Lie algebroids and Lie pseudoalgebras.

*(English)*Zbl 0829.22001This paper presents a survey of the generalizations of the Lie theory of Lie groups and Lie algebras which have been introduced since the 1950’s. Lie algebroids and Lie pseudoalgebras arise from an enormous variety of constructions in differential geometry. Motivation for their introduction has come from geometry, physics, and algebra, and the author claims that they have been introduced under fourteen or more different terminologies. The purpose of this paper is to collect, summarize, and clarify an unrecognized part of the folklore of differential geometry.

The first part of this survey discusses the four principal classes of geometric constructions in which Lie algebroids and Lie pseudoalgebras arise (from principal bundles, Lie groupoids, symplectic groupoids and Poisson manifolds, and various types of foliations), and emphasizes how each arises as a generalization of basic Lie theory. The second part of the survey focuses on the algebraic aspects, describes various algebraic constructions for Lie algebroids and Lie pseudoalgebras, and comments on their geometrical significance.

The first part of this survey discusses the four principal classes of geometric constructions in which Lie algebroids and Lie pseudoalgebras arise (from principal bundles, Lie groupoids, symplectic groupoids and Poisson manifolds, and various types of foliations), and emphasizes how each arises as a generalization of basic Lie theory. The second part of the survey focuses on the algebraic aspects, describes various algebraic constructions for Lie algebroids and Lie pseudoalgebras, and comments on their geometrical significance.

Reviewer: W.J.Satzer jun.(St.Paul)

##### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

17B65 | Infinite-dimensional Lie (super)algebras |

53D05 | Symplectic manifolds (general theory) |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

58H05 | Pseudogroups and differentiable groupoids |