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

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jason P. Bell

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Wing Hong Leung

Authors
  1. Jason P. Bell
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  2. Wing Hong Leung
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Correspondence to Jason P. Bell.

Additional information

Presented by Susan Montgomery

The research of the first-named author was supported by NSERC grant 611456.

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Bell, J.P., Leung, W.H. The Dixmier-Moeglin Equivalence for Cocommutative Hopf Algebras of Finite Gelfand-Kirillov Dimension. Algebr Represent Theor 17, 1843–1852 (2014). https://doi.org/10.1007/s10468-014-9474-y

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  • DOI: https://doi.org/10.1007/s10468-014-9474-y

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