symplectomorphism group

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.

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 $\phi^* X \simeq X$.

- 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)

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

(extension are listed for sufficiently connected $X$)

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