Reflexive Ideals in Iwasawa Algebras
Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $\Omega_G$ is a unit...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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2007
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| _version_ | 1797065312050348032 |
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| author | Ardakov, K Wei, F Zhang, J |
| author_facet | Ardakov, K Wei, F Zhang, J |
| author_sort | Ardakov, K |
| collection | OXFORD |
| description | Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $\Omega_G$ is a unit. We show that these conditions hold in the case when $G$ is an open subgroup of $\SL_2(\Zp)$ and $p$ is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in $\Omega_G$ when $G$ is a congruence subgroup of $\SL_2(\Zp)$: the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown. |
| first_indexed | 2024-03-06T21:26:51Z |
| format | Journal article |
| id | oxford-uuid:4365d6df-258d-4bc4-9214-6dcd55371a85 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:26:51Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:4365d6df-258d-4bc4-9214-6dcd55371a852022-03-26T14:55:02ZReflexive Ideals in Iwasawa AlgebrasJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4365d6df-258d-4bc4-9214-6dcd55371a85EnglishSymplectic Elements at Oxford2007Ardakov, KWei, FZhang, JLet $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $\Omega_G$ is a unit. We show that these conditions hold in the case when $G$ is an open subgroup of $\SL_2(\Zp)$ and $p$ is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in $\Omega_G$ when $G$ is a congruence subgroup of $\SL_2(\Zp)$: the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown. |
| spellingShingle | Ardakov, K Wei, F Zhang, J Reflexive Ideals in Iwasawa Algebras |
| title | Reflexive Ideals in Iwasawa Algebras |
| title_full | Reflexive Ideals in Iwasawa Algebras |
| title_fullStr | Reflexive Ideals in Iwasawa Algebras |
| title_full_unstemmed | Reflexive Ideals in Iwasawa Algebras |
| title_short | Reflexive Ideals in Iwasawa Algebras |
| title_sort | reflexive ideals in iwasawa algebras |
| work_keys_str_mv | AT ardakovk reflexiveidealsiniwasawaalgebras AT weif reflexiveidealsiniwasawaalgebras AT zhangj reflexiveidealsiniwasawaalgebras |