: Unifying Internal and External Symmetry A Tour through Some Examples

Alan Weinstein

Introduction Some History Mathematicians tend to think of the notion of The following historical remarks are not in- symmetry as being virtually synonymous with tended to be complete but merely to indicate the the theory of groups and their actions, perhaps breadth of areas where groupoids have been largely because of the well-known Erlanger pro- used. An extensive survey of groupoids as of gram of F. Klein and the related work of S. Lie, 1986 can be found in R. Brown’s article [2]. which virtually defined geometric structures by their groups of automorphisms. (See, for exam- Alan Weinstein is professor of mathematics at the Uni- ple, Yaglom’s account in [27].) In fact, though versity of California at Berkeley. His e-mail address is groups are indeed sufficient to characterize ho- [email protected]. mogeneous structures, there are plenty of objects which exhibit what we clearly recognize as sym- metry, but which admit few or no nontrivial au- Research partially supported by NSF Grant DMS-93- tomorphisms. It turns out that the symmetry, 09653. This article is based on a talk given first at UCB and hence much of the structure, of such objects (the University of California at Berkeley) in April 1995 can be characterized algebraically if we use during Mathematics Awareness Week, the theme of which was Mathematics and Symmetry. I would like to groupoids and not just groups. thank the Department of Mathematics of another UCB The aim of this paper is to explain, mostly (the University of Colorado at Boulder) for an invitation through examples, what groupoids are and how to Boulder in November 1995, where a similar talk was they describe symmetry. We will begin with el- most recently given and where the first draft of this man- ementary examples, with discrete symmetry, uscript was largely written in the downtown cafés. and end with examples in the differentiable set- Thanks should also be given to a third UCB (Université ting which involve Lie groupoids and their cor- Claude-Bernard, Lyon), where I started my intensive responding infinitesimal objects, Lie algebroids. work on groupoids in the summer of 1986. George These objects play a role in the study of general Bergman, Ronnie Brown, Pierre Dazord, Jack Feldman, differentiable and partial differential Kirill Mackenzie, George Mackey, Saunders Mac Lane, Jean Pradines, Arlan Ramsay, Sasha Voronov, and the equations, which is, in a sense, an extension of referees made helpful comments on the original man- the role which Lie groups play in the geometry uscript. Finally, I would like to thank Paul Brown for and analysis of highly symmetric manifolds. preparing the figures.—A. Weinstein

744 NOTICES OF THE AMS VOLUME 43, NUMBER 7 Groupoids were introduced (and named) by H. Brandt [1] in a 1926 paper on the composi- tion of quadratic forms in four variables. C. Ehresmann [6] added further structures (topo- logical and differentiable as well as algebraic) to groupoids, thereby introducing them as a tool in differentiable topology and geometry. In algebraic geometry, A. Grothendieck used groupoids extensively and, in particular, showed how they could be used to tame the unruly equivalence x relations which arise in the con- Figure 1. Tiling of the plane by 2 1 rectangles. struction of moduli spaces. (See better-known of differentiable [8], with further details in [7].) As the role of mod- transformations. (See the brief remark at the uli spaces expands in physics as well as math- end of the following section or [8] for more de- ematics, groupoids continue to play an essential tails.) role. Finally, the author’s own interest in groupoids In analysis, G. Mackey [16] used groupoids arose from the discovery [24, 25], made inde- under the name of virtual groups to allow the pendently at around the same time by Karasev treatment of ergodic actions of groups, “as if’’ [10] and Zakrzewski [28], that a with they were transitive, while it was observed that compatible symplectic structure is the appro- the convolution operation, extended from groups priate geometric model for a family of non- to groupoids, made possible the construction of commutative algebras obtained by deformation a multitude of interesting noncommutative al- from the algebra of functions on a gebras. (See [21] for a survey.) In this context, the with a Poisson bracket structure. use of groupoid convolution algebras as a sub- stitute for the algebras of functions on badly be- Global and Local Symmetry Groupoids haved quotient spaces is a central theme in the We begin our exposition with a simple example noncommutative geometry of A. Connes [4]. The which will lead to the definition of a groupoid. book just cited also shows the extent to which 2 Consider a tiling of the euclidean plane R by groupoids provide a framework for a unified identical 2 1 rectangles, specified by the set study of operator algebras, , and index X (idealized× as 1-dimensional) where the grout theory. between the tiles lies: X = H V, where In , the fundamental ∪ H = R Z and V =2Z R . (See Figure 1.) We groupoid of a topological space has been ex- × × 2 will call each connected component of R X a ploited by P. Higgins, R. Brown, and others (see tile. \ Brown’s textbook [3], or Higgins’ book [9], which The symmetry of this tiling is traditionally de- is a good general introduction to groupoids as scribed by the consisting of those rigid well) in situations where the use of a fixed base 2 motions of R which leave X invariant. It con- point as imposed by the usual fundamental sists of the normal subgroupΓ of translations by group would be too restrictive. Groupoid meth- elements of the lattice = H V =2Z Z(cor- ods are thus well adapted to disconnected spaces responding to corner points∩ of the tiles),× to- (which arise frequently when connected spaces gether with reflections throughΛ each of the points are cut into pieces, as in the van Kampen theo- 1 1 in 2 = Z 2 Z and across the horizontal and rem) and to spaces with fixed-point-free group vertical lines× through those points. actions. NowΛ consider what is lost when we pass from The extension of Lie theory from differen- X to . tiable groups to groupoids was carried out for • The same symmetry group would arise if we the most part by J. Pradines, as described in a hadΓ replaced X by the lattice of corner series of notes ending with [20]. An exposition points, even though looks quite different of this work, together with a detailed study of Λ the role of groupoids and Lie algebroids in dif- 1We note that, before theΛ appearance of [5], Lie ferential geometry, can be found in the book of groupoids were called differentiable groupoids, the K. Mackenzie [13].1 There is also a close relation term “Lie” being reserved for those satisfying a certain between Lie algebroids and Lie groupoids and the transitivity property.

JULY 1996 NOTICES OF THE AMS 745 (g,h) gh satisfying the conditions 1–4 above.7→ We may think of each element g of G as an arrow pointing from β(g) to α(g) in B; arrows are multiplied by placing them head to tail, as in Figure 2. Many properties of groupoids suggested by this picture are easily proven. For in- 1 1 stance, α(g− )=β(g) (since gg− is de- Figure 2. Multiplication in a groupoid. fined), and α(gh)=α(g) (since (λgg)h = λg(gh) implies that λg(gh) is de- from X. (Of course, there are reasons why fined). One also shows easily that α(g) and λg this loss of detail in passing from X to is (or β(g) and ρg) determine one another, thus welcome, as is the case for many math- producing a bijective mapping ι from the base ematical abstractions, but read on.) Γ B to the subset G(0) G consisting of the iden- • The group retains no information about tity elements. The reader⊆ familiar with cate- the local structure of the tiled plane, such gories will probably have realized by now that as the fact Γthat neighborhoods of all the a groupoid is just a in which every mor- points inside the tiles look identical if the phism has an inverse.3 tiles are uniform, while they may look dif- Now we return to our transformation ferent if the tiles are painted with a design 2 groupoid G( , R ) and form its restriction to (which could still be invariant under ). 2 B =[0,2m] [0,n](or any other subset B of R ) × • If, as is the case on real bathroom floors, the by defining Γ tiling is restricted to a finite part ofΓ the 2 2 plane, such as B =[0,2m] [0,n], the sym- G( , R ) B = g G( , R ) α(g) metry group shrinks drastically.× The sub- | { ∈ | and β(g) belong to B . group of leaving X B invariant contains Γ Γ } just 4 elements, even though∩ a repetitive pat- The following concepts from groupoid theory, tern on theΓ bathroom floor is clearly visible. applied to this example, now show that 2 Our first groupoid will enable us to describe G( ,R ) B indeed captures the symmetry which the symmetry of the finite tiled rectangle. We we see in| the tiled floor. first define the transformation groupoid of the •Γ An orbit of the groupoid G over B is an 2 action of on R to be the set2 equivalence class for the relation

Γ x G y if and only if there is a groupoid 2 2 2 ∼ G( , R )= (x, γ, y) x R ,y R ,γ , { | ∈ ∈ ∈ element g with α(g)=xand β(g)=y. and x = γy Γ } Γ In the example, two points are in the same with the partially defined binary operation orbit if they are similarly placed within their tiles or within the grout pattern. (We are al- (x, γ, y)(y,ν,z)=(x, γν, z). lowed to turn the tiles over, as well as to 2 This operation on G = G( , R ) has the follow- translate them or rotate them by 180◦.) ing properties. • The isotropy group of x B consists of ∈ 1. It is defined only forΓ certain pairs of ele- those g in G with α(g)=x=β(g). In the ex- ments: gh is defined only when β(g)=α(h) ample, the isotropy group is trivial for every 1 for certain maps α and β from G onto 2; point except those in B, for which it is R 2 ∩ here α :(x, γ, y) x and β :(x, γ, y) y. Z2 Z2. × 2. It is associative:7→ if either of the products7→ We can describe local symmetriesΛ of our tiling (gh)k or g(hk) is defined, then so is the by introducing another groupoid. Consider the 2 other, and they are equal. plane R as the disjoint union of P1 = B X (the ∩ 2 3. For each g in G, there are left and right iden- grout), P2 = B P1 (the tiles), and P3 = R B \ \ tity elements λg and ρg such that (the exterior of the tiled room). Let E be the λgg = g = gρg. group of all euclidean motions of the plane, and 1 4. Each g in G has an inverse g− for which define the local symmetry groupoid Gloc as the gg 1 = λ and g 1g = ρ . set of triples (x, γ, y) in B E B for which − g − g × × More generally, a groupoid with base B is a x = γy, and for which y has a neighborhood U set G with mappings α and β from G onto B and a partially defined binary operation 3Actually, not quite. The elements of a category can con- stitute a class rather than a set, so a groupoid is a small 2 2 2As a set, G( , R ) is isomorphic to R , but we pre- category with every morphism invertible. This is just fer the following more symmetric description,× which a question of terminology, though; many authors allow makes the compositionΓ law more transparent.Γ their groupoids to be classes.

746 NOTICES OF THE AMS VOLUME 43, NUMBER 7 2 in R such that γ( Pi) Pi for i =1,2,3. The composition is givenU∩ by the⊆ same formula as for 2 G( , R ). For this groupoid, there are only a finite num- berΓ of orbits (see Figure 3). 1 = interior points of tiles O 2 = interior edge points O 3 = interior crossing points O 4 = boundary edge points O 5 = boundary “T” points O 6 = boundary corner points O The isotropy group of a point in 1 is now O isomorphic to the entire rotation group O(2); it Figure 3. The point labelled by j belongs to the orbit j of the O 2 is Z2 Z2 for 2, the 8-element dihedral group groupoid G( , R ) B. × O | D4 for 3, and Z2 for 4, 5, and 6. O O O O Gloc L now has just three orbits, with isotropy Γ | groups isomorphic to D4, Z2, and Z2 as before. (See Figure 4.) At this point, we have something like the smile which remains after the Cheshire Cat has disappeared. The set L has no structure at all except for that provided by the groupoid Gloc L, which is a sort of relic of the symmetry structure| which was given by the original de- composition into grout, tiles, and exterior. We could say that the points of L have “internal sym- metry” specified by the isotropy groups of Gloc L and that the “types” of these points are classi-| fied by the orbit structure of the groupoid. This internal symmetry and type classification can be Figure 4. Orbit decomposition of the finite groupoid Gloc L. viewed either as a relic of the original tiling or | simply as a “geometric structure” on L. The situation here is similar to that in gauge Groupoids and Equivalence Relations theory, where the points of space-time are If B is any set, the product B B is a groupoid × equipped with internal symmetry groups whose over B with α(x, y)=x, β(x, y)=y, and representations correspond to physical parti- (x, y)(y,z)=(x, z). The identities are the (x, x), 1 cles. But gauge groups and gauge transforma- and (x, y)− =(y,x). We call this the pair tions are applicable only because all the points groupoid of B.5 Note that a subgroupoid of B B, × of space-time look alike. To describe nonhomo- i.e., a subset closed under product and inversion geneous structures like the tiled floor requires and containing all the identity elements,6 is noth- groupoids rather than groups.4 ing but an on B. The orbits To close this section, we mention that an- are the equivalence classes, and the isotropy other approach to local symmetry is through subgroups are all trivial. the theory of pseudogroups, whose elements If G is any groupoid over B, the map typically consist of selected homeomorphisms (α, β):G B Bis a morphism from G to the → × between open subsets of a space of interest, pair groupoid of B. (A groupoid morphism from with the composition of two transformations G over B to G0 over B0 is a pair of maps G G0 → defined on the largest possible domain. The and B B0 compatible with the multiplication, → “germs” of elements of a form a “head,” and “tail” maps of the two groupoids.) groupoid, while in the other direction certain sub- The image of (α, β) is the orbit equivalence re- sets of a groupoid admit a multiplication which lation G, and the kernel is the union of the ∼ makes them the abstract counterpart of a isotropy groups. pseudogroup (in the same sense that a group is The morphism (α, β) suggests another way of the abstract counterpart of a set of invertible looking at groupoids. Until now, we have been mappings closed under composition) called an describing them as generalized groups, but now inverse semigroup. We refer to [21] for an ex- position of inverse semigroups and their relation 5Other authors use the terms coarse groupoid or banal to groupoids. groupoid for this object. 6One often omits this last condition from the definition 4Actually, groupoids are useful even in the realm of of a subgroupoid. To emphasize it, we can refer to a gauge theory. (See [14].) wide subgroupoid.

JULY 1996 NOTICES OF THE AMS 747 we want to think of groupoids as generalized Groupoid Convolution and Matrices equivalence relations as well. From this point of The convolution of two complex valued func- view, a groupoid over B tells us not only which tions on a group G is defined as the sum elements of B are equivalent to one another (or “isomorphic”), but it also parametrizes the dif- 1 (a b)(g)= a(k)b(k− g), ∗ ferent ways (“isomorphisms”) in which two ele- k G ments can be equivalent. This leads us to the fol- X∈ lowing guiding principle of Grothendieck, at least when the support of each function (i.e., Mackey, Connes, Deligne,… the set where it is not zero) is finite. When the supports are infinite, the sum may still be de- Almost every interesting equivalence fined if the functions are absolutely summable. relation on a space B arises in a nat- One obtains in this way, for instance, the rules ural way as the orbit equivalence re- for multiplying Fourier series as functions on the lation of some groupoid G over B. In- additive group . stead of dealing directly with the Z To extend the definition of convolution from orbit space B/G as an object in the groups to groupoids, one replaces the range of category of sets and mappings, map summation G by the α-fibre G = α 1(α(g)). one shouldS consider instead the g − Alternatively, one may use the more symmetric groupoid G itself as an object in the formula for convolution: category htp of groupoids and ho- G motopy classes of morphisms. (a b)(g)= a(k)b(`). ∗ Here is the definition of . If (k,`) k`=g { X| } For instance, if G = B B is the pair groupoid over B = 1, 2,... ,n , the× convolution opera- tion (defined{ on all functions)} is

(a b)(k, j)= a(i,k)b(i,j); ∗ (i,k)(k,jX)=(i,j)

and if we write aij for a(i,j), we get n (a b)ij = aikbkj , are morphisms of groupoids for i =1,2, a ho- ∗ k=1 motopy (or natural transformation in the lan- X guage of categories) from f 1 to f 2 is a map which exhibits ( 1, 2,... ,n 1,2,... ,n ) F { }×{ } h : B G0 with the following properties. as the algebra of n n matrices. → 1 × • For each x B, α0(h(x)) = fB (x) and This may look like a peculiar way of viewing 2 ∈ β0(h(x)) = fB (x). the algebra of matrices, but Connes ([4], pp. • For each g G, h(α(g))f 2(g)=f1(g)h(β(g)). 33–39, and elsewhere) has made the point that ∈ G G An isomorphism in htp between the it was precisely as a groupoid algebra that G groupoids G and G0 (sometimes called an equiv- Heisenberg constructed his original formula- alence of groupoids) gives a bijection between the tion of quantum mechanics. (The elements of the orbit spaces B/G and B/G0, as well as group iso- groupoid were transitions between excited states morphisms between corresponding isotropy sub- of an atom. Only later were the elements of the groups in G and G0. In the absence of further groupoid algebra identified as matrices.) The structure on the groupoids, an equivalence is es- noncommutativity of the algebra of observables sentially no more than this, but when the in quantum mechanics is then seen to be a di- groupoids (and their bases) have measurable, rect consequence of the noncommutativity of the topological, differentiable, algebraic, or sym- pair groupoid B B, as compared with the com- × plectic (and Poisson) structures, the quotient mutativity of the group Z underlying the Fourier spaces can be rather nasty objects, outside the series representation of oscillatory motion in category originally under consideration, and it classical mechanics. is essential to focus attention on the groupoids themselves. One tool which makes this focus Topological Groupoids and Groupoid effective is an algebra associated to the groupoid Algebras G over B which is a useful substitute for the For many interesting groupoids, such as the pair space of functions on B/G when the quotient is groupoid of the real numbers, the functions a “bad” space. We begin to describe this algebra with finite or even countable support are rather in the next section. unnatural objects, and it is more common to con- sider continuous functions. To define the con-

748 NOTICES OF THE AMS VOLUME 43, NUMBER 7 volution of such functions, it is necessary to re- (including unbounded ones) satisfying reason- place the sum in the definition by an integral. able continuity properties can be realized by An appropriate general setting for this extended kernels which are distributions, or generalized n n notion of convolution is that of topological functions on R R . (See [23].) groupoids. To define convolution× operations on func- A groupoid G is a topological groupoid over tions on a general locally compact topological B if G and B are topological spaces and α, β, groupoid, we need to start with a family µx of and multiplication are continuous maps. It turns measures along the α-fibres, which will {fill }the out that this notion is useful for many purposes role played by the measure dz in (1) above. Con- beyond the definition of convolution. A criti- volution is then defined9 by cism sometimes applied to the theory of groupoids is that their classification up to iso- 1 (a b)(g)= a(k)b(k− g)dµg(k). morphism is nothing other than the classifica- ∗ ZGg tion of equivalence relations (via the orbit equiv- alence relation) and groups (via the isotropy This operation is associative if the measures µg groups). The imposition of a compatible topo- satisfy a suitable left-invariance property. logical structure produces a nontrivial interac- From continuous functions on G, one can tion between the two structures. (The theory of pass by various processes of completion and lo- principal bundles is just a special case of this.7) calization to larger groupoid algebras, which play the role of continuous functions, measur- For instance, the groupoid Gloc L of local symmetries introduced in the section| entitled able functions, or even distributions on the orbit “Global and Local Symmetry Groupoids” can be space of G. We refer to the books [4, 19], and entirely decomposed into its restrictions to the [21] for more details. A more intrinsic construction of groupoid al- orbits 3, 5, and 6. In other words, the groupoidO containsO no Oinformation about the re- gebras, free of the choice of measures, can also lation of the orbits to one another. This relation be given. It includes as special cases the convo- can be encoded, however, in a suitable topology lution of measures on a group G and the mul- on the groupoid. tiplication of kernels of operators on Mackey’s To return to our discussion of convolution on “intrinsic Hilbert spaces” [15]. The definition groupoids, we consider an example. If can be found on p. 101 of [4] for the case of Lie n n groupoids (see next section “Lie Groupoids and G = R R , we can define a convolution oper- × Lie Algebroids” below). A construction modeled ation on the space Cc (G) of continuous functions with proper support (i.e., for each compact sub- on the definition of the intrinsic Hilbert spaces n n should extend this idea to a general locally com- set K of R , the intersections of K R and n × pact groupoid with a prescribed left-invariant R K with (x, y) a(x, y) =0 have compact × { | 6 } family of measure classes along the α-fibres. closure). Multiplication in Cc (G) is given by the integral formula Lie Groupoids and Lie Algebroids (1) (a b)(x, y)= a(x, z)b(z,y) dz, A groupoid G over B is a Lie groupoid if G and n ∗ ZR where dz is Lebesgue measure.8 B are differentiable manifolds, and α, β, and 10 This “continuous matrix multiplication” is pre- multiplication are differentiable maps. For ex- cisely the composition law which we obtain by ample, the pair groupoid M M over any dif- ferentiable manifold M is a Lie× groupoid, and any thinking of each a Cc (G) as the “kernel” (here the term is used in∈ the sense of Schwartz [23]) is a Lie groupoid over a single point. 2 n The infinitesimal object associated with a Lie of an operator a˜ on L (R ): groupoid is a . A Lie algebroid over a manifold B is defined to be a A (a˜)(x)= a(x, y)(y)dy. n over B with a structure [ , ] on its ZR space of smooth sections, together with a bun- Not every operator (not even the identity) is re- dle map ρ (called the anchor of the Lie algebroid) alized by a continuous kernel, but all operators from A to the TB satisfying the conditions: 7Of course, a completely different retort could be made 1. [ρ(X),ρ(Y)] = ρ([X,Y]); to the criticism above. By the same criterion, finite-di- 2. [X,ϕY]=ϕ[X,Y]+(ρ(X) ϕ)Y. · mensional vector spaces should be completely unin- teresting, but the study of their morphisms is a very rich subject. 9This definition seems to be due to J. Westman [26]. 8Dirac’s notation x a y for a(x, y) emphasizes the 10A further technical condition is included in the def- idea that the valuesh |of| aiare “transition amplitudes” inition to insure that the domain of multiplication is a between “states” and so fits nicely with the groupoid smooth submanifold of G G; the derivatives of α and interpretation of (1). β are required to have maximal× rank everywhere.

JULY 1996 NOTICES OF THE AMS 749 there may be no such G0 for which the induced morphism G0 G is globally one-to-one. More→ seriously (and contrary to an incorrect assertion in [20]), there exist Lie algebroids which are “nonintegrable” in the sense that they are not the Lie alge- broid of any global Lie groupoid at all. (There is a local Lie groupoid corresponding to any Lie algebroid, though.) We refer to Chapter V of [13] for a dis- cussion of all this, together with further references.

Boundary Lie Algebroids In this section we will explain Figure 5. The groupoid G with an element of and its image under ρ. how Lie algebroids provide a nat- A ural setting for understanding Here, X and Y are smooth sections of A, ϕ is some recent work by R. Melrose [17, 18] on the a smooth function on B, and the bracket on the analysis of (pseudo)differential operators on left-hand side of the first condition is the usual manifolds with boundary. (Some prior knowledge Jacobi-Lie bracket of vector fields. For example, about such operators on manifolds without the tangent bundle TB with the usual bracket, boundary will be assumed here.) and the identity map as anchor, is a Lie algebroid The space (M,∂M) of vector fields which are over B, while any Lie algebra is a Lie algebroid tangent to theX boundary ∂M of a manifold M over a single point. forms a Lie algebra over R and a module over The Lie algebroid of a Lie groupoid is defined C∞(M), and the condition [X,ϕY]= via left-invariant vector fields, just as it is in the ϕ[X,Y]+(X ϕ)Y is satisfied (since it is satis- case of Lie groups. Since the left translation by fied for all vector· fields and functions). Re- b a groupoid element g maps only from Gβ(g) to markably, there is a vector bundle TM over M Gα(g), we must first restrict the values of our vec- whose sections are in one-to-one correspon- tor fields to lie in the subbundle T αG = ker(Tα) dence with the elements of (M,∂M) via a bun- X of TG consisting of vectors tangent to the α-fi- dle map ρ : bTM TM which is an isomor- → bres. (Here, Tαis the derivative of α.) For every phism over the interior of M and whose image 1 x in B and every g in the β-fibre β− (x), the fibre over a boundary point is the tangent space to ∂M α α b Tg G can be identified with Tι(x)G via left trans- at that point. These structures make TM into lation by g. Thus, the space of left-invariant sec- a Lie algebroid. We refer to [17] for a precise de- α 1 b tions of T G along β− (x) can be identified with finition of TM, mentioning here only that, α the single space xG = T G, and Tβ gives a when (y1,... ,yn 1,x) are local coordinates for A ι(x) − well-defined map ρx : xG TxB. The (disjoint) M defined on an open subset of the half space A → union of these objects for all x in B forms a vec- x 0, a local basis over C∞(M) for the sections of≥ bTM is given by ( ∂ ,... , ∂ ,x ∂ ). tor bundle G over B with a bundle map ∂y1 ∂yn 1 ∂x A − ρ : G TB. (See Figure 5.) Identifying the sec- Melrose develops analysis on M by regarding A → tions of G with left-invariant sections of the b-tangent bundle bTM as the “correct” tan- A T αG TG and using the usual bracket on sec- gent bundle for this manifold with boundary. ⊆ tions of TG defines a bracket of sections on G Thus, the algebra of b-differential operators is A making it into a Lie algebroid with anchor ρ. the algebra of operators on C∞(M) generated by For instance, the Lie algebroid of B B is TB, the sections of bTM (acting on functions via ρ) × 11 and by extension of a well-known idea from Lie and C∞(M) (acting on itself by multiplication). group theory, one may think of the Jacobi-Lie The principal symbols of these operators are ho- bracket on sections of TB as being the infini- mogeneous functions on the dual, or b-cotan- b tesimal remnant of the noncommutativity of the gent bundle T ∗M. Inverting (modulo smooth- composition law (x, y)(y,z)=(x, z). ing operators) elliptic b-differential operators The local versions of Lie’s three fundamental leads to the notion of b-pseudodifferential op- theorems all extend from groups to groupoids, but there are some interesting differences at the global level. Although every subalgebroid 11Such a “universal enveloping algebra” was already A of G is indeed G for a Lie groupoid G , defined for general Lie algebroids by Rinehart [22]. 0 A A 0 0

750 NOTICES OF THE AMS VOLUME 43, NUMBER 7 erators, whose symbols are again func- tions on the b-cotangent bundle. Since the b-pseudodifferential op- erators act on C∞(M), their Schwartz kernels are distributions on M M, which is a manifold with corners. Analy-× sis of these kernels is facilitated, though, by lifting them to a certain manifold with boundary, b(M M), ob- tained from M M by a blowing-up× op- eration along ×the corners. The case where M is a half-open interval is il- Figure 6. The groupoids G = M M and bG =b (M M),where M =(0,1). lustrated in Figure 6. The general case × × is locally the product of this one with a mani- Conclusion fold without boundary. This article has presented a small sample of the It turns out that b(M M) is a groupoid whose applications of groupoids in analysis, algebra, Lie algebroid is bTM and× that the composition and topology. R. Brown has reported in [2] the law for b-kernels ([17], Equation (5.66)) is pre- suggestion of F. W. Lawvere that groupoids cisely the convolution operation in the groupoid should perhaps be renamed “groups” and those algebra of b(M M). Thus, we can think of the special groupoids with just one base point be groupoid b(M ×M) and its Lie algebroid bTM as given a new name to reflect their singular nature. providing a new× “internal symmetry structure” Even if this is too far to go, I hope to have con- appropriate to our manifold with boundary. vinced the reader that groupoids are worth know- ing about and worth looking out for.12 It is therefore interesting to understand the structure of the groupoid b(M M). In fact, it de- References composes algebraically (though× not topologi- cally) as the disjoint union of the pair groupoid W. Brandt over the interior of M and the groupoid G over [1] , Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann. 96 (1926), 360–366. ∂M for which (α, β) 1(x, y) consists of the ori- − [2] R. Brown, From groups to groupoids: A brief sur- entation-preserving linear isomorphisms be- vey, Bull. London Math. Soc. 19 (1987), 113–134. tween the normal lines T M/T (∂M) and y y [3] ———, Topology: A geometric account of general TxM/Tx(∂M). Thus, with the “geometric struc- topology, homotopy types, and the fundamental ture” on M provided by the enlarged pair groupoid, Halsted Press, New York, 1988. groupoid b(M M), each point of the boundary [4] A. Connes, Noncommutative geometry, Acade- ∂M which has the× internal symmetries are those mic Press, San Diego, 1994. of an oriented 1-dimensional vector space. We [5] A. Coste, P. Dazord, and A. Weinstein, should think of the vectors in this space as being Groupoïdes symplectiques, Publications du Dé- points “infinitesimally far” from the boundary, partement de Mathématiques, Université Claude Bernard-Lyon I 2A (1987), 1–67. so that the boundary of M becomes an “infini- C. Ehresmann ∂ b [6] , Oeuvres complètes et commen- tesimal boundary layer”. The section x ∂x of TM tées, (A. C. Ehresmann, ed.) Suppl. Cahiers Top. can now be seen as pointing into this layer in a Géom. Diff., Amiens, 1980–1984. nontrivial way along ∂M; that is why it is not zero [7] P. Gabriel, Construction de préschémas quotient, there. Schémas en Groupes, Sém. Géométrie Algébrique, At this point, our use of Lie algebroids and Inst. Hautes Études Scientifiques, 1963/64, Fasc. Lie groupoids adds only a geometric picture to 2a, Exposé 5. [8] A. Grothendieck, Techniques de construction et Melrose’s analysis. It does, however, suggest the théorèmes d’existence en géométrie algébrique III: possibility of extending the theory of pseudo- préschemas quotients, Séminaire Bourbaki 13e differential operators to algebras of operators année 1960/61, no. 212 (1961). associated with other Lie algebroids. In fact, the [9] P. J. Higgins, Notes on categories and groupoids, theory of longitudinal operators on foliated man- Van Nostrand Reinhold, London, 1971. ifolds (see Section I.5.γ of [4]) is an example of [10] M. V. Karasev, Analogues of objects of Lie group such an extension, but the class of Lie algebroids theory for nonlinear Poisson brackets, Math. USSR which can be treated in this way remains to be Izvestia 28 (1978) 497–527. determined. An interesting test case should be the Lie algebroids associated with the Bruhat- 12Any reader stimulated to look further may wish to Poisson structures on flag manifolds [12], since consult the Groupoid Home Page, http://amath- the corresponding orbit decomposition on a flag www.colorado.edu:80/math/department/ manifold is the Bruhat decomposition, which groupoids/groupoids.shtml (under construction in plays a central role in representation theory. May 1996).

JULY 1996 NOTICES OF THE AMS 751 A. Kumpera D. Spencer [11] and , Lie equations, Prince- [21] J. Renault, A groupoid approach to C∗ algebras, ton University Press, Princeton, NJ, 1972. Lecture Notes in Math. 793 (1980). [12] J.-H. Lu and A. Weinstein, Poisson Lie groups, [22] G. S. Rinehart, Differential forms on general com- dressing transformations, and the Bruhat decom- mutative algebras, Trans. Amer. Math. Soc. 108 position, J. Differential Geom. 31 (1990), 501–526. (1963), 195–222. [13] K. Mackenzie, Lie groupoids and Lie algebroids in [23] L. Schwartz, Théorie des distributions, Nouvelle differential geometry, LMS Lecture Note Ser., vol. edition, Hermann, Paris, 1966. 124, Cambridge Univ. Press, 1987. [24] A. Weinstein, Symplectic groupoids and Poisson [14] ———, Classification of principal bundles and Lie manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), groupoids with prescribed gauge group bundle, 101–104. J. Pure Appl. Algebra 58 (1989), 181–208. [25] ———, Noncommutative geometry and geometric [15] G. W. Mackey, The mathematical foundations of quantization, Symplectic Geometry and Math- quantum mechanics, W. A. Benjamin, New York, ematical Physics: Actes du Colloque en l’Honneur 1963. de Jean-Marie Souriau (P. Donato et al. eds.), [16] ———, Ergodic theory and virtual groups, Math. Birkhäuser, Boston, 1991, pp. 446–461. Ann. 166 (1966), 187–207. [26] J. Westman, Harmonic analysis on groupoids, Pa- [17] R. B. Melrose, The Atiyah-Patodi-Singer index the- cific J. Math. 27 (1968), 621–632. orem, A. K. Peters, Wellesley, 1993. [27] I. M. Yaglom, Felix Klein and Sophus Lie: Evolution [18] ———, Geometric scattering theory, Cambridge of the idea of symmetry in the nineteenth century, Univ. Press, Cambridge, 1995. Birkhäuser, Boston, 1988. [19] C. C. Moore and C. Schochet, Global analysis on [28] S. Zakrzewski, Quantum and classical foliated spaces, MSRI Publications, vol. 9, Springer- pseudogroups, I and II, Commun. Math. Phys. 134 Verlag, New York, 1988. (1990), 347–370, 371–395. [20] J. Pradines, Troisième théorème de Lie sur les groupoïdes différentiables, C. R. Acad. Sci. Sér. I Math. Paris 267 (1968), 21–23.

752 NOTICES OF THE AMS VOLUME 43, NUMBER 7