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Definition

Integrated

Given a symplectic manifold $(X,\omega)$ , there is the group ofgroup 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 $\hat{\varphi }::G\to \mathrm{Quant}(X,\omega )$ \hat \phi : \colon G \to Quant(X, \omega)

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

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

which is a homomorphism ofhomomorphism of Poisson manifolds . , This is called themoment 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 not a unique. 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 made. clear.

Properties

Characterization

By ( Atiyah-Bott 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 tobasic, 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 account: is in (see around section 2.1)

• Viktor Ginzburg, Victor Guillemin , Yael Karshon,Yael Karshon, around section 2.1 of: Moment maps, cobordisms and Hamiltonian group actions ( [[pdf](http://kolxo3.tiera.ru/M_Mathematics/MD_Geometry%20and%20topology/MDdg_Differential%20geometry/Ginzburg.pdf)]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 , 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, Topology Vol 23, No. 1 (1984) (pdf23 ) 1 (1984) [[pdf](https://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.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](http://arxiv.org/abs/math.SG/0604027)]arXiv:math.SG/0604027)

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