Abstract

The problem of simultaneous linearization of Lie algebras of (smooth, real-analytic) vector fields on a manifold vanishing at a point can be formulated in terms of Lie algebra representations. In the case the Lie algebra is abelian and the vector fields are Hamiltonian vector fields on a symplectic manifold, the linearization problem is related to the existence of normal forms for completely integrable Hamiltonian systems in a neighbourhood of a non-degenerate singular point of the first integrals. In this talk, we will present some linearization results for Lie algebras of real-analytic vector fields vanishing at a point given by a representation of a semisimple Lie algebra and preserving an additional geometric structure on the manifold (symplectic, Poisson, contact). These results generalize previous results by Guillemin-Sternberg, Kushnirenko and Chaperon. We will also show that these linearization results do not hold , in general, in the smooth setting. This last problem was proposed by Eliasson in relation to normal forms for Hamiltonian systems with linear part of semisimple type.