By Victor Guillemin
ISBN-10: 1461202698
ISBN-13: 9781461202691
ISBN-10: 1461266874
ISBN-13: 9781461266877
The motion of a compact Lie staff, G, on a compact sympletic manifold supplies upward push to a few impressive combinatorial invariants. the easiest and finest 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 ambitions of this monograph is to explain what forms of geometric details are encoded during this polytope. for example, the 1st bankruptcy is essentially dedicated to the Delzant theorem, which says that there's a one-one correspondence among specific sorts of second polytopes and likely kinds of symplectic G-spaces. (One of the main demanding unsolved difficulties in symplectic geometry is to figure out to what quantity Delzant’s theorem is correct of each compact symplectic G-Space.)
The second polytope additionally encodes quantum information regarding the activities of G. utilizing the tools of geometric quantization, you can still 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 photo of the irreducible representations of G which take place as subrepresentations of p. unique models of this merchandise of folklore are mentioned in Chapters three and four. additionally, halfway via bankruptcy 2 a extra complex 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 offers 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 new tremendous fascinating generalizations of those effects via Witten, Jeffrey-Kirwan, Lalkman, and others.
The final chapters of this e-book are a self-contained and a bit of unorthodox therapy of the idea of toric types within which the standard hierarchal relation of advanced to symplectic is reversed. This ebook is addressed to researchers and will be used as a semester textual content.
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Extra info for Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
Sample text
3). As in Chapter 1, let's denote by 0- the symplectic manifold (0, -wo) and recall that Mo is, by definition, the reduction of the product manifold M x 0-. By axioms I and II and Q(M x 0-) = Q(M) ® Q(O)* = Hom(Q(O), Q(M)). Thus, in particular Q(M X O-)a is the space of G-invariant linear mappings of Q(O) into Q(M), and these are just the intertwining operators from Q(O) to Q(M). 3) dim Q(Mo) = multiplicity of Po in Q(M). In order for us to make computations with this formula we need a recipe for computing the left-hand side, and what I will describe next is a recipe for computing dim Q(X) for any compact symplectic manifold X.
In order for us to make computations with this formula we need a recipe for computing the left-hand side, and what I will describe next is a recipe for computing dim Q(X) for any compact symplectic manifold X. This recipe has the virtue that it doesn't require one to know anything about how Q(X) is defined, but it does involve characteristic classes and numbers of "almost complex" manifolds. So I will pause for a moment and review what we will need to know about this subject: Let us equip X with a Riemannian metric B.
By definition, Now let nO be the annihilator of n in (JRd)*. 8. C E H2 (Xl:,. , JRd). , c is really an n-valued form! Proof. Take any element ~ E (Zd)* and let X~: Td --+ Sl be the character of the group Td corresponding to~. (In other words, xdx) = exp27ri(~,x) for all x E Td. ) Let x~ be the restriction of x~ to N and let L~ be the line bundle over Xl:,. 10). If ~ = (6, ... , ~d) then so the Chern class of L~ is c(L~) = 6C1 + ... + ~dCd = (~, c). However, if ~ E nO, x~ = 1; so L~ is trivial. Thus c(L~) = o.
Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces by Victor Guillemin
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