Lie Subalgebras of vector fields and the Jacobian Conjecture

TL;DR

This paper explores the structure of certain Lie subalgebras of vector fields on affine 2-space and establishes their connection to the 2-dimensional Jacobian Conjecture, linking algebraic properties to a major open problem.

Contribution

It classifies Lie subalgebras of vector fields with constant divergence and relates their properties to the Jacobian Conjecture in dimension two.

Findings

Classification of Lie subalgebras isomorphic to _2

Equivalence of Jacobian Conjecture with conjugacy and algebraicity of these subalgebras

Connection between algebraic structure of subalgebras and the Jacobian Conjecture

Abstract

We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the group $\mathrm{Aff}_{2}(K)$ of affine transformations of ${\mathbb A}^{2}$. We then show that the following three statements are equivalent: (i) The Jacobian Conjecture holds in dimension 2; (ii) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are conjugate under $\mathrm{Aut}({\mathbb A}^{2})$; (iii) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are algebraic.