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

TL;DR

The article investigates how three finiteness properties—finite Gelfand-Kirillov dimension ($\mathrm{GKdim}$), the Dixmier-Moeglin Equivalence (DME), and noetherian/group-structure finiteness—interact in group algebras and pointed Hopf algebras. It proves that for group algebras of polycyclic-by-finite groups, $\mathrm{GKdim}$ finite, DME, and $G(H)$ being nilpotent-by-finite are all equivalent, and extends this equivalence to certain cocommutative Hopf algebras in characteristic zero using the Cartier–Gabriel–Kostant decomposition. The paper further analyzes finiteness conditions via Goldie theory, establishing chains of implications between amenability, bounded finite-subgroup orders, finite uniform dimension, and the existence of semisimple artinian quotient rings, with consequences for zero-divisor conjectures. Finally, it explores the noetherian group-algebra question, proving that noetherian $kG$ enforces amenability and max-condition, and outlining a Krull-dimension based conjecture (Krullqn) that a finite Krull-dimension noetherian group algebra should imply $G\in\mathcal{P}$, along with a detailed profile of a potential minimal counterexample $\widehat{G}$ and its structural properties. The work highlights deep links between algebraic growth, representation-theoretic finiteness, and group-theoretic structure, with implications for both theory and potential applications in algebraic studies of group actions and Hopf-algebra representations.

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 $§$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.

Theorems & Definitions (55)

• Conjecture 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3