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

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.