Roots of the affine Cremona group

Let k ^n] = kx ₁,…, x _n ] be the polynomial algebra in n variables and let \mathbbAⁿ = \textSpec  \boldk ^{n ]} {\mathbb{A}^n} = {\text{Spec}}{{\bold{k}}^{\left n \right]}} . In this note we show that the root vectors of \textAu\textt^*( \mathbbAⁿ ) {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) , the subgroup of volume preserving automorphisms in the affine Cremona group \textAut( \mathbbAⁿ ) {\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.