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 $\mathrm{Spec}(H)$ we have the equivalences $$P\mathrm{primitive}⟺P\mathrm{rational}⟺P\mathrm{locally}\mathrm{closed}\mathrm{in}\mathrm{Spec}(H).$$ 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.

10 pages

References in corpus (1)

• Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two

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