Convexity Properties of Hamiltonian Group Actions

Convexity Properties of Hamiltonian Group Actions

by Victor Guillemin, Reyer Sjamaar
     
 

ISBN-10: 0821842366

ISBN-13: 9780821842362

Pub. Date: 09/08/2006

Publisher: American Mathematical Society

This is a monograph on convexity properties of moment mappings in symplectic geometry. The fundamental result in this subject is the Kirwan convexity theorem, which describes the image of a moment map in terms of linear inequalities. This theorem bears a close relationship to perplexing old puzzles from linear algebra, such as the Horn problem on sums of Hermitian

…  See more details below

Overview

This is a monograph on convexity properties of moment mappings in symplectic geometry. The fundamental result in this subject is the Kirwan convexity theorem, which describes the image of a moment map in terms of linear inequalities. This theorem bears a close relationship to perplexing old puzzles from linear algebra, such as the Horn problem on sums of Hermitian matrices, on which considerable progress has been made in recent years following a breakthrough by Klyachko. The book presents a simple local model for the moment polytope, valid in the ''generic'' case, and an elementary Morse-theoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinite-dimensional manifestations of moment convexity, such as the Kostant type theorems for orbits of a loop group (due to Atiyah and Pressley) or a symplectomorphism group (due to Bloch, Flaschka and Ratiu). Finally, it gives an account of a new convexity theorem for moment map images of orbits of a Borel subgroup of a complex reductive group acting on a Kahler manifold, based on potential-theoretic methods in several complex variables.

Read More

Product Details

ISBN-13:
9780821842362
Publisher:
American Mathematical Society
Publication date:
09/08/2006
Series:
CRMM, #26
Pages:
82
Product dimensions:
6.90(w) x 9.90(h) x 0.20(d)

Table of Contents

Ch. 1The convexity theorem for Hamiltonian G-spaces7
Ch. 2A constructive proof of the non-abelian convexity theorem21
Ch. 3Some elementary examples of the convexity theorem31
Ch. 4Kahler potentials and convexity55
Ch. 5Applications of the convexity theorem71

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >