Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$, and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (at the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$). We also show that all primitive ideals of $\mathrm{S}(\mathfrak{n})$ from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that $\mathfrak{n}$ is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals $I(\lambda )\subset \mathrm{S}(\mathfrak{n})$ and $J(\lambda )\subset \mathrm{U}(\mathfrak{n})$, $\lambda \in {\mathfrak{n}}^{⁎}$, to be nonzero. Most of these results generalize known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals $I(\lambda )$, $J(\lambda )$ are nonzero).

让 $\mathfrak{n}$ 是一个局部幂零无限维李代数 $\mathbb{C}$. 让$\mathrm{你}(\mathfrak{n})$ 和 $\mathrm{秒}(\mathfrak{n})$是其各自的通用包络代数和对称代数。考虑原始谱上的雅各布森拓扑$\mathrm{你}(\mathfrak{n})$，以及原始泊松谱上的泊松拓扑 $\mathrm{秒}(\mathfrak{n})$. 我们在相应的拓扑空间之间提供了同胚（在点的水平上，它给出了原始理想之间的双射）$\mathrm{你}(\mathfrak{n})$ 和 $\mathrm{秒}(\mathfrak{n})$）。我们还表明，所有原始理想$\mathrm{秒}(\mathfrak{n})$从一个正确选择的拓扑中的开放集由与泊松中心的交叉口产生。在假设$\mathfrak{n}$ 是一个 nil-Dynkin Lie 代数，我们给出了原始理想的两个标准 $\mathrm{一世}(\lambda )\subset \mathrm{秒}(\mathfrak{n})$ 和 $J(\lambda )\subset \mathrm{你}(\mathfrak{n})$, $\lambda \in {\mathfrak{n}}^{⁎}$，非零。大多数这些结果概括了有限维幂零李代数的原始和泊松谱的已知事实（但请注意，对于有限维幂零李代数，所有原始理想$\mathrm{一世}(\lambda )$, $J(\lambda )$ 非零）。