Lie Groupoids and Lie Algebroids in Differential Geometry

K. Mackenzie

doi:10.1017/CBO9780511661839

This book provides a striking synthesis of the standard theory of connections in principal bundles and the Lie theory of Lie groupoids. The concept of Lie groupoid is a little-known formulation of the concept of principal bundle and corresponding to the Lie algebra of a Lie group is the concept of Lie algebroid: in principal bundle terms this is the Atiyah sequence. The author's viewpoint is that certain deep problems in connection theory are best addressed by groupoid and Lie algebroid methods. After preliminary chapters on topological groupoids, the author gives the first unified and detailed account of the theory of Lie groupoids and Lie algebroids. He then applies this theory to the cohomology of Lie algebroids, re-interpreting connection theory in cohomological terms, and giving criteria for the existence of (not necessarily Riemannian) connections with prescribed curvature form. This material, presented in the last two chapters, is work of the author published here for the first time. This book will be of interest to differential geometers working in general connection theory and to researchers in theoretical physics and other fields who make use of connection theory.

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Lie Groupoids and Lie Algebroids in Differential Geometry

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

INTRODUCTION
pp vii-xvi

CHAPTER I - The algebra of groupoids
pp 1-15

CHAPTER II - Topological groupoids
pp 16-81

CHAPTER III - Lie groupoids and Lie algebroids
pp 82-184

CHAPTER IV - The cohomology of Lie algebroids
pp 185-258

CHAPTER V - An obstruction to the Integrability of transitive Lie algebroids
pp 259-271

APPENDIX A - On principal bundles and Atiyah sequences
pp 272-307

APPENDIX B - On Lie groups and Lie algebras
pp 308-312

APPENDIX C - On vector bundles
pp 313-316

REFERENCES
pp 317-322

INDEX
pp 323-327

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