Poisson algebras via model theory and differential-algebraic geometry

Bell, Jason P.; Launois, Stéphane; Sánchez, Omar León; Moosa, Rahim

doi:10.4171/jems/712

Cited by 40 publications

(73 citation statements)

References 33 publications

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“…Now in order to prove that

M

is countable, it suffices to show that

Z

contains only countably many minimal non‐zero

normalΔ

‐prime ideals. This follows from the argument given in [, Theorem 3.2], using the fact that

C_{normalΔ} false(Q (Z) false) = C

. Now we may enumerate

M = {P_{1}, P_{2}, P_{3}, \dots}

and write

{frakturp}_{i} : = P_{i} \cap Z

for all

i = 1, 2, 3, \dots

.…”

Section: The Weak Poisson–dixmier–moeglin Equivalencementioning

confidence: 99%

“…First of all we observe that (0) is a

normalΔ

‐rational ideal of

Z

, thanks to the identification

Q false(A false) = A \otimes_{Z} Q false(Z false)

. Thanks to [, Theorem 7.1] we know that (0) is

normalΔ

‐weakly locally closed in

Spec (Z)

. Suppose that

{frakturp}_{1}, \dots, {frakturp}_{l}

are the set of those minimal non‐zero prime ideals of

Z

which are

normalΔ

‐stable.…”

Section: The Weak Poisson–dixmier–moeglin Equivalencementioning

confidence: 99%

“…Another example of a Poisson algebra

Z

with non‐algebraic symplectic leaves for which the PDME holds is the polynomial ring

C [x_{1}, x_{2}, x_{3}]

with brackets

{x, y} = a x, {y, z} = - y, {x, y} = 0

. According to [, Example 3.8], the leaves are non‐algebraic but the PDME holds since the Gelfand–Kirillov (GK) dimension is 3.…”

Section: Proof Of the First Theorem And Some Applicationsmentioning

confidence: 99%

“…The equivalence of (i), (ii) and (iii) is known as the weak Poisson Dixmier-Moeglin equivalence (PDME). It has recently been proven for Poisson algebras by Bell et al [3], and our approach is to lift the theorem to the setting of Poisson orders using the close relationships between prime and primitive ideals in finite centralising extensions. Part (a) is similarly a well-known fact in the setting of Poisson algebras, and the same proof works here.…”

Section: The Poisson-dixmier-moeglin Equivalence For Poisson Ordersmentioning

confidence: 99%

“…When the Poisson primitive ideals of a Poisson algebra Z are Poisson locally closed, we say that the PDME holds for Z. Generalising this rubric, we shall say that the PDME holds for A when conditions (I) and (II) hold for A. It was an open question from [6] as to whether every affine Poisson algebra satisfies the PDME; however, recently counterexamples have been discovered [3]. We may now rephrase the last sentence of the second theorem: we have shown that if the PDME holds for a complex affine Poisson algebra Z, then it holds for every Poisson order A over Z.…”

Section: The Poisson-dixmier-moeglin Equivalence For Poisson Ordersmentioning

confidence: 99%

See 4 more Smart Citations

The orbit method for Poisson orders

Launois

Topley

2019

Proc. London Math. Soc.

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A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation A of a Poisson algebra Z should correspond bijectively to the symplectic leaves of Spec(Z). In this article, we consider a Poisson order A over a complex affine Poisson algebra Z. We stratify the primitive spectrum Prim(A) into symplectic cores, which should be thought of as families of non‐commutative symplectic leaves. We then introduce a category A-P-Mod of A‐modules adapted to the Poisson structure on Z, and we show that when Spec(Z) is smooth with locally closed symplectic leaves, there is a natural homeomorphism from the spectrum of annihilators of simple objects in A-P-Mod to the set of symplectic cores in Prim(A) with its quotient topology. Several applications are given to Poisson representation theory. Our main tool is the Poisson enveloping algebra Ae of a Poisson order A, which captures the Poisson representation theory of A. For Z regular and affine, we prove a Poincarée–Birkhoff–Witt (PBW) theorem for Ae and use this to characterise the annihilators of simple Poisson modules: they coincide with the Poisson weakly locally closed, the Poisson primitive and the Poisson rational ideals. We view this as a generalised weak Poisson Dixmier–Mœglin equivalence.

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“…Now in order to prove that