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arXiv · 1403.7190

The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension

Jason P. Bell·Wing Hong Leung
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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 ${\rm Spec}(H)$ we have the equivalences $$P~{\rm primitive}\iff P~{\rm rational}\iff P ~{\rm locally~closed~in}~{\rm Spec}(H).$$ 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.

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Jason P. Bell, Wing Hong Leung. 2014-03-27. The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension. https://doi.org/10.1007/s10468-014-9474-y

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