Poisson Deleting Derivations Algorithm and Poisson Spectrum
St\'ephane Launois, C\'esar Lecoutre

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…
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