Poisson deleting derivations algorithm and Poisson spectrum

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 Poisson spectra of 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.

On the Semi-centre of a Poisson Algebra

“…(iii) These Poisson algebras are all Poisson iterated Ore extensions to which the Poisson deleting derivations algorithm [10] can be applied, and therefore they localise to Poisson tori [11,Theorem 5.3.2]. The result then follows from the first claim along with part (ii).…”

On the importance of being primitive

“…Armed with these notions, we can then define the Poisson Dixmier-Moeglin equivalence for a Poisson algebra, as being an algebra for which the notions of Poisson rationality, Poisson primitivity, and being Poisson locally closed are equivalent for all Poisson prime ideals of the algebra. Brown and Gordon [BG03, Question 3.2] whether the Poisson Dixmier-Moeglin equivalence holds for all affine complex Poisson algebras, and it has been shown to hold in numerous cases: mainly those coming via the semiclassical limit construction applied to a quantum algebra and closely related algebras (see, for example, [LL17,JO14,Oh17,Oh8,GL11]). A negative answer to the question of Brown and Gordan was given in [BLLM17] although it is shown in this paper that the answer is affirmative when the Krull dimension is at most two.…”

The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

“…Many tools developed originally for quantum algebras (for example, the ‐stratification mentioned above, and Cauchon's deleting derivations algorithm) turn out to transfer naturally to the Poisson case with very little modification. These two pictures — prime ideals in quantum algebras for not a root of unity, and Poisson prime ideals in their semiclassical limits — are often remarkably alike, and there have been many recent results exploring these similarities (for example, see ). In light of this, Goodearl makes the following conjecture in .…”