# 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)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$

| $\infty$ | higher prequantum geometry | cohesive ∞-group | Hamiltonian symplectomorphism ∞-group | moduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$ | quantomorphism ∞-group | 1 | symplectic geometry | Lie algebra | Hamiltonian vector fields | real numbers | Hamiltonians under Poisson bracket | | 1 | | Lie group | Hamiltonian symplectomorphism group | circle group | quantomorphism group | | 2 | 2-plectic geometry | Lie 2-algebra | Hamiltonian vector fields | line Lie 2-algebra | Poisson Lie 2-algebra | | 2 | | Lie 2-group | Hamiltonian 2-plectomorphisms | circle 2-group | quantomorphism 2-group | | $n$ | n-plectic geometry | Lie n-algebra | Hamiltonian vector fields | line Lie n-algebra | Poisson Lie n-algebra | | $n$ | | smooth n-group | Hamiltonian n-plectomorphisms | circle n-group | quantomorphism n-group |

(extension are listed for sufficiently connected $X$)

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