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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.00756 |
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| _version_ | 1866912620965003264 |
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| author | Pham, Tuan Anh Timmins, James |
| author_facet | Pham, Tuan Anh Timmins, James |
| contents | The Witt algebra $W_{\geq -1}$ is the Lie algebra of algebraic vector fields on a line. We investigate the two-sided ideal structure of its universal enveloping algebra, by studying the orbit homomorphisms $Ψ_n: U(W_{\geq -1}) \rightarrow T_n$, an infinite family of homomorphisms to noncommutative Noetherian algebras. The orbit homomorphisms lift primitive ideals from solvable Lie algebras to $U(W_{\geq -1})$, thereby playing a central role in the orbit method for the Witt algebra. We prove that the kernel of any orbit homomorphism is generated by an infinite set of differentiators as a one-sided ideal, whilst being generated by any single element of this set as a two-sided ideal. One consequence is an explicit description of primitive and semi-primitive ideals of $U(W_{\geq -1})$ corresponding to one-point local functions. We also prove that the image $B_n$ of the nth orbit homomorphism is both non-Noetherian and birational to the Noetherian algebra $T_n$. On the other hand, the degree zero subring of $B_n$ is left and right Noetherian, and we conjecture that the same holds for $U(W_{\geq -1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00756 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Kernel and Image of Orbit Homomorphisms for the Witt Algebra Pham, Tuan Anh Timmins, James Rings and Algebras Representation Theory 16S30, 17B66, 16D25, 17B10 The Witt algebra $W_{\geq -1}$ is the Lie algebra of algebraic vector fields on a line. We investigate the two-sided ideal structure of its universal enveloping algebra, by studying the orbit homomorphisms $Ψ_n: U(W_{\geq -1}) \rightarrow T_n$, an infinite family of homomorphisms to noncommutative Noetherian algebras. The orbit homomorphisms lift primitive ideals from solvable Lie algebras to $U(W_{\geq -1})$, thereby playing a central role in the orbit method for the Witt algebra. We prove that the kernel of any orbit homomorphism is generated by an infinite set of differentiators as a one-sided ideal, whilst being generated by any single element of this set as a two-sided ideal. One consequence is an explicit description of primitive and semi-primitive ideals of $U(W_{\geq -1})$ corresponding to one-point local functions. We also prove that the image $B_n$ of the nth orbit homomorphism is both non-Noetherian and birational to the Noetherian algebra $T_n$. On the other hand, the degree zero subring of $B_n$ is left and right Noetherian, and we conjecture that the same holds for $U(W_{\geq -1})$. |
| title | The Kernel and Image of Orbit Homomorphisms for the Witt Algebra |
| topic | Rings and Algebras Representation Theory 16S30, 17B66, 16D25, 17B10 |
| url | https://arxiv.org/abs/2510.00756 |