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.