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:
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)
“It has lately been drawn to your correspondent’s attention that, at social gatherings, she is not the human magnet she would be. Indeed, it turns out that as a source of entertainment, conviviality, and good fun, she ranks somewhere between a sprig of parsley and a single ice- skate. It would appear, from the actions of the assembled guests, that she is about as hot company as a night nurse.”
—Dorothy Parker (1893–1967)