Lie Groupoids

TL;DR

Lie groupoids are a generalization of Lie groups that encode complex symmetries and have significant applications across geometry, physics, and mathematical structures, with a comprehensive overview of their theory and key concepts.

Contribution

This paper provides an extensive overview of Lie groupoids, covering fundamental definitions, examples, and advanced topics like actions, Morita equivalence, and cohomology.

Findings

Lie groupoids generalize Lie groups with partial multiplication.

They are connected to Poisson geometry and non-commutative geometry.

Applications include classical mechanics, quantization, and topological field theories.

Abstract

A Lie groupoid can be thought of as a generalization of a Lie group in which the multiplication is only defined for certain pairs of elements. From another perspective, Lie groupoids can be regarded as manifolds endowed with a type of action codifying internal and external symmetries. The vigorous development of their theory in the last few decades has been largely stimulated by their connections with such areas as Poisson geometry and non-commutative geometry, as well as several topics in mathematical physics, including classical mechanics, quantization and topological field theories. This article is an overview on Lie groupoids, including basic definitions, key examples and constructions, and topics such as actions and representations, local models, Morita equivalence and cohomology.