Centralizers of locally nilpotent derivations
A \(G_a\)-action on the polynomial ring \(\mathbb C[x_1, \ldots, x_n]\) is called triangulable in case there is a coordinate system \(\{u_1, \ldots, u_n\}\) such that the group action has the form \(\sigma_t(u_1) = u_1, \sigma_t(u_i) = u_i + Q_i\) with \(Q_i \in \mathbb C[u_1, \ldots, u_{i-1}]\) for \(i > 1.\) One consequence of the Jung-van der Kulk theorem is that all \(G_a\)-actions on \(\mathbb C [x_1,x_2]\) are triangulable. Moreover algebraic actions of the additive group \(G_a\) on \(\mathbb C[x_1, \ldots, x_n]\), locally nilpotent derivations of \(\mathbb C[x_1, \ldots, x_n],\) and vector fields on \(\mathbb C^n\) admitting a strictly polynomial flow, are equivalent objects. The authors investigate the polynomial centralizer of the vector field corresponding to a triangulable locally nilpotent derivation. In particular this yields a criterion for triangulability. This allows to present several new examples of nontriangulable \(G_a\)-actions extending those previously known in the literature.
scientific article; zbMATH DE number 1061804