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THE DIXMIER-MOEGLIN EQUIVALENCE AND A GEL'FAND-KIRILLOV PROBLEM
 

Summary: THE DIXMIER-MOEGLIN EQUIVALENCE
AND A GEL'FAND-KIRILLOV PROBLEM
FOR POISSON POLYNOMIAL ALGEBRAS
K. R. Goodearl and S. Launois
Abstract. The structure of Poisson polynomial algebras of the type obtained as semiclas-
sical 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 estab-
lishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings,
such as semiclassical limits of quantum matrices, quantum symplectic and euclidean spaces,
quantum symmetric and antisymmetric matrices. For a similarly large class of Poisson poly-
nomial rings, it is proved that the quotient field of the algebra (respectively, of any Poisson
prime factor ring) is a rational function field F(x1, . . . , xn) over the base field (respectively,
over an extension field of the base field) with {xi, xj} = ijxixj for suitable scalars ij,
thus establishing a quadratic Poisson version of the Gel'fand-Kirillov problem. Finally, par-
tial solutions to the isomorphism problem for Poisson fields of the type just mentioned are
obtained.
0. Introduction
Fix a base field k of characteristic zero throughout. All algebras are assumed to be over
k, and all relevant maps (automorphisms, derivations, etc.) are assumed to be k-linear.

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics