On complete generators of certain Lie algebras on Danielewski surfaces

TL;DR

This paper explicitly constructs generators for the Lie algebra of polynomial vector fields on Danielewski surfaces, revealing finite generation and infinite transitivity properties.

Contribution

It provides explicit generators for the Lie algebra of polynomial vector fields and volume-preserving fields on Danielewski surfaces, including higher-dimensional generalizations.

Findings

The Lie algebra of polynomial vector fields is generated by 6 complete vector fields.

The volume-preserving polynomial vector fields are finitely generated, depending on the polynomial degree.

A sub-algebra generated by 4 LNDs acts infinitely transitively on the surface.

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