Roots of the affine Cremona group

Abstract

Let k ^[n] = k[x ₁,…, x _n ] be the polynomial algebra in n variables and let ${\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}}$. In this note we show that the root vectors of ${\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right)$, the subgroup of volume preserving automorphisms in the affine Cremona group ${\text{Aut}}\left( {{\mathbb{A}^n}} \right)$, with respect to the diagonal torus are exactly the locally nilpotent derivations x ^α(∂/∂x _i ), where x ^α is any monomial not depending on x _i . This answers a question posed by Popov.