Integrable generators of Lie algebras of vector fields on $\mathrm{SL}_2(\mathbb{C})$ and on $xy = z^2$

TL;DR

This paper explicitly constructs finite sets of polynomial vector fields that generate the entire Lie algebra of vector fields on $ ext{SL}_2( ext{C})$ and the Danielewski surface, and identifies automorphisms acting transitively.

Contribution

It provides explicit generators for the Lie algebra of polynomial vector fields on these algebraic varieties and describes automorphisms with infinite transitivity.

Findings

Finite generators for Lie algebra of vector fields on $ ext{SL}_2( ext{C})$ and $xy=z^2$

Explicit unipotent automorphisms acting transitively on $xy=z^2$

Construction of automorphism subgroup with infinite transitivity

Abstract

For the special linear group $\mathrm{SL}_2(\mathbb{C})$ and for the singular quadratic Danielewski surface $x y = z^2$ we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on $x y = z^2$.