Lie Subalgebras of vector fields 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.

arXiv comment10 pages