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© 1994

Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces

Book

Part of the Progress in Mathematics book series (PM, volume 122)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Victor Guillemin
    Pages 1-4
  3. Victor Guillemin
    Pages 5-23
  4. Victor Guillemin
    Pages 24-53
  5. Victor Guillemin
    Pages 54-80
  6. Victor Guillemin
    Pages 81-108
  7. Back Matter
    Pages 109-152

About this book

Introduction

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 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.

Keywords

Lie algebra Partition algebra cohomology homology

Authors and affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

Bibliographic information

  • Book Title Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
  • Authors Victor Guillemin
  • Series Title Progress in Mathematics
  • Series Abbreviated Title Progress in Mathematics(Birkhäuser)
  • DOI https://doi.org/10.1007/978-1-4612-0269-1
  • Copyright Information Birkhäuser Boston 1994
  • Publisher Name Birkhäuser, Boston, MA
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-8176-3770-5
  • Softcover ISBN 978-1-4612-6687-7
  • eBook ISBN 978-1-4612-0269-1
  • Series ISSN 0743-1643
  • Series E-ISSN 2296-505X
  • Edition Number 1
  • Number of Pages VII, 152
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Topological Groups, Lie Groups
    Combinatorics
    Algebra
  • Buy this book on publisher's site

Reviews

"...carefully written book [on] the so-called Delzant spaces... Many of the statements in the body of the book are left as exercises (with copious hints). At the end of each chapter there are very interesting bibliographical notes. The reader who works through this book will acquire a solid knowledge of this beautiful and elegant part of mathematics."

--Mathematical Reviews