Geometria em Lisboa

15 October 2002, 17h00

Linearizing actions and Poisson brackets
Sala P3.10-Dep. Mat./IST

Rui Loja Fernandes (Instituto Superior Técnico)

Abstract:

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.
Many examples of such equations arise in statistics, algebraic geometry, representation theory, etc. The function $P_A(b)$ is "locally quasi polynomial", a result due to Ehrhart, and which is based on the relation between the number $P_A(b)$ and the Riemann-Roch formula on a toric variety. I will present here a more efficient approach based on the cohomology of the complement of hyperplanes. Due to a multidimensional residue formula, this number $P_A(b)$ may be calculated as an integral over a cycle $C$ in the complement. This is joint work with Velleda Baldoni-Silva and Andras Szenes.

| back