Moment map

In mathematics, specifically in symplectic geometry, the moment map (or momentum 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 moment 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 ${\displaystyle {\mathfrak {g}}}$ be the Lie algebra of G, ${\displaystyle {\mathfrak {g}}^{*}}$ its dual, and

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