions, while in the final section we discuss what can be said about a minimal counterexample to the conjecture that if $kG$ is noetherian then $G$ is polycyclic-by-finite.