# nLab Hamiltonian action

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

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

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

### 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 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 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.

## References

A comprehensive account:

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

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]

Last revised on February 6, 2024 at 07:58:04. See the history of this page for a list of all contributions to it.