Groupoids: Unifying Internal and External Symmetry
نویسندگان
چکیده
744 NOTICES OF THE AMS VOLUME 43, NUMBER 7 Introduction Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the well-known Erlanger program of F. Klein and the related work of S. Lie, which virtually defined geometric structures by their groups of automorphisms. (See, for example, Yaglom’s account in [27].) In fact, though groups are indeed sufficient to characterize homogeneous structures, there are plenty of objects which exhibit what we clearly recognize as symmetry, but which admit few or no nontrivial automorphisms. It turns out that the symmetry, and hence much of the structure, of such objects can be characterized algebraically if we use groupoids and not just groups. The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which involve Lie groupoids and their corresponding infinitesimal objects, Lie algebroids. These objects play a role in the study of general differentiable manifolds and partial differential equations, which is, in a sense, an extension of the role which Lie groups play in the geometry and analysis of highly symmetric manifolds. Some History The following historical remarks are not intended to be complete but merely to indicate the breadth of areas where groupoids have been used. An extensive survey of groupoids as of 1986 can be found in R. Brown’s article [2].
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