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  • Print publication year: 2005
  • Online publication date: April 2013

Introduction

Summary

As with many books, this one is best read piecewise backwards. In describing the contents, I accordingly begin with Part III.

Groupoid theory was transformed in the mid 1980s by the introduction of the notion of symplectic groupoid and the methods of Poisson geometry. The announcement by Weinstein [1987] and the seminar notes of Coste, Dazord, and Weinstein [1987] on symplectic groupoids and Poisson geometry became available about late 1986. In fact a similar approach to the use of groupoid structures in Poisson geometry had been given by Karasëv [1989] in papers deposited in VINITI in Moscow in 1981 but not generally available until much later. The two papers of Zakrzewski [1990a,b], gave a third and independent treatment. In most of the discussion in this Introduction I will treat these three very-different approaches as if they were a single body of work.

The work of these authors transformed both the subject and the applications of Lie groupoid and Lie algebroid theory. Until that time only the case of locally trivial Lie groupoids and transitive Lie algebroids was well-understood. Despite the very general programme and results announced by Pradines in four short notes [1966], [1967a], [1967b], [1968], and some isolated work on specific aspects of general Lie groupoids and Lie algebroids, there seemed to be little compelling reason to understand the very difficult general theory.

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