Moment map

In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. Let M be a manifold with symplectic form ω. Suppose that a Lie group G acts on M via symplectomorphisms (that is, the action of each g in G preserves ω). Let g {displaystyle {mathfrak {g}}} be the Lie algebra of G, g ∗ {displaystyle {mathfrak {g}}^{*}} its dual, and the pairing between the two. Any ξ in g {displaystyle {mathfrak {g}}} induces a vector field ρ(ξ) on M describing the infinitesimal action of ξ. To be precise, at a point x in M the vector ρ ( ξ ) x {displaystyle ho (xi )_{x}} is where exp : g → G {displaystyle exp :{mathfrak {g}} o G} is the exponential map and ⋅ {displaystyle cdot } denotes the G-action on M. Let ι ρ ( ξ ) ω {displaystyle iota _{ ho (xi )}omega ,} denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that ι ρ ( ξ ) ω {displaystyle iota _{ ho (xi )}omega ,} is closed (for all ξ in g {displaystyle {mathfrak {g}}} ). Suppose that ι ρ ( ξ ) ω {displaystyle iota _{ ho (xi )}omega ,} is not just closed but also exact, so that ι ρ ( ξ ) ω = d H ξ {displaystyle iota _{ ho (xi )}omega =dH_{xi }} for some function H ξ {displaystyle H_{xi }} . Suppose also that the map g → C ∞ ( M ) {displaystyle {mathfrak {g}} o C^{infty }(M)} sending ξ ↦ H ξ {displaystyle xi mapsto H_{xi }} is a Lie algebra homomorphism. Then a momentum map for the G-action on (M, ω) is a map μ : M → g ∗ {displaystyle mu :M o {mathfrak {g}}^{*}} such that for all ξ in g {displaystyle {mathfrak {g}}} . Here ⟨ μ , ξ ⟩ {displaystyle langle mu ,xi angle } is the function from M to R defined by ⟨ μ , ξ ⟩ ( x ) = ⟨ μ ( x ) , ξ ⟩ {displaystyle langle mu ,xi angle (x)=langle mu (x),xi angle } . The momentum map is uniquely defined up to an additive constant of integration. A momentum map is often also required to be G-equivariant, where G acts on g ∗ {displaystyle {mathfrak {g}}^{*}} via the coadjoint action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the momentum map coadjoint equivariant. However, in general the coadjoint action must be modified to make the map equivariant (this is the case for example for the Euclidean group). The modification is by a 1-cocycle on the group with values in g ∗ {displaystyle {mathfrak {g}}^{*}} , as first described by Souriau (1970). The definition of the momentum map requires ι ρ ( ξ ) ω {displaystyle iota _{ ho (xi )}omega } to be closed. In practice it is useful to make an even stronger assumption. The G-action is said to be Hamiltonian if and only if the following conditions hold. First, for every ξ in g {displaystyle {mathfrak {g}}} the one-form ι ρ ( ξ ) ω {displaystyle iota _{ ho (xi )}omega } is exact, meaning that it equals d H ξ {displaystyle dH_{xi }} for some smooth function If this holds, then one may choose the H ξ {displaystyle H_{xi }} to make the map ξ ↦ H ξ {displaystyle xi mapsto H_{xi }} linear. The second requirement for the G-action to be Hamiltonian is that the map ξ ↦ H ξ {displaystyle xi mapsto H_{xi }} be a Lie algebra homomorphism from g {displaystyle {mathfrak {g}}} to the algebra of smooth functions on M under the Poisson bracket.