By Victor Guillemin
The motion of a compact Lie staff, G, on a compact sympletic manifold supplies upward thrust to a couple impressive combinatorial invariants. the best and best of those is the moment polytope, a convex polyhedron which sits contained in the twin of the Lie algebra of G. one of many major objectives of this monograph is to explain what sorts of geometric details are encoded during this polytope. for example, the 1st bankruptcy is basically dedicated to the Delzant theorem, which says that there's a one-one correspondence among particular types of second polytopes and likely forms of symplectic G-spaces. (One of the main tough unsolved difficulties in symplectic geometry is to figure out to what volume Delzant’s theorem is right of each compact symplectic G-Space.)
The second polytope additionally encodes quantum information regarding the activities of G. utilizing the equipment of geometric quantization, it is easy to often convert this motion right into a representations, p , of G on a Hilbert house, and in a few experience the instant polytope is a diagrammatic photograph of the irreducible representations of G which take place as subrepresentations of p. special models of this merchandise of folklore are mentioned in Chapters three and four. additionally, halfway via bankruptcy 2 a extra advanced item is mentioned: the Duistermaat-Heckman degree, and the writer explains in bankruptcy four how you can learn off from this degree the approximate multiplicities with which the irreducible representations of G happen in p. this provides an excuse to the touch on a few effects that are in themselves of serious present curiosity: the Duistermaat-Heckman theorem, the localization theorems in equivariant cohomology of Atiyah-Bott and Berline-Vergne and the hot super intriguing generalizations of those effects by way of Witten, Jeffrey-Kirwan, Lalkman, and others.
The final chapters of this e-book are a self-contained and slightly unorthodox remedy of the speculation of toric kinds during which the standard hierarchal relation of advanced to symplectic is reversed. This publication is addressed to researchers and will be used as a semester textual content.