Cambridge University Press London Mathematical Society Lecture Note Series 213.

2005, xxxviii + 501 pages, ISBN 0541499283

Review by Th. Th. Voronov in Bull. London Math. Soc.

- Contents, Prologue, Introduction and Preface (30 pages)
- PART I: The general theory
- Lie groupoids: fundamental theory
- Chapter 2:- Lie groupoids: algebraic constructions (31 pages)
- Lie algebroids: fundamental theory
- Lie algebroids: algebraic constructions

- PART II: The locally trivial theory
- Infinitesimal connection theory
- Path connections
- Cohomology and Schouten Calculus for Lie algebroids
- The cohomological obstruction

- PART III: The symplectic and Poisson theory
- Double vector bundles and their duality
- Poisson structures and Lie algebroids
- Poisson groupoids and symplectic groupoids
- Lie bialgebroids

- Appendix

**Cover blurb:** This is a comprehensive and up to date
account of the theory
of Lie groupoids and Lie algebroids, and their importance in
differential geometry, in particular their relations with Poisson
geometry and general connection theory. It covers much work done
since the mid
1980s and includes the first treatment in book form of Poisson
groupoids, Lie bialgebroids and double vector bundles, as well as
a revised account of the relations between locally trival Lie
groupoids, Atiyah sequences, and connections in principal bundles.
As such, this book will be of great interest to all those concerned
with the use of Poisson geometry as a semi-classical limit of
quantum
geometry, as well as to all those working in or wishing to learn
the modern theory of Lie groupoids and Lie algebroids.

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April 2010

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