The quadratic Poisson Gel’fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polynomial algebra K [ X 1 , … , X n ] \mathbb {K}[X_1, \dots , X_n] with Poisson bracket defined by { X i , X j } = λ i j X i X j \{X_i,X_j\}=\lambda _{ij} X_iX_j for some skew-symmetric matrix ( λ i j ) ∈ M n ( K ) (\lambda _{ij}) \in M_n(\mathbb {K}) . This problem was studied in 2011 by Goodearl and Launois over a field of characteristic 0 0 by using a Poisson version of the deleting derivation homomorphism of Cauchon. In this paper, we study the quadratic Poisson Gel’fand-Kirillov problem over a field of arbitrary characteristic. In particular, we prove that the quadratic Poisson Gel’fand-Kirillov problem is satisfied for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings. We introduce the concept of higher Poisson derivation, which allows us to extend the Poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic. When a torus is acting rationally by Poisson automorphisms on a Poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring, we prove (under some technical assumptions) that quotients by Poisson prime torus-invariant ideals also satisfy the quadratic Poisson Gel’fand-Kirillov problem. In particular, we show that coordinate rings of determinantal varieties satisfy the quadratic Poisson Gel’fand-Kirillov problem.
Abstract. Let k be a field of characteristic zero. For any positive integer n and any scalar a ∈ k, we construct a family of Artin-Schelter regular algebras R(n, a), which are quantisations of Poisson structures on k[x 0 , . . . , xn]. This generalises an example given by Pym when n = 3. For a particular choice of the parameter a we obtain new examples of Calabi-Yau algebras when n ≥ 4. We also study the ring theoretic properties of the algebras R(n, a). We show that the point modules of R(n, a) are parameterised by a bouquet of rational normal curves in P n , and that the prime spectrum of R(n, a) is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe Spec R(n, a) as a union of commutative strata.
In [5] Cauchon introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus-orbits of symplectic leaves in matrix Poisson varieties and totally nonnegative cells in totally nonnegative matrix varieties [12]. This led to recent progress in the study of totally nonnegative matrices such as new recognition tests, see for instance [18]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class P of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier-Moeglin equivalence does not hold for all polynomial Poisson algebras [2]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.
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