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

Abstract:

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 $\mathbb {K}[X_1, \dots , X_n]$ with Poisson bracket defined by $\{X_i,X_j\}=\lambda _{ij} X_iX_j$ for some skew-symmetric matrix $(\lambda _{ij}) \in M_n(\mathbb {K})$. This problem was studied in 2011 by Goodearl and Launois over a field of characteristic $0$ 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.