The Dixmier-Moeglin Equivalence for Cocommutative Hopf Algebras of Finite Gelfand-Kirillov Dimension

Abstract

Let k be an algebraically closed field of characteristic zero and let H be a noetherian cocommutative Hopf algebra over k. We show that if H has polynomially bounded growth then H satisfies the Dixmier-Moeglin equivalence. That is, for every prime ideal P in Spec(H) we have the equivalences $$P\; \text{primitive}\iff P\; \text{rational}\iff P\; \text{locally closed in}~\text{Spec}(H).$$ We observe that examples due to Lorenz show that this does not hold without the hypothesis that H have polynomially bounded growth. We conjecture, more generally, that the Dixmier-Moeglin equivalence holds for all finitely generated complex noetherian Hopf algebras of polynomially bounded growth.