nLab Hamiltonian action

Definition

Integrated

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

quantomorphisms$\to$ Hamiltonian symplectomorphisms $\to$ diffeomorphisms .

A Hamiltonian action of a Lie group $G$ on $(X,\omega)$ is an action by quantomorphisms, hence a Lie group homomorphism $\hat \phi \colon G \to Quant(X, \omega)$

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

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 $\mu \colon \mathfrak{g} \to (C^\infty(X), \{-,-\})$ to the Poisson bracket-algebra:

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

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

$$\tilde \mu \colon X \to \mathfrak{g}^\ast$$

which is a homomorphism of Poisson manifolds, called the momentum map of the (infinitesimal) Hamiltonian $G$-action.

Warning The lift from $\phi$ to $\hat \phi$ above, hence from the existence of Hamiltonians to an actual choice of Hamiltonians is in general not unique. In the literature the difference between $\hat \phi$ and $\phi$ (or of their Lie theoretic analogs) is not always made clear.

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.

Related concepts

• momentum map

• symplectic quotient

• symplectic reduction

• Guillemin-Sternberg geometric quantization conjecture

• projective Hamiltonian action: classical anomaly

References

A comprehensive account:

• Viktor Ginzburg, Victor Guillemin, Yael Karshon, around section 2.1 of: Moment maps, cobordisms and Hamiltonian group actions [pdf]

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

• Jean-Luc Brylinski, Prop. 2.4.10 of: Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)

The characterization in equivariant cohomology is due to

• Michael Atiyah, Raoul Bott, The moment map and equivariant cohomology, Topology 23 1 (1984) [pdf]

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]