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

TL;DR

This paper extends the quadratic Poisson Gel'fand-Kirillov problem to arbitrary characteristic fields, proving it for a broad class of Poisson algebras related to quantum coordinate rings using a new higher Poisson derivation concept.

Contribution

It introduces higher Poisson derivations to generalize the deleting-derivations algorithm to arbitrary characteristic fields, solving the quadratic Poisson Gel'fand-Kirillov problem in new cases.

Findings

Proves the problem for semiclassical limits of quantized coordinate rings.

Shows quotients by Poisson prime torus-invariant ideals also satisfy the problem.

Demonstrates the result for coordinate rings of determinantal varieties.

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 $\K[X_1,..., X_n]$ with Poisson bracket defined by $\{X_i,X_j\}=\lambda_{ij} X_iX_j$ for some skew-symmetric matrix $(\lambda_{ij}) \in M_n(\K)$. This problem was studied in \cite{GL} over a field of characteristic 0 by using a Poisson version of the deleting-derivations algorithm 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. For, we introduce the concept of {\it higher Poisson derivation} which allows us to extend the Poisson version of the deleting-derivations algorithm 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.