Editing Symplectomorphism (section)

== The group of (Hamiltonian) symplectomorphisms ==
The symplectomorphisms from a manifold back onto itself form an infinite-dimensional [[pseudogroup]]. The corresponding [[Lie algebra]] consists of symplectic vector fields.
The Hamiltonian symplectomorphisms form a subgroup, whose Lie algebra is given by the Hamiltonian vector fields. 
The latter is isomorphic to the Lie algebra of smooth
functions on the manifold with respect to the [[Poisson bracket]], modulo the constants.

The group of Hamiltonian symplectomorphisms of <math>(M,\omega)</math> usually denoted as <math>\operatorname{Ham}(M,\omega)</math>.

Groups of Hamiltonian diffeomorphisms are [[simple Lie group|simple]], by a theorem of [[Augustin Banyaga|Banyaga]].<ref>McDuff & Salamon 1998, Theorem 10.25</ref> They have natural geometry given by the [[Hofer norm]]. The [[homotopy type]] of the symplectomorphism group for certain simple symplectic [[four-manifold]]s, such as the product of [[sphere]]s, can be computed using [[Mikhail Gromov (mathematician)|Gromov]]'s theory of [[pseudoholomorphic curve]]s.