# nLab symplectomorphism group

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

For $(X,\omega)$ a symplectic manifold, the symplectomorphism group

$Diff(X,\omega) \hookrightarrow Diff(X)$

is the subgroup of the diffeomorphism group of $X$ on the diffeomorphisms.

### In higher symplectic geometry

Analogous constructions apply when symplectic manifolds are generalized to n-plectic infinity-groupoids: for $(X, \omega)$ an n-plectic manifold, and $n$-plectomorphism is a diffeomorphism $\phi : X \to X$ that preserves the $n$-plectic form $\phi^* X \simeq X$.

## Examples

• The linear part of the 2-plectomorphism group/3-plectomorphism group of the Cartesian space $\mathbb{R}^7$ equipped with its associative 3-form $\omega = \lanlge (-), (-) \times (-)\rangle$ is the exceptional Lie group G2. See there for more details.

A further subgroup is that of Hamiltonian symplectomorphisms. The group extension of that whose elements are pairs consisting of a Hamiltonian diffeomorphism and a choice of Hamiltonian for this is the quantomorphism group.

The Lie algebra of the symplectomorphism group is that of symplectic vector fields.

higher and integrated Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$quantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)

Revised on December 15, 2012 01:24:59 by Urs Schreiber (71.195.68.239)