Infinite transitivity and polynomial vector fields

TL;DR

This paper investigates the conditions under which certain automorphism groups of complex affine planes act transitively on multiple copies, using Lie algebra and Poisson bracket techniques.

Contribution

It establishes new results on the transitivity of automorphism groups generated by root subgroups on complex affine spaces.

Findings

Many pairs of root subgroups induce open orbits on $( ext{C}^2)^m$

The study of polynomial Lie algebras with Poisson brackets underpins the results

Provides conditions for infinite transitivity in automorphism groups

Abstract

We prove that for many pairs $H_1, H_2$ of root subgroups of the automorphism group $\text{Aut}(\mathbb{C}^2)$ the diagonal action of the group generated by $H_1, H_2$ on $(\mathbb{C}^2)^m$ has an open orbit for any positive integer $m$. The result is based on the study of the Lie algebra of polynomials in two variables with the standard Poisson bracket.