nLab
symplectomorphism group

Contents

Idea

Traditional

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

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

is the subgroup of the diffeomorphism group of XX on the diffeomorphisms.

In higher symplectic geometry

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

Examples

  • The linear part of the 2-plectomorphism group/3-plectomorphism group of the Cartesian space 7\mathbb{R}^7 equipped with its associative 3-form ω=lanlge(),()×()\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 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-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
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

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