Hamiltonian pseudo-representations | Zendy

Vincent Humilière | Zendy

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Hamiltonian pseudo-representations

Author(s) -

Vincent Humilière

Publication year - 2009

Publication title -

commentarii mathematici helvetici

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.603

H-Index - 46

eISSN - 1420-8946

pISSN - 0010-2571

DOI - 10.4171/cmh/173

Subject(s) - mathematics , poisson bracket , symplectic geometry , pure mathematics , lie group , poisson algebra , symplectomorphism , invariant (physics) , hamiltonian (control theory) , symplectic manifold , adjoint representation , algebra over a field , lie algebra , mathematical physics , mathematical optimization

The question studied here is the behavior of the Poisson bracket underC^0-perturbations. In this purpose, we introduce the notion ofpseudo-representation and prove that for a normed Lie algebra, it converges toa representation. An unexpected consequence of this result is that for manynon-closed symplectic manifolds (including cotangent bundles), the group ofHamiltonian diffeomorphisms (with no assumptions on supports) has no C^{-1}bi-invariant metric. Our methods also provide a new proof of Gromov-EliashbergTheorem, it is to say that the group of symplectic diffeomorphisms isC^0-closed in the group of all diffeomorphisms.Comment: 16 pages. Main result extended to a large class of Lie algebra

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