On complete generators of certain Lie algebras on Danielewski surfaces

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...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Author: Andrist, Rafael B
Format: Article
Language:English
Subjects:
Algebra
Fields (mathematics)
Lie groups
Polynomials
Vectors (mathematics)
Online Access:Get full text
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Summary: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 \(z \in \mathbb{C}^N\).
ISSN:2331-8422