Progress in Mathematics

Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces

Authors: Guillemin, Victor

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About this book

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.

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


Table of contents (5 chapters)

Table of contents (5 chapters)

Buy this book

eBook $59.99
price for USA in USD
  • ISBN 978-1-4612-0269-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
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  • Immediate eBook download after purchase
Hardcover $169.99
price for USA in USD
  • ISBN 978-0-8176-3770-5
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
Softcover $79.99
price for USA in USD
  • ISBN 978-1-4612-6687-7
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
Authors
Series Title
Progress in Mathematics
Series Volume
122
Copyright
1994
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4612-0269-1
DOI
10.1007/978-1-4612-0269-1
Hardcover ISBN
978-0-8176-3770-5
Softcover ISBN
978-1-4612-6687-7
Series ISSN
0743-1643
Edition Number
1
Number of Pages
VII, 152
Topics