Lie groupoids and Lie algebroids in physics and noncommutative geometry

Abstract

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 ∗ ( Γ ) to a Lie groupoid Γ : in noncommutative geometry C ∗ ( Γ ) 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 Γ . Moreover, Connes’s map Γ ↦ C ∗ ( Γ ) has a classical analogue Γ ↦ A ∗ ( Γ ) 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 ( Γ ) of the Lie groupoid Γ , an object generalizing both Lie algebras and tangent bundles. Only a handful of physicists appear to be familiar with Lie groupoids and Lie algebroids, whereas the latter are practically unknown even to mathematicians working in noncommutative geometry: so much the worse for its relationship with symplectic geometry! Thus the aim of this review paper is to explain the relevance of both objects to both audiences. We do so by outlining the road from canonical quantization to Lie groupoids and Lie algebroids via Mackey’s imprimitivity theorem and its symplectic counterpart. This will also lead the reader into symplectic groupoids, which define a ‘classical’ category on which quantization may speculatively be defined as a functor into the category K K defined by Kasparov’s bivariant K-theory of C ∗ -algebras. This functor unifies deformation quantization and geometric quantization, the conjectural functoriality of quantization counting the “quantization commutes with reduction” conjecture of Guillemin and Sternberg among its many consequences.