A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras

TL;DR

This paper establishes conditions under which the symmetric algebra of certain infinite-dimensional Lie algebras satisfies the ascending chain condition on Poisson ideals, with applications to graded simple Lie algebras and the Virasoro algebra.

Contribution

It introduces the Dicksonian condition on graded Lie algebras and proves that it ensures the ACC on radical Poisson ideals in their symmetric algebras.

Findings

Symmetric algebra of Dicksonian graded Lie algebras satisfies ACC on radical Poisson ideals.

ACC holds for symmetric algebra of graded simple Lie algebras of polynomial growth.

ACC also applies to the symmetric algebra of the Virasoro algebra.

Abstract

We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $\mathbb N^k$. Our main result is: Theorem. If $\mathfrak g$ is a Dicksonian graded Lie algebra over a field of characteristic zero, then the symmetric algebra $S(\mathfrak g)$ satisfies the ACC on radical Poisson ideals. As an application, we establish this ACC for the symmetric algebra of any graded simple Lie algebra of polynomial growth over an algebraically closed field of characteristic zero, and for the symmetric algebra of the Virasoro algebra. We also derive some consequences connected to the Poisson primitive spectrum of finitely Poisson-generated algebras.