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arXiv · 2406.14702

On complete generators of certain Lie algebras on Danielewski surfaces

Rafael B. Andrist
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Abstract

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 $z \in \mathbb{C}^N$.

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Rafael B. Andrist. 2024-06-20. On complete generators of certain Lie algebras on Danielewski surfaces. https://arxiv.org/abs/2406.14702

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