Hamiltonian Group Actions
The definition of the moment map requires to be exact. 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 the one-form is exact, meaning that it equals for some smooth function
If this holds, then one may choose the to make the map linear. The second requirement for the G-action to be Hamiltonian is that the map be a Lie algebra homomorphism from to the algebra of smooth functions on M under the Poisson bracket.
If the action of G on (M, ω) is Hamiltonian in this sense, then a moment map is a map such that writing defines a Lie algebra homomorphism satisfying . Here is the vector field of the Hamiltonian, defined by
Read more about this topic: Moment Map
Famous quotes containing the words group and/or actions:
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)
“Throughout my life, I have seen narrow-shouldered men, without a single exception, committing innumerable stupid acts, brutalizing their fellows and perverting souls by all means. They call the motive for their actions fame.”
—Isidore Ducasse, Comte de Lautréamont (1846–1870)