Triangulability criteria for additive group actions on affine space

Abstract

This paper concerns algebraic one-parameter subgroups of GA n ( k), the group of k-algebra automorphisms of the polynomial ring in n variables over a field k: R n = k[ X 1, …, X n ]. These subgroups are of the form exp( tD) ( t ϵ k), where D is a locally nilpotent derivation of R n . The rank of D is defined; this reduces to the usual notion of rank when D is a linear derivation. A characterization is given of all rank 1 derivations. In addition, the rank of D is used to give two criteria for the triangulability of certain actions induced by D (Propositions 2 and 3). These criteria are used to show the existence of tame non-triangulable actions in dimension 4 or greater.