Poisson Deleting Derivations Algorithm and Poisson Spectrum

TL;DR

This paper develops a Poisson version of Cauchon's deleting derivations algorithm to analyze polynomial Poisson algebras, establishing the Poisson Dixmier-Moeglin equivalence for a broad class and comparing quantum and Poisson spectra.

Contribution

It introduces a Poisson deleting derivations algorithm and applies it to prove the Poisson Dixmier-Moeglin equivalence for certain polynomial Poisson algebras.

Findings

Proves Poisson Dixmier-Moeglin equivalence for a significant class of polynomial Poisson algebras.

Establishes a comparison between quantum matrix spectra and Poisson spectra.

Develops a new tool linking Poisson algebra structures with quantum algebra results.

Abstract

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 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.