Lie Groupoids and Lie Algebroids in Differential Geometry

This book provides a striking synthesis of the standard theory of connections in principal bundles and the Lie theory of Lie groupoids.

K. Mackenzie (Author)

9780521348829, Cambridge University Press

Paperback, published 25 June 1987

344 pages
22.8 x 15.2 x 2.1 cm, 0.544 kg

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.

Introduction
1. The algebra of groupoids
2. Topological groupoids
3. Lie groupoids and Lie algebroids
4. The cohomology of Lie algebroids
5. An obstruction to the integrability of transitive Lie algebroids
Appendix A: On principal bundles and Atiyah sequences
Appendix B: On Lie groups and Lie algebras
Appendix C: On vector bundles
References
Index.

Subject Areas: Geometry [PBM]