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I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any single place to learn about Hamiltonian groups.

I have found some books (even available online from the author!) that come highly recommended, specifically: Introduction to Symplectic and Hamiltonian Geometry, A. C. da Silva.

As the title suggests however, this seems to come from more of a geometric standpoint. Which books are recommended that might focus on the group-theory and topology end of this subject? The project description specifically mentions cohomological obstructions, something that I think is related to group cohomology? (At this point I'm getting all of this from Wikipedia...)

I have had basic, introductory courses in Differential Geometry (in $\mathbb{R}^n$) and in topology (up to calculating the first fundamental group of a topological space).

Thank you in advance for any input!

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Cohomological obstructions to what? To existence of (equivariant) moment maps? If so then e.g. Weinstein's Lecture 4 has more details than da Silva's §4.9. Equivariance is well-treated in Ortega-Ratiu or originally Souriau. –  Francois Ziegler May 26 at 18:04
You may wish to check out some of the discussion/references in this question: mathoverflow.net/questions/55442 –  José Figueroa-O'Farrill May 26 at 23:20

2 Answers 2

up vote 2 down vote accepted

The best Book about Hamiltonian action is

Moment Maps, Cobordisms, and Hamiltonian Group Actions Par Victor Guillemin,Yael Karshon,Viktor L. Ginzburg

The second excellent lecture note is from Heckman

Lecture notes on Geometry of the momentum map, written with Gert Heckman,

Moreover, from geometric point of view this book is excellent

Convexity Properties of Hamiltonian Group Actions Victor Guillemin,Reyer Sjamaar

Following note is also good and introductory

Hamiltonian group actions, Sara Grundel

Also the master thesis entitled "The Momentum Map, Symplectic Reduction and an Introduction to Brownian Motion" which supervised by Alan Weinstein is very good

And in final if you know french as Francois said , Structure des systèmes dynamiques

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The project is now finished, and for anybody else looking to do something similar, I would like to add the following book as an excellent source for an introduction to the material:

An Introduction to Symplectic Geometry, Berndt

This was found more helpful than any of the others, (save perhaps da Silva's lectures) as a short-term introduction to the subject.

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