Roots of the affine Cremona group (Q413945)

Here \(\mathrm{Aut}^*(\mathbb A^n)\) denotes the algebraic group of the volume preserving automorphisms of \(\mathbb A^n\), the affine space over an algebraically closed field \(K\). It is formed by the automorphisms \(\gamma\) such that \(\det({\frac{\partial}{\partial x_i}}\gamma(x_j))_{ij}=1\). The author gives a characterization of the root vectors of \(\mathrm{Aut}^*(\mathbb A^n)\) with respect to the diagonal torus \(T\). They coincide with the locally nilpotent \(K\)--derivations on the polynomial algebra that can be written in the form \(\partial=\lambda x^\alpha {\frac{\partial}{\partial x_i}}\), where \(\lambda \in K^*\) and \(\alpha\) is an \(n\)--tuple of non--negative integer numbers with \(\alpha_i=0\). Moreover a character \(\chi\), given by \(\mathrm{diag}(\gamma_1, \dots ,\gamma_n)\rightarrow \gamma_1^{\beta_1}\dots \gamma_n^{\beta_n}\), is a root of \(\mathrm{Aut}^*(\mathbb A^n)\) with respect to \(T\) if and only if the minimum of the set \(\{ \beta_1, \dots, \beta_n\}\) is achieved by only one \(\beta_i\). The proof is based on a combinatorial description of normal affine domains, graded over the group \(M\) of characters of \(T\), and on a description of the locally nilpotent derivations on such a domain \(A\) in the case where the transcendence degree of \(A\) is equal to \(\mathrm{rk} M+1\).