Hamiltonian pseudo-representations
Symplectic Geometry
2013-06-27 v2 Mathematical Physics
math.MP
Abstract
The question studied here is the behavior of the Poisson bracket under C^0-perturbations. In this purpose, we introduce the notion of pseudo-representation and prove that for a normed Lie algebra, it converges to a representation. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C^{-1} bi-invariant metric. Our methods also provide a new proof of Gromov-Eliashberg Theorem, it is to say that the group of symplectic diffeomorphisms is C^0-closed in the group of all diffeomorphisms.
Cite
@article{arxiv.math/0703335,
title = {Hamiltonian pseudo-representations},
author = {Vincent Humilière},
journal= {arXiv preprint arXiv:math/0703335},
year = {2013}
}
Comments
16 pages. Main result extended to a large class of Lie algebras