The Dixmier–Moeglin conjecture for noetherian pointed Hopf algebras

Let $k$ be an algebraically closed field of characteristic zero, and let $H$ be a noetherian pointed Hopf $k$-algebra. Write $G(H)$ for the group of group-like elements of $H$, and let $\operatorname{GKdim} H$ denote the Gelfand–Kirillov dimension of $H$.

Dixmier–Moeglin conjecture for pointed Hopf algebras. The following conditions are equivalent:

1. $\operatorname{GKdim} H$ is finite.
2. $H$ satisfies the Dixmier–Moeglin equivalence.
3. $G(H)$ is nilpotent-by-finite.

The conjecture extends a statement known for group algebras of polycyclic-by-finite groups to noetherian pointed Hopf algebras. The paper investigates this equivalence and proves it in several cases, while the full assertion is not established here.