Lie, symplectic and Poisson groupoids and their Lie algebroids

Abstract

Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional structures (topological and differentiable) as essential tools in topology and differential geometry. In recent years, Mickael Karasev, Alan Weinstein and Stanis{\l}aw Zakrzewski independently discovered that symplectic groupoids can be used for the construction of noncommutative deformations of the algebra of smooth functions on a manifold, with potential applications to quantization. Poisson groupoids were introduced by Alan Weinstein as generalizations of both Poisson Lie groups and symplectic groupoids. We present here the main definitions and first properties relative to groupoids, Lie groupoids, Lie algebroids, symplectic and Poisson groupoids and their Lie algebroids.