The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras

K. R. Goodearl; S. Launois

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 1, page 1-39
  • ISSN: 0037-9484

Abstract

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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 } = λ i j x i x j for suitable scalars λ i j , 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.

How to cite

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Goodearl, K. R., and Launois, S.. "The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras." Bulletin de la Société Mathématique de France 139.1 (2011): 1-39. <http://eudml.org/doc/272384>.

@article{Goodearl2011,
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 $F(x_1,\dots ,x_n)$ over the base field (respectively, over an extension field of the base field) with $\lbrace x_i,x_j\rbrace = \lambda _\{ij\} x_ix_j$ for suitable scalars $\lambda _\{ij\}$, 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.},
author = {Goodearl, K. R., Launois, S.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Poisson polynomial algebras; Dixmier-Moeglin equivalence; Gel’fand-Kirillov problem},
language = {eng},
number = {1},
pages = {1-39},
publisher = {Société mathématique de France},
title = {The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras},
url = {http://eudml.org/doc/272384},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Goodearl, K. R.
AU - Launois, S.
TI - The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 1
SP - 1
EP - 39
AB - 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 $F(x_1,\dots ,x_n)$ over the base field (respectively, over an extension field of the base field) with $\lbrace x_i,x_j\rbrace = \lambda _{ij} x_ix_j$ for suitable scalars $\lambda _{ij}$, 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.
LA - eng
KW - Poisson polynomial algebras; Dixmier-Moeglin equivalence; Gel’fand-Kirillov problem
UR - http://eudml.org/doc/272384
ER -

References

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