Hamiltonian Lie group action - Imported Knowlpedia Content

Definition

Let $G$ be a Lie group acting by a symplectic Lie group action on $(M,\omega)$ . With the convention $\iota_{X_f}\omega=df$ , the action is Hamiltonian if it admits a moment map $\mu:M\to\mathfrak g^*$ such that

d\langle\mu,\xi\rangle=\iota_{\xi_M}\omega

for every $\xi\in\mathfrak g$ , and $\mu(gx)=\operatorname{Ad}_g^*\mu(x)$ for every $g\in G$ . The triple $(M,\omega,\mu)$ , together with the $G$ -action, is a Hamiltonian $G$ -space. Thus Hamiltonianity requires compatible global Hamiltonians for all infinitesimal generators, not merely preservation of $\omega$ . Equivariance couples those Hamiltonians to the full group action.

Infinitesimal characterization

Symplecticity gives $d(\iota_{\xi_M}\omega)=0$ . A moment map exists only if these closed one-forms are exact and their primitives can be chosen linearly in $\xi$ . Equivariance further requires

\{\langle\mu,\xi\rangle,\langle\mu,\eta\rangle\} =\langle\mu,[\xi,\eta]\rangle

with the compatible Poisson-bracket convention. For connected $G$ , this infinitesimal bracket identity is equivalent to global coadjoint equivariance Ortega and Ratiu, §4.2.

Examples and non-examples

The standard counterclockwise rotation action of $S^1$ on $\mathbb R^2$ with $\omega=dx\wedge dy$ is Hamiltonian; a moment map is $(x,y)\mapsto -(x^2+y^2)/2$ , after identifying $\mathfrak{s}^1{}^*$ with $\mathbb R$ . Cotangent-lifted actions provide a canonical family of examples.

Translations on a symplectic torus preserve its symplectic form but need not be Hamiltonian: contraction with a generating translation field can be a closed nonexact one-form. The failed condition is global exactness, not symplecticity.

Conventions and scope

References

Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. DOI record. Relevant: Chapter 3, Hamiltonian group actions and moment maps.
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: §4.2, momentum maps and equivariance.