A quadratic Poisson Gel'fand-Kirillov problem in prime characteristic

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 polyno-mial algebra K[X1,..., Xn] with Poisson bracket defined by {Xi, Xj} = λijXiXj for some skew-symmetric matrix (λij) ∈Mn(K). This problem was studied in [9] over a field of charac-teristic 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 co-ordinate rings. For, 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

On the dimension of H-strata in quantum matrices

We study the topology of the prime spectrum of an algebra supporting a rational torus action. More precisely, we study inclusions between prime ideals that are torus-invariant using the $H$-stratification theory of Goodearl and Letzter on one hand and the theory of deleting derivations of Cauchon on the other. We apply the results obtained to the algebra of $m \times n$ generic quantum matrices to show that the dimensions of the $H$-strata described by Goodearl and Letzter are bounded above by the minimum of $m$ and $n$, and that moreover all the values between 0 and this bound are achieved.Comment: New introduction; results improve