发表机构
The University of Edinburgh（爱丁堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Witt 代数包络代数的同态像，证明其在轨道同态下满足理想升链条件，并分类其素谱与本原谱，为原包络代数的理想结构提供新见解。
AI 中文摘要
设 $W_{\geq -1} = \mathbb{C}[t]\partial$ 和 $W = \mathbb{C}[t, t^{-1}]\partial$ 分别为 $\mathbb{C}$ 和 $\mathbb{C}^*$ 上代数向量场的 Witt 代数。本文针对一个开放猜想取得了重大进展，该猜想断言包络代数 $\mathrm{U}(W_{\geq -1})$ 和 $\mathrm{U}(W)$ 满足双边理想的升链条件（ACC）。我们证明了在任意 Gelfand-Kirillov 维数的“轨道同态”族下，$\mathrm{U}(W_{\geq -1})$ 和 $\mathrm{U}(W)$ 的所有同态像均满足理想的 ACC。这些轨道同态是我们近期工作 [Pham, 2025, arXiv:2504.14670] 中的关键要素，使我们能够将 Dixmier 映射从有限维可解情形“提升”到 Witt 和 Virasoro 代数的无限维情形。由此，我们完全分类了这些同态像的素谱和本原谱。由于随着 GK 维数的增加，这些像能更好地逼近 $\mathrm{U}(W_{\geq -1})$，这一分类为 $\mathrm{U}(W_{\geq -1})$ 的双边理想和素理想结构提供了新的启示。最后，我们讨论了所得结果在 $W_{\geq -1}$ 的 Dixmier 映射上的若干应用。
英文摘要
Let $W_{\geq -1} = \mathbb{C}[t]\partial$ and $W = \mathbb{C}[t, t^{-1}]\partial$ be the Witt algebra of algebraic vector fields on $\mathbb{C}$ and $\mathbb{C}^*$ respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(W)$ satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(W)$ under the family of ``orbit homomorphisms'' of arbitrary Gelfand-Kirillov dimension satisfy ACC on ideals. These orbit homomorphisms were the key ingredient allowing us to ``lift'' the Dixmier map from finite-dimensional solvable settings to infinite-dimensional settings of the Witt and Virasoro algebras in our recent work [Pham, 2025, arXiv:2504.14670]. As a result, we completely classify the prime and primitive spectra of these homomorphic images. As these images approximate $\mathrm{U}(W_{\geq -1})$ better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of $\mathrm{U}(W_{\geq -1})$. Finally, we discuss several applications of our results to the Dixmier map for $W_{\geq -1}$.
Comments32 pages; comments welcome
