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| Main Authors: | , |
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| Format: | Preprint |
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2014
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| Online Access: | https://arxiv.org/abs/1403.7190 |
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| _version_ | 1866914049819672576 |
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| author | Bell, Jason P. Leung, Wing Hong |
| author_facet | Bell, Jason P. Leung, Wing Hong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1403_7190 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension Bell, Jason P. Leung, Wing Hong Rings and Algebras 16W30, 16T05, 16S30, 16P90 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. |
| title | The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension |
| topic | Rings and Algebras 16W30, 16T05, 16S30, 16P90 |
| url | https://arxiv.org/abs/1403.7190 |