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.

中文翻译：

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无限维李代数对称代数的泊松基定理

我们考虑配备自然泊松括号的无限维李代数的对称代数何时满足泊松理想上的升链条件（ACC）。我们在分级李代数上定义了一个组合条件，我们将其称为Dicksonian，因为它与 $\mathbb{N}^k$ 有限子集上的 Dickson 引理相关。我们的主要结果是：定理。 如果 $\mathfrak{g}$ 是特征零域上的迪克森分级李代数，则对称代数 $S(\mathfrak{g})$ 满足激进泊松理想上的 ACC。作为一个应用，我们为特征零代数闭域上多项式增长的任何分级简单李代数的对称代数以及 Virasoro 代数的对称代数建立了这个 ACC。我们还得出了一些与有限泊松生成代数的泊松本原谱相关的结果。