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 $\hat \phi : G \to Quant(X, \omega)$

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

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 $\mu : \mathfrak{g} \to (C^\infty(X), \{-,-\})$

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

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

$$\tilde \mu : X \to \mathfrak{g}^*$$

which is a homomorphism of Poisson manifolds. This is called the moment 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 indeed a choice. There may be different choices. In the literature the difference between $\hat \phi$ and $\phi$ (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.

Related concepts

• moment map

• symplectic quotient

• symplectic reduction

• Guillemin-Sternberg geometric quantization conjecture

• projective Hamiltonian action: classical anomaly

References

A comprehensive account is in (see around section 2.1)

• Viktor Ginzburg, Victor Guillemin, Yael Karshon, 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 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

• Michael Atiyah, Raoul Bott, The moment map and equivariant cohomology, Topology, Vol 23, No. 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)

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