Title:

Polynomial Poisson algebras : Gel'fandKirillov problem and Poisson spectra

We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras. First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fandKirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fandKirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fandKirillov problem. Our proof relies on the socalled characteristicfree Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator. Next we develop a method, the characteristicfree Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson DixmierMoeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.
