Group symplectomorphisms

The group of the symplectomorphisms of a symplectic Variety $(M, \ Omega)$ , noted $Symp (M, \ Omega)$ , indicates the whole of the symplectic Symplectomorphisme S or diffeomorphisms of $(M, \ Omega)$ , provided with the law of composition.

Interest

Algebraic properties

The group of the symplectomorphisms is not a normal sub-group of $Diff (M, \ Omega)$ .

If F is a diffeomorphism of the variety M , the conjugation by F makes correspond bijectivement the symplectomorphisms of $(M, \ Omega)$ and of $(M, f^* \ Omega)$ :

f.Symp (M, \ Omega) .f^ {- 1} =Symp (M, f^* \ Omega)

In particular, the conjugation by a diffeomorphism F preserves the sub-group

Symp (M, \ Omega)

if and only if F is a symplectomorphism.

Topology

It is practical to provide the group with the symplectomorphisms of topology $C^ {\ infty}$ . In this case, $Symp (M, \ Omega)$ seems a sub-group closed of the group of the diffeomorphisms $Diff (M, \ Omega)$ . Incidentally, the group of the diffeomorphisms can in all legitimacy being seen as a group of Dregs of infinite size. More precisely, tangent space in the associated identity is the space of Fréchet $X (M)$ of the fields of vectors of class $C^ {\ infty}$ on M .

The group of the symplectomorphisms is a sub-group closed for topology $C^0$ of the group of the homeomorphisms of the variety M .

The proof rests on the use of the symplectic capacities. A symplectic capacity is the data for any symplectic variety $(M, \ Omega)$ of a positive reality $C (M, \ Omega)$ , checking the following properties:

In particular, for a symplectic variety $(M, \ Omega)$ given, reality $C (U)$ is associated each opened U .

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Group symplectomorphisms

Interest

Algebraic properties

Topology

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