Rui Loja Fernandes, Instituto Superior Técnico
Linearizing actions and Poisson brackets

A classical result of H. Cartan states that if a compact Lie group acts by analytic transformations and has a fixed point then in suitably chosen local analytic coordinates around the fixed point the action is linear. Bochner extended this result to smooth compact Lie group actions. Guillemin and Sternberg proved that any semisimple Lie algebra action by analytic transformations can be analytically linearized, and gave a counter-example in the smooth case. We will show that any compact Lie algebra action by smooth transformations can be smoothly linearized, and we apply this result to prove the Levi decomposition of Poisson brackets. This is joint work with Philippe Monnier.