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Bulletin de la SMF - Parutions - 139 (2011) 1-39

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The Dixmier-Moeglin Equivalence and a Gel'fand-Kirillov Problem for Poisson Polynomial Algebras
K. R. Goodearl, S. Launois
Bulletin de la SMF 139, fascicule 1 (2011), 1-39

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Résumé :
L'équivalence de Dixmier-Moeglin et un analogue du problème de Gel'fand-Kirillov pour certaines algèbres de Poisson polynômiales
Nous étudions la structure de certaines algèbres de Poisson polynômiales obtenues comme limites semi-classiques de certaines déformations quantiques d'anneaux de fonctions régulières. Lorsqu'un tore agit rationnellement sur une telle algèbre de Poisson, nous donnons une condition suffisante pour que cette algèbre n'ait qu'un nombre fini d'idéaux premiers de Poisson invariants sous cette action. Ce résultat, combiné à des résultats antérieurs de K.R. Goodearl, permet d'établir l'équivalence de Dixmier-Moeglin pour une large classe d'algèbres de Poisson polynômiales incluant les limites semi-classiques des matrices quantiques, des espaces Euclidiens and symplectiques quantiques, des matrices symétriques et antisymétriques quantiques. De plus, nous démontrons que le corps des fractions de ces algèbres (respectivement, de leurs quotients premiers de Poisson) est un corps de fractions rationnelles F(x_1,,x_n) sur le corps de base (respectivement, sur une certaine extension du corps de base) dont la structure de Poisson est de la forme x_i,x_j= _ij x_ix_j pour certains scalaires _ij convenablement choisis. Ce résultat est un analogue quadratique du problème de Gel'fand-Kirillov pour la structure de Poisson de ces corps. Finallement, nous présentons des résultat partiels quant à la classification de tels corps de fractions à isomorphisme (de Poisson) près.

Mots-clefs : Algèbres de Poisson polynômiales, équivalence de Dixmier-Moeglin, problème de Gel'fand-Kirillov

Abstract:
The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum matrices, quantum symplectic and euclidean spaces, quantum symmetric and antisymmetric matrices. For a similarly large class of Poisson polynomial rings, it is proved that the quotient field of the algebra (respectively, of any Poisson prime factor ring) is a rational function field F(x_1,,x_n) over the base field (respectively, over an extension field of the base field) with x_i,x_j= _ij x_ix_j for suitable scalars _ij , thus establishing a quadratic Poisson version of the Gel'fand-Kirillov problem. Finally, partial solutions to the isomorphism problem for Poisson fields of the type just mentioned are obtained.

Keywords: Poisson polynomial algebras, Dixmier-Moeglin equivalence, Gel'fand-Kirillov problem

Class. math. : 17B63

ISSN : 0037-9484
Publié avec le concours de : Centre National de la Recherche Scientifique

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