nLab
Hamiltonian action

Contents

Definition

Integrated

Given a symplectic manifold (X,ω)(X,\omega), there is the group of Hamiltonian symplectomorphisms HamSympl(X,ω)HamSympl(X,\omega) acting on XX. If (X,ω)(X,\omega) is prequantizable this lifts to the group of quantomorphisms, both of them covering the diffeomorphisms of XX:

quantomorphisms \to Hamiltonian symplectomorphisms \to diffeomorphisms .

A Hamiltonian action of a Lie group GG on (X,ω)(X,\omega) is an action by quantomorphisms, hence a Lie group homomorphism ϕ^:GQuant(X,ω)\hat \phi : G \to Quant(X, \omega)

Quant(X,ω) ϕ^ G ϕ HamSympl(X,ω) Diff(X). \array{ && Quant(X, \omega) \\ & {}^{\mathllap{\hat \phi}}\nearrow & \downarrow \\ G &\stackrel{\phi}{\to}& HamSympl(X, \omega) \\ & {}_{\mathllap{}}\searrow & \downarrow \\ && Diff(X) } \,.

See (Brylinski, prop. 2.4.10).

Differentially

In the literature this is usually discussed at the infinitesimal level, hence for the corresponding Lie algebras:

smooth functions+Poisson bracket \to Hamiltonian vector fields \to vector fields

Now an (infinitesimal) Hamiltonian action is a Lie algebra homomorphism μ:𝔤(C (X),{,})\mu : \mathfrak{g} \to (C^\infty(X), \{-,-\})

(C (X),{,}) μ 𝔤 HamVect(X,ω) Vect(X). \array{ && (C^\infty(X),\{-,-\}) \\ & {}^{\mathllap{\mu}}\nearrow & \downarrow \\ \mathfrak{g} &\stackrel{}{\to}& HamVect(X, \omega) \\ & {}_{\mathllap{}}\searrow & \downarrow \\ && Vect(X) } \,.

Dualizing, the homomorphism μ\mu is equivalently a linear map

μ˜:X𝔤 * \tilde \mu : X \to \mathfrak{g}^*

which is a homomorphism of Poisson manifolds. This is called the moment map of the (infinitesimal) Hamiltonian GG-action.

Warning The lift from ϕ\phi to ϕ^\hat \phi above, hence from the existence of Hamiltonians to an actual choice of Hamiltonians is in general indeed a choice. There may be different choices. In the literature the difference between ϕ^\hat \phi and ϕ\phi (or of their Lie theoretic analogs) is not always clearly made.

Properties

Characterization

By (Atiyah-Bott), the action of a Lie algebra on a symplectic manifold is Hamiltonian if and only if the symplectic form has a (basic, closed) extension to equivariant de Rham cohomology.

References

A comprehensive account is in (see around section 2.1)

The perspective on Hamiltonian actions in terms of maps to extensions, infinitesimally and integrally, is made explicit in prop. 2.4.10 of

  • Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)

The characterization in equivariant cohomology is due to

Generalization to Hamiltonian actions by a Lie algebroid (instead of just a Lie algebra) is discussed in

  • Rogier Bos, Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids (arXiv:math.SG/0604027)

\,

\,

\,

\,

\,

\,

\,

\,

\,

Revised on September 13, 2013 02:08:21 by Urs Schreiber (77.251.114.72)