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Bücher Englische Bücher Wissen & Bildung Naturwissenschaft, Medizin, Informatik, Technik Mathematik Arithmetik & Algebra Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
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Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
Victor Guillemin (Broschiert, Englisch)
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Beschreibung
The 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 . Using 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. The last two chapters of this book are a self-contained and somewhat unorthodoxtreatment of the theory of toric varieties in which the usual hierarchal relation of complex to symplectic is reversed. This book is addressed to researchers 106,99 €
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Technische Daten
Erscheinungsdatum
08.10.2012
Sprache
Englisch
EAN
9781461266877
Herausgeber
Birkhäuser Boston
Serien- oder Bandtitel
Progress in Mathematics
Sonderedition
Nein
Autor
Victor Guillemin
Seitenanzahl
152
Einbandart
Broschiert
Schlagwörter
Lie algebra, Partition, algebra, cohomology, homology, combinatorics
Thema-Inhalt
PBG - Gruppen und Gruppentheorie
PBD - Diskrete Mathematik
PBF - Algebra
Höhe
235 mm
Breite
15.5 cm
Transparenz & Sicherheit
Hersteller: Springer Nature Customer Service Center GmbH, Europaplatz 3, Heidelberg, Deutschland, 69115, ProductSafety@springernature.com
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