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LEADER: 03553nam a22005055a 4500
001 013839882-8
005 20131206201327.0
008 121227s1994 xxu| s ||0| 0|eng d
020 $a9781461202691
020 $a9781461202691
020 $a9781461266877
024 7 $a10.1007/978-1-4612-0269-1$2doi
035 $a(Springer)9781461202691
040 $aSpringer
050 4 $aQA252.3
050 4 $aQA387
072 7 $aPBG$2bicssc
072 7 $aMAT014000$2bisacsh
072 7 $aMAT038000$2bisacsh
082 04 $a512.55$223
082 04 $a512.482$223
100 1 $aGuillemin, Victor,$eauthor.
245 10 $aMoment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces /$cby Victor Guillemin.
264 1 $aBoston, MA :$bBirkhäuser Boston :$bImprint: Birkhäuser,$c1994.
300 $aVII, 152 p.$bonline resource.
336 $atext$btxt$2rdacontent
337 $acomputer$bc$2rdamedia
338 $aonline resource$bcr$2rdacarrier
347 $atext file$bPDF$2rda
490 1 $aProgress in Mathematics ;$v122
520 $aThe action of a compact Lie group, G, on a compact sympletic manifold gives rise to some remarkable combinatorial invariants. The simplest and most interesting of these is the moment polytope, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope. For instance, the first chapter is largely devoted to the Delzant theorem, which says that there is a one-one correspondence between certain types of moment polytopes and certain types of symplectic G-spaces. (One of the most challenging unsolved problems in symplectic geometry is to determine to what extent Delzant’s theorem is true of every compact symplectic G-Space.) The moment polytope also encodes quantum information about the actions of G.
520 $aUsing the methods of geometric quantization, one can frequently convert this action into a representations, p , of G on a Hilbert space, and in some sense the moment polytope is a diagrammatic picture of the irreducible representations of G which occur as subrepresentations of p. Precise versions of this item of folklore are discussed in Chapters 3 and 4. Also, midway through Chapter 2 a more complicated object is discussed: the Duistermaat-Heckman measure, and the author explains in Chapter 4 how one can read off from this measure the approximate multiplicities with which the irreducible representations of G occur in p. This gives an excuse to touch on some results which are in themselves of great current interest: the Duistermaat-Heckman theorem, the localization theorems in equivariant cohomology of Atiyah-Bott and Berline-Vergne and the recent extremely exciting generalizations of these results by Witten, Jeffrey-Kirwan, Lalkman, and others.
520 $aThe last two chapters of this book are a self-contained and somewhat unorthodox treatment of the theory of toric varieties in which the usual hierarchal relation of complex to symplectic is reversed. This book is addressed to researchers and can be used as a semester text.
650 20 $aAlgebra.
650 20 $aCombinatorial analysis.
650 10 $aMathematics.
650 0 $aCombinatorial analysis.
650 0 $aMathematics.
650 0 $aAlgebra.
650 0 $aTopological Groups.
650 24 $aTopological Groups, Lie Groups.
776 08 $iPrinted edition:$z9781461266877
830 0 $aProgress in Mathematics ;$v122.
988 $a20131119
906 $0VEN