momentum map

Let (M,ω) be a symplectic manifoldMathworldPlanetmath, G a Lie group acting on that manifoldMathworldPlanetmath, 𝔤 its Lie algebraMathworldPlanetmath, and 𝔤* the dual of the Lie algebra. This action induces a map α:𝔤𝔛(M) where 𝔛(M) is the Lie algebra of vector fields on M, such that exp(tX)(m)=ρt(m) where ρ is the flow of α(X). Then a moment map μ:M𝔤* for the action of G is a map such that


Here μ(X)(m)=μ(m)(X), that is, μ(m) is a covector, so we apply it to the vector X and get a scalar function μ(X), and Hμ(X) is its Hamiltonian vector field.

Generally, the moment maps we are interested in are equivariant with respect to the coadjoint action, that is, they satisfy

Title momentum map
Canonical name MomentumMap
Date of creation 2013-03-22 13:14:36
Last modified on 2013-03-22 13:14:36
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 4
Author bwebste (988)
Entry type Definition
Classification msc 53D20
Synonym moment map