Chapter

Symplectic group actions

Dusa McDuff and Dietmar Salamon

in Introduction to Symplectic Topology

Published in print March 2017 | ISBN: 9780198794899
Published online June 2017 | e-ISBN: 9780191836411 | DOI: http://dx.doi.org/10.1093/oso/9780198794899.003.0006

Series: Oxford Graduate Texts in Mathematics

Symplectic group actions

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The chapter begins with a discussion of circle actions and their relation to 2-sphere bundles. It continues with a section on general Hamiltonian group actions and moment maps, then proceeds to discuss various explicit examples in both finite and infinite dimensions, and introduces the Marsden–Weinstein quotient, together with new examples that explain its relation to the construction of generating functions for Lagrangians. Further sections give a proof of the Atiyah–Guillemin–Sternberg convexity theorem about the image of the moment map in the case of torus actions, and use equivariant cohomology to prove the Duistermaat–Heckman localization formula for circle actions. It closes with an overview of geometric invariant theory which grows out of the interplay between the actions of a real Lie group and its complexification.

Keywords: Hamiltonian group action; moment map; convexity; localization; geometric invariant theory

Chapter.  35154 words.  Illustrated.

Subjects: Mathematics ; Geometry

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