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arXiv · 0705.3486

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

K. R. Goodearl·S. Launois
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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, such as 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 $\{x_i,x_j\}= λ_{ij} x_ix_j$ for suitable scalars $λ_{ij}$, 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.

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K. R. Goodearl, S. Launois. 2007-05-24. The Dixmier-Moeglin equivalence and a Gel'fand-Kirillov problem for Poisson polynomial algebras. https://arxiv.org/abs/0705.3486

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