Lie Groupoids and Lie algebroids in physics and noncommutative geometry

TL;DR

This review explores how Lie groupoids and Lie algebroids serve as fundamental tools in noncommutative geometry and physics, linking symmetries, quotient spaces, and quantization through algebraic and geometric structures.

Contribution

It clarifies the role of Lie groupoids and algebroids in connecting noncommutative geometry with physical symmetries and classical systems, emphasizing the unifying framework they provide.

Findings

Lie groupoids generalize groups, spaces, and equivalence relations.

C*-algebras of groupoids model noncommutative spaces in geometry.

Lie algebroids relate to classical Poisson structures and symplectic geometry.

Abstract

The aim of this review paper is to explain the relevance of Lie groupoids and Lie algebroids to both physicists and noncommutative geometers. Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is the importance of Connes's idea of associating a C*-algebra C*(G) to a Lie groupoid G: in noncommutative geometry C*(G) replaces a given singular quotient space by an appropriate noncommutative space, whereas in physics it gives the algebra of observables of a quantum system whose symmetries are encoded by G. Moreover, Connes's map G -> C*(G) has a classical analogue G -> A*(G) in symplectic geometry due to Weinstein, which defines the Poisson manifold of the corresponding classical system as the dual of the so-called Lie algebroid A(G) of the Lie groupoid G, an object generalizing both Lie algebras and tangent bundles. This will also lead into symplectic groupoids and the conjectural functoriality of quantization.