The prime spectrum of the Drinfeld double of the Jordan plane

TL;DR

This paper completely characterizes the prime and primitive spectra of a specific Hopf algebra related to the Jordan plane and confirms it satisfies the Dixmier-Moeglin Equivalence, proposing a broader conjecture.

Contribution

It provides a full description of the prime spectrum of the Drinfeld double of the Jordan plane and establishes the Dixmier-Moeglin Equivalence for this algebra.

Findings

Prime and primitive spectra are fully determined.

The algebra satisfies the Dixmier-Moeglin Equivalence.

Conjecture on the equivalence for pointed Noetherian Hopf algebras.

Abstract

The Hopf algebra $\mathcal{D}$ which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime and primitive spectra are completely determined. As a corollary of this analysis it is shown that $\mathcal{D}$ satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.