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.