!AS/Park City Mathematics Series
Volume 7, 1999Hamiltonian Group Actions and
Symplectic Reduction
Lisa C. Jeffrey
LECTURE 1
Introduction to Hamiltonian Group Actions
The idea of a Hamiltonian flow on a symplectic manifold has its roots in Hamil-
ton's equations, which govern the trajectory of a particle in phase space (the space
parametrizing coordinates and momenta of a classical particle). A fundamental
idea in theoretical physics is that to every symmetry in a physical system (such
as a group action), there is an associated conserved quantity: invariance under
translation corresponds to conservation of linear momentum, rotational symmetry
corresponds to conservation of angular momentum and so on, and these momenta
are functions on the phase space. The mathematical formulation of this idea is the
idea of the moment map associated to a group action on a symplectic manifold:
the group action is obtained from the Hamiltonian flow of the moment map.
These lectures will describe some basic features of moment maps associated
to Hamiltonian group actions, and some recent results about the geometry and
topology of symplectic manifolds which have such group actions.
Let ( M, w) be a symplectic manifold. The Hamiltonian vector field ~H gener-
ated by a function H is defined by
Wm(~H, Y) = dHm(Y)for any YE TmM· If XE g f-> X# are the vector fields generated by the symplectic
action of a compact Lie group G with Lie algebra g, then the moment mapμ : M -t
g* is defined by two properties:
(a)
(^1) Dept. of Mathematics, University of Toronto, 100 St. G eorge St., Toronto, ON,
Canada M5S 3G3.
E-mail address: j effrey©math. toronto. edu.
The author acknowledges support from NSERC and FCAR.
@ 1999 American Mathe matical Socie t y
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