Normalization of nonlinear representations of Lie algebras

The principal goal of this work is the simultaneous normalization of a set of vector fields having a Lie algebra structure. Thus, we generalize the Poincaré-Dulac theorem concerning the normalization of a single vector field. We first study the nonlinear representations of nilpotent Lie algebras (or nilpotent Lie groups) in a complex finite-dimensional or infinite-dimensional vector space. We define a normal form for such representations (the simplest form), we prove that these representations are formally normalizable, and we define the necessary conditions for analyticity of the normalization (or linearization) operator. As an application, we linearize Schrodinger's nonlinear equation in the Schwarz space. We also consider the nonlinear representations of any finite-dimensional Lie algebras, and we define normal forms for these representations depending on a Levi-Malcev decomposition of the Lie algebra.