Ideals of homomorphic images of the enveloping algebra of the Witt algebra
Abstract
Let W≥ −1 = ℂ[t]∂ and W = ℂ[t, t−1]∂ be the Witt algebra of algebraic vector fields on ℂ and ℂ* respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras U(W≥ −1) and U(W) satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of U(W≥ −1) and 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 U(W≥ −1) better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of U(W≥ −1). Finally, we discuss several applications of our results to the Dixmier map for W≥ −1.
Related
- Ideals in Enveloping Algebras of Loop Algebras
- Finite-dimensional Lie subalgebras of the universal enveloping algebra of sl(2,ℂ)
- Transposed Poisson structure on the Witt-type algebra 𝒲(a,−1): Derivations, Automorphisms, and Rota--Baxter operators
- DG Algebra Structures on Mapping Cones with an Application to Edge Ideals
- Lie Ideals and Lie Derivations of the Algebraic Toeplitz Algebra
BibTeX
@misc{pham2026ideals,
title = {{Ideals of homomorphic images of the enveloping algebra of the Witt algebra}},
author = {Tuan Anh Pham},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01913},
eprint = {2610.01913},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01913},
}
RIS
TY - UNPB
TI - Ideals of homomorphic images of the enveloping algebra of the Witt algebra
AU - Tuan Anh Pham
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01913
UR - https://arxiv.org/abs/2610.01913
AB - Let W_≥ −1 = ℂ[t]∂ and W = ℂ[t, t^−1]∂ be the Witt algebra of algebraic vector fields on ℂ and ℂ^* respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras U(W_≥ −1) and U(W) satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of U(W_≥ −1) and 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 U(W_≥ −1) better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of U(W_≥ −1). Finally, we discuss several applications of our results to the Dixmier map for W_≥ −1.
ER -