On complete generators of certain Lie algebras on Danielewski surfaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915927973429248 |
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| author | Andrist, Rafael B. |
| author_facet | Andrist, Rafael B. |
| contents | We study the Lie algebra of polynomial vector fields on a smooth Danielewski surface of the form $x y = p(z)$ with $x,y,z \in \mathbb{C}$. We provide explicitly given generators to show that: 1. The Lie algebra of polynomial vector fields is generated by $6$ complete vector fields. 2. The Lie algebra of volume-preserving polynomial vector fields is generated by finitely many vector fields, whose number depends on the degree of the defining polynomial. 3. There exists a Lie sub-algebra generated by $4$ LNDs whose flows generate a group that acts infinitely transitively on the Danielewski surface. The latter result is also generalized to higher dimensions where $z \in \mathbb{C}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14702 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On complete generators of certain Lie algebras on Danielewski surfaces Andrist, Rafael B. Complex Variables Algebraic Geometry 32M17, 32E30, 32Q56, 14R10, 14R20 We study the Lie algebra of polynomial vector fields on a smooth Danielewski surface of the form $x y = p(z)$ with $x,y,z \in \mathbb{C}$. We provide explicitly given generators to show that: 1. The Lie algebra of polynomial vector fields is generated by $6$ complete vector fields. 2. The Lie algebra of volume-preserving polynomial vector fields is generated by finitely many vector fields, whose number depends on the degree of the defining polynomial. 3. There exists a Lie sub-algebra generated by $4$ LNDs whose flows generate a group that acts infinitely transitively on the Danielewski surface. The latter result is also generalized to higher dimensions where $z \in \mathbb{C}^N$. |
| title | On complete generators of certain Lie algebras on Danielewski surfaces |
| topic | Complex Variables Algebraic Geometry 32M17, 32E30, 32Q56, 14R10, 14R20 |
| url | https://arxiv.org/abs/2406.14702 |