J.F. Cari~nena structures and that of groupoid. So we will nd topological groupoids, Lie groupoids, symplectic groupoids, Poisson groupoids and so on.

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1 Lie groupoids and algebroids in Classical and Quantum Mechanics 1 Jose F. Cari~nena Departamento de Fsica Teorica. Facultad de Ciencias. Universidad de Zaragoza, E-50009, Zaragoza, Spain. Abstract The concept of Lie groupoid and algebroid is quickly reviewed and the theory is illustrated with several examples. Some applications of these concepts in generalized geometrical mechanics are pointed out and use is made of the Connes' approach to the tangent groupoid for the strict quantization of a system. 1 Introduction The idea of symmetry has usually been realized in mathematics by the concept of group: the set of transformations preserving some structure or property is characterized by group properties. So, Reexivity implies that there is a neutral element, Symmetry means that every symmetry transformation has an inverse, and Transitivity means that any two symmetry transformations can be composed giving rise to a new symmetry transformation. There are objects, spaces and structures that seem to have a number of symmetries lower than the expected one and then they cannot close on a group. The concept generalizing the idea of group is that of groupoid (for a review see [1]). This new concept plays an important role in the study of general dierentiable manifolds and partial dierential equations. In many aspects a groupoid is like a group but with several neutral elements. Actually, a grupoid with only one neutral element is a group. As we will see in a groupoid the composition law is only dened for some ordered pairs of elements, and then an element g can only be a neutral element for those elements that are composable with g. A groupoid can be endowed with other algebraic, geometric or topological structures and in this case we should study the compatibility among these 1 Talk given at the Workshop \Symmetries in Quantum Mechanics and Quantum Optics", Burgos (Spain), September, 1998

2 J.F. Cari~nena structures and that of groupoid. So we will nd topological groupoids, Lie groupoids, symplectic groupoids, Poisson groupoids and so on. The innitesimal version of the Lie groupoid structure is that of a Lie algebroid, which in some sense is a generalization of the concept of tangent bundle. We will also give the algebraic denition of Lie algebroid and illustrate the theory by means of examples. Poisson structures have also a direct relationship with Lie algebroid structures: a Poisson structure on Q is a particular kind of Lie algebroid structure on T Q satisfying an integrability condition. The mentioned relations point out the interest of this Lie algebroid structure in the geometrical approach to Classical Mechanics [, 3]. Some particular examples and applications will be presented in Section 7. Finally, as an application in Quantum Mechanics we will also see how the tangent groupoid (see [4]) is the appropriate framework for understanding the quantization process [5, 6]. Notation and basic denitions Let :? X! X be a left action of the group? on the set X and G be the set G = f(y; ; x) j x; y X;?; y = (; x)g : Let us dene two maps : G! X and : G! X, by (y; ; x) = y and (y; ; x) = x. It is possible to dene a partial composition law, internal in G, in the set G of G determined by as follows: G = f(g ; g 1 ) G G j (g ) = (g 1 )g ; : G G G! G ; If g 1 = (y 1 ; 1 ; x 1 ) and g = (y ; ; x ) are such that (g ; g 1 ) G, i.e., x = y 1, then, g g 1 = (y ; 1 ; x 1 ) : This partial composition law is associative. Moreover, if e denotes the neutral element of the group?, then (y; e; y) G satises (y; e; y)(y; ; x) = (y; ; x) ; or in other words, (y; e; y) behaves like a left neutral element for those g G such that (g) = y. Similarly, (x; e; x) behaves like a right neutral element for those g G such that (g) = x.

3 Lie groupoids and algebroids in Classical and Quantum Mechanics 3 Moreover, if (y; ; x) G, then (x;?1 ; y) is such that (y; ; x) (x;?1 ; y) = (y; e; y) ; (x;?1 ; y) (y; ; x) = (x; e; x) and therefore, we can consider that (x;?1 ; y) is an inverse of the element (y; ; x), usually denoted (y; ; x)?1. We also remark that the set X can be identied to the set G (0) of G made up by the elements (x; e; x), namely, the neutral elements of G. Let us suppose now that we choose a subset M of X, which may be noninvariant under the action of G, and dene G M G by G M = fg = (y; ; x) G j y = (g) M; x = (g) Mg : We can consider the restrictions of and onto G M, and then the image of both restrictions will be in M, and dene the subset G M in a a similar to what we did before, i.e., G M = f(g ; g 1 ) G j (g 1 ); (g ); (g 1 ) Mg : If M is not invariant under the action of the group?, we cannot dene an action of? on M, but most of the properties we mentioned still hold, up to minor modications, and this suggests us the following algebraic structure which includes a lot of interesting cases: Denition 1 A groupoid G with base B is a set G endowed with maps : G! B and : G! B and a partial composition law : G G G! G, internal in G, but that is only dened in the subset such that: G = f(g ; g 1 ) G G j (g ) = (g 1 )g ; i) is associative, i.e., if one of the two products (g 3 g ) g 1 and g 3 (g g 1 ) is dened, then the other product is also dened and both coincide, (g 3 g ) g 1 = g 3 (g g 1 ) : ii) For every g G there exist a neutral element for g on the left, g, and a neutral element for g on the right, g, g g = g = g g : iii) Each g G has one inverse, to be denoted g?1 G, such that g?1 g = g ; g g?1 = g :

4 4 J.F. Cari~nena The denition of g?1 shows that (g?1 )?1 = g. Moreover, the preceding relations show that g?1 = g ; 8g G. As indicated before, the set of neutral elements of the groupoid, is usually denoted G (0). We will say that the elements g and g 1 of the groupoid G can be composed when (g ) = (g 1 ), namely, (g ; g 1 ) G. We can think of g as an arrow starting from (g) and arriving to (g) and that the multiplication g g 1 in the groupoid is made by juxtaposition of the tail of g with the head of g 1 when both coincide. It is also very common to use the letters r (for range) and s (for source) instead of the Greek letters and for denoting the two projections of the groupoid onto the base. (g) is called head of g and (g), is called tail of g. Denition Given a groupoid G with base B, we will call subgroupoid (with the same base B) to any subset of G that is stable under the composition law and under taking the inverse and such that it contains all neutral elements. Proposition 1 Let G be a groupoid with base B. Then, i) For each g G, ( g ) = (g). Similarly, (g) = ( g ). ii) For each g G, (g) = (g?1 ). iii) For each pair of elements g; h G, such that (g) = (h), it happens that (g h) = (g), and (g h) = (h). Dem.- i) The product g g is well dened, and thus, ( g ) = (g). Similarly, as the product g g is well dened, (g) = ( g ). ii) The product g g?1 being dened, (g) = (g?1 ). iii) Taking into account that for every such a pair of elements of the groupoid, g; h G, the two products ( g g) h and g (g h) are dened and both coincide, it should be (g h) = ( g ), and then, i) shows that (g h) = (g). In an analogous way, from (g h) h = g (h h ), we can conclude that (g h) = ( h ) and using i) we nd that (g h) = (h). Notice that (gh) = (g) means that the bre of g is made by all products g h when h is composable on the right with g. Similarly, (g h) = (h) means that the {bre of h is made up by the products g h when g is composable on the left with h. Moreover, from g (g h) = g h we see that g = gh. In an analogous way, hg = g. Let us remark that (g) determines g and conversely, and the same can be said of g and (g). This implies that there is a bijective map i : B! G (0) G, i((g)) = g. A similar assertion is true for the map i 0 : B! G (0) G, i 0 ((g)) = (g) = g?1. Notice that i = i = id B. Finally, when the product g g 1 does exist, i.e., (g ) = (g 1 ), then (g?1 ) = 1 (g?1 ), and therefore the product g?1 1 g?1 also exists and it turns out that g?1 1 g?1 is the inverse of g g 1.

5 Lie groupoids and algebroids in Classical and Quantum Mechanics 5 3 Some interesting examples a) Given a group?, one can choose G =?, B = feg and the maps ; : G! B given by (g) = (g) = e; 8g G, and then G can be considered as a groupoid with base the set of the neutral element of the group, with g = g = e, 8g?, i.e., G (0) = feg, and g?1 being the inverse element of g G in the group. Here any two elements can be composed, and then G = G G. b) Let f? j g be a family of groups. The set G = [? ; disjoint union of groups, can be endowed with a groupoid structure with base by considering the maps (g) = (g) = ; if g? ; together with the partial composition law only dened in G = f(g 1 ; g ) j 9 ; such that g 1? ; g? g by means of the corresponding composition law each one of the groups?. The neutral elements, both sides, are the respective neutral elements e of each group?, and same for the inverse element: if g?, then g?1 is the inverse of g in the subgroup?. c) If B is an arbitrary set, then the Cartesian product B B, endowed with the canonical projections on each factor as maps and, together with the composition law (x; y) = x ; (x; y) = y ; (z; y) (y; x) = (z; x) ; is a groupoid onto B, very often called the pair groupoid or banal groupoid. The neutral elements are those of the form (x; x), and the inverse of the element (x; y) is (x; y)?1 = (y; x). A subgroupoid of the banal groupoid B B is nothing but an equivalence relation in B, because it is a subset of B B, i.e., a relation, containing all the neutral elements, the property of reexivity of the equivalence relations, it is closed under the composition law, which corresponds to the property of transitivity, and under passing to the inverse, which is the property of symmetry for equivalence relations.

6 6 J.F. Cari~nena d) The previously considered example of the action of a group? on a set X, is a groupoid called semidirect product of X by? relative to the action. Now G = X? X, and if g = ((; x); ; x), then g?1 = (x;?1 ; (; x)). e) Let X be a set and R XX an equivalence relation on X. Such a structure encloses a groupoid structure where G is the subset R, the set of pairs of related elements, the set G (0) is the diagonal subset X X, and the projections and are the projections onto each element. The product reproduces the property of transitivity of the equivalence relations: and : G! G is given by G = f((x; y); (y; z)) j (x; y); (y; z) Rg (x; y) (y; z) = (x; z) : Finally, the inverse element of (x; y) is: (x; y)?1 = (y; x); 8(x; y) R: f) Another very interesting example for its applications in Dierential Geometry and Geometrical Mechanics is that of a vector bundle : E! M. In particular we will be interested in the tangent bundle, T M, the cotangent bundle, T M, or the normal bundle of M. In this case G is E and G (0) is M. The projections and coincide with the projection of the bundle. The base M of E can be identied with the zero section: G (0) f(p; 0) j p Mg : The product is dened in G = f((p; A p ); (p; B p )) j p M; A p ; B p?1 (p)g by means of (p; A p ) (p; B p ) = (p; A p + B p ) : Finally, the inverse element of (p; A p ) is: (p; A p )?1 = (p;?a p ) : 4 Morphisms of groupoids Denition 3 A morphism of the groupoid G with base B into the groupoid G 0 with base B 0 is a pair of maps : G! G 0, : B! B 0, such that if g 1 and g

7 Lie groupoids and algebroids in Classical and Quantum Mechanics 7 are two composable elements of G, (g ; g 1 ) G, then the same us true for the corresponding elements of the groupoid G 0, (g ) and (g 1 ), and furthermore (g g 1 ) = (g ) (g 1 ) ; with 0 = ; 0 = : As a rst instance, if G is a groupoid with base B, then (; ) dene a morphism of the groupoid G into the pair groupoid B B. Denition 4 The groupoid G with base B is said to be principal if the morphism (; ) of the groupoid G into the pair groupoid B B is injective. Remark that the groupoid is principal if G is isomorphic to the image (; )(G), an equivalence relation on B. We recall that given an action of a group? on a set X the two important concepts of concepts of orbit and isotopy subgroup of a point x X were introduced as follows: i) The orbit of the point x X is its equivalence class for the relation: x y when there exists an element g of the groupoid G such that (g) = y and (g) = x. ii) The isotopy subgroup of x X consists on the elements g G such that (g) = (g) = x, where we identify? with the element (x; ; x) of the groupoid. These concepts can be generalized to the framework of groupoids. Denition 5 If G is a groupoid with base B, the orbit of an element x B is made up by the elements y B for which there exists an element g of the groupoid G such that (g) = y and (g) = x. The isotopy subgroup of x B consists on the elements g G such that (g) = (g) = x. In the particular example of the subgroupoid of the pair groupoid dened by an equivalence relation on the base B the orbits are the equivalence classes and the isotopy subgroups are trivial. 5 Convolution in groupoids and matrices We recall that if G is a nite group, we can dene a composition law in the space of real o complex functions X in G by means X of (a b)(g) = a(g 0 ) b(g 0?1 g) = a(gg 0?1 g) b(g 0 ) : g 0 G g 0 G

8 8 J.F. Cari~nena When G is a topological or Lie group we can proceed similarly by considering measures in G and replacing the sum by an integral on the group. The generalization for groupoids is obtained by replacing the summation on the group by the sum on the bre G g =?1 ((g)), (a b)(g) = X f(k;l)jkl=gg a(k)b(l) : Let us remark that if G is a groupoid, for each g G, the left translation by g is a map among -bres `g :?1 ((g))!?1 ((g)); h 7! g h ; an similarly, the right translation by g is a map r g :?1 ((g))!?1 ((g)) : As an example of the convolution product, when G is the pair groupoid constructed from B, assuming it to be nite, B = f1; : : : ; ng, then the convolution product (a b)(i; j) = nx k=1 a(i; k)b(k; j) ; reduces to the usual rule for matrix multiplication. In other words, the non{ commutativity of the algebra of quantum observables is a consequence of the non{commutativity of the product in the pair groupoid. 6 Lie groupoids and algebroids Denition 6 When G is a topological space that is a groupoid with base another topological space B and the maps and and the partial composition law are continuous, we will say that G is a topological groupoid. Similarly, when G and B are dierentiable manifolds and, and the partial composition law are dierentiable maps, G is said to be a Lie groupoid. Examples: 1.- When M is a dierentiable manifold, then the pair groupoid is a Lie groupoid. In particular, if G is a Lie group, then G G is a Lie groupoid with ((g ; g 1 )) = g, ((g ; g 1 )) = g 1 and the partial composition law (g 3 ; g ) (g ; g 1 ) = (g 3 ; g 1 ) :.- When G is a Lie group, then G can be seen as a Lie groupoid with base the neutral element of G, feg.

9 Lie groupoids and algebroids in Classical and Quantum Mechanics 9 The convolution product can be generalized to Lie groupoids by choosing some measures g on the bres for the projections and : Z (a b)(g) = a(k) b(k?1 g) d g (k) : G g The innitesimal object associate with a Lie groupoid is a Lie algebroid. We rst give the abstract denition of a Lie algebroid structure: Denition 7 A Lie algebroid with base B is a vector bundle A with base B together with a Lie algebra structure in the space of its sections given by a Lie product [; ] A and a map, called anchor, : A! T B, inducing a map between the corresponding spaces of sections, to be denoted with the same name and symbol, such that [(X); (Y )] = ([X; Y ] A ) ; [X; ' Y ] A = ' [X; Y ] A + ((X)')Y ; for any pair of sections of A, X; Y, and each continuous function ' dened in B. Let us remark that for each section X of the Lie algebroid A we can dene the Lie derivative L X of both sections in A and functions in the base B by means of L X Y = [X; Y ] A ; L X f = (X)f ; and then the condition dening the Lie algebroid structure is nothing but the Leibniz rule. When we consider in A adapted coordinates, (x 1 ; : : : ; x n ; 1 ; : : : ; r ), where (x 1 ; : : : ; x n ) are coordinates in the base and ( 1 ; : : : ; r ) coordinates in the bres associated with a basis of local sections, f g, of the Lie algebroid, then [ ; ] A = X c ; ; ; and the coordinate expression of the anchor map is ( ) = nx i=1 i ; with c and a i being the so called structure functions of the algebroid. Examples: 1.- The simplest example is that of a Lie algebra g considered as a vector bundle over a single point. Now the sections are the elements of g and the Lie product is the one of g.

10 10 J.F. Cari~nena.- Another simple example is the tangent bundle T B, when choosing the identity as anchor map and the commutator of vector elds as [; ] A. With the usual coordinates (q i ; v i ) in T B induced from coordinates (q i ) in the base B, the structure functions are c ij k = 0 ; a ij = ij : However, in arbitrary coordinates in T B the structure functions do not vanish. 3.- An integrable subbundle of T B with the inclusion as anchor map and the Lie product on the space of sections induced from that of T B is also an interesting example of Lie algebroid. 4.- Let P (G; B) a principal bundle. G acts on T P by right translations and the quotient manifold T P=G is a vector bundle whose sections are the G{invariant vector elds in P (innitesimal gauge transformations). The differential of the projection of P onto B passes to the quotient and denes the anchor of the Lie algebroid. Such an algebroid is called gauge algebroid of P (G; B). 5.- The cotangent bundle of a Poisson manifold (P; ) is a Lie algebroid when we dene f; g = L ()? L ()? d[(; )] ; and the anchor is given by ()f = (; df). This Poisson bracket veries for any pair of functions in P. fdf; dgg = dff; gg ; The pairing of forms so dened is very useful: if two 1{forms and are invariant under a Hamiltonian vector eld X H, then f; g provides a new invariant 1{form. 6.- Let : g! X(B) be an action of the of the Lie algebra g on the manifold B. Then, we can dene a Lie algebroid structure in the trivial bundle B g! B by dening : A! T B and [; ] A as follows: (x; v) = (v)(x) ; [; ] A (x) = [(x); (x)] + ()? () ; where we have identied sections of B g with g{valued functions. A remarkable property is that if A! B is a vector bundle with a structure of Lie algebroid, then the dual bundle A! B is endowed with a homogeneous of degree minus one Poisson structure, L Z =?, with Z being the vector eld generating dilations along the bres of A, and the converse property is also true[7]:

11 Lie groupoids and algebroids in Classical and Quantum Mechanics 11 Theorem 1 The vector bundle : A! B is a Lie algebroid with base B and anchor : A! T B if and only if the dual bundle A! B is a Poisson manifold whose linear functions form a Lie subalgebra. Examples: 1.- If B reduces to a point and A = g is a Lie algebra, we nd the well{ known case of the canonical Poisson structure on the dual of the Lie algebra, g..- When A = T B, then A = T B and the Poisson structure in T B is the canonical symplectic structure on the cotangent bundle. In order to describe the Poisson structure of A it suces to give the Poisson brackets of a class of functions such that their dierentials span the cotangent space at each point of A. Such a class of functions is given by functions which are ane in the bres. Those functions which are constant on the bres, basic functions, correspond to pull{back of functions on B. On the other hand, functions that are linear in the bres can be identied with sections of A. If f and g are functions on B and and are sections in A, their bracket relations as functions on A are ff; gg = 0 ; ff; g = ()f ; f; g = [; ] : Given adapted coordinates in A, (q; ), and with the corresponding dual coordinates, (q; ), the dening relations of the Poisson structure are fq i ; q j g = 0 ; f ; g = X c ; fq i ; g = a i : 7 Applications in Geometrical Mechanics We start by mentioning two important cases in which groupoids arise in Classical Mechanics. One fundamental ingredient in Classical Mechanics is the theory of second order dierential equations describing the dynamics. For mechanical systems with holonomic constraints the conguration space is a dierentiable manifold Q and the tangent bundle Q : T Q! Q is the velocity-phase space. T Q : T (T Q)!T Q is endowed with another vector bundle structure given by T Q : T (T Q)!T Q. Vector elds on T Q that are also sections for T Q are called SODE (second order dierential equations). The set of such SODE vector elds will be denoted X s (T Q). Given a? X s (T Q), a dieomorphism : T Q! T Q such that (?) X s (T Q) is said to be Newtonoid with respect to?. Those which are Newtonoid for any SODE X are the point transformations = T with : Q! Q.

12 1 J.F. Cari~nena The set G = f(y; ; X) j X; Y X s (T Q); Di (T Q); (X) = Y g ; is a groupoid over X s (T Q), with (Y; ; X) = Y and (Y; ; X) = X. The partial composition law is (Z; ; Y ) (Y; 1 ; X) = (Z; 1 ; X) : In the Hamiltonian approach to Classical Mechanics, one considers a symplectic manifold (M;!) and the dynamics is given by a locally-hamiltonian vector eld? (L?! = 0). The set of locally-hamiltonian vector elds will be denoted X LH (M;!). A dieomorphism : M! M is said to be a canonoid transformation with respect to? if (?) is also locally Hamiltonian. If is canonoid for any locally-hamiltonian vector eld, then there exists a real number c such that! =! and is canonical. The set G = f(y; ; X) j X; Y X LH (M;!); Di (M); (X) = Y g ; is a groupoid over X LH (M;!), with (Y; ; X) = Y and (Y; ; X) = X. The partial composition law is (Z; ; Y ) (Y; 1 ; X) = (Z; 1 ; X) : Next we look for situations in which Lie algebroids arise in Classical Mechanics. We want to remark that it is very useful to think of a Lie algebroid over B as a new \tangent bundle" for B. As the tangent bundle T Q of a dierentiable manifold, the conguration space Q, plays a relevant role in the geometrical approach to the Lagrangian formalism of Classical mechanics, we can try to extend the formalism to this new framework of Lie algebroids. If L is a real-valued function on the Lie algebroid A over M, the Legendre mapping FL : A! A of L is dened as the bre derivative of L, i.e., if b B and L b : A! R is given by the restriction to?1 (b), dlb (v + tw) FL(v)(w) = L bv (w) = : dt The action A is the function in A dened by A (v) = hfl (v) ; vi, (using the natural dual pairing) and the energy E is given by A? L. When FL is a local dieomorphism we will say that L is a regular Lagrangian. In this case FL can be used to pull back the Poisson structure from jt=0

13 Lie groupoids and algebroids in Classical and Quantum Mechanics 13 A to a Poisson structure on A which we will call the Lagrange Poisson structure. The Hamiltonian vector eld of the energy function, f; Eg, is called the Lagrangian vector eld associated with the regular Lagrangian L. In standard coordinates, if the Lagrangian is given by L(q; ), then its bre derivative FL is dened so the bracket relations for the Lagrange Poisson structure on A fq i ; q j g = 0 ; ; and q = a i : = c The action and the energy functions are, respectively, A ; E L : The equations of motion in Poisson bracket form are Using the bracket relation dq i dt = fqi ; Eg d dt = f ; Eg: X fq i ; Lg = fq i ; we can rewrite the rst Lagrange equation as follows: dq i dt = fq i ; Eg = fq i Lg X = fq i g + fq i fqi ; Lg X = fq i g = a i 0 X = ( ) q? i X 1 A q i or in other words, the Lagrangian vector eld is? L = X 1 A :

14 14 J.F. Cari~nena A tangent vector v to a Lie algebroid A at a point is called admissible when T A (v) = (v), where T A : T A! T M is the derivative of the vector bundle projection A : A! M. A curve in A is called admissible if its tangent vectors are all admissible. A vector eld X on A is called a second order dierential equation if its values are all admissible vectors. (Thus, a vector eld on A is a second order dierential equation i all its integral curves are admissible.) The explicit expression for the Lagrangian vector eld? L shows that the Lagrangian vector eld associated with any regular Lagrangian is a second order dierential equation in the preceding sense. 8 The Tangent Groupoid Let us study the example we are at most interested in. Let M be a dierentiable manifold and let us dene G = M M]0; 1[ [T M f0g as a groupoid on G 0 = M [0; 1[. The inclusion map G 0! G is dened as: (x; )! (x; x; ) M M]0; 1[ for x M; > 0 (x; 0)! x M T M as the zero section; for = 0 The range and source maps are dened as: (x; y; ) = (x; ) for x M; > 0 ; (z; X z ) = (z; 0) for z M; X z T z M (x; y; ) = (y; ) for y M; > 0 ; (z; X z ) = (z; 0) for z M; X z T z M And nally the composition map is dened as: (x; y; ) (y; z; ) = (x; z; ) for > 0 and x; y; z M (z; X z ) (z; Y z ) = (z; X z + Y z ) for z M; X z ; Y z T z M The tangent groupoid is the disjoint union of two smooth ones, the groupoids G 1 = M M]0; 1[ and G = T M f0g. The remarkable fact is that we can endow this union with a suitable topology in order to obtain G as the border of the whole set G, and G 1 as an open subset of G. This topology is given by specifying the convergence of sequences: Every convergent sequence on G 1 or G will converge in our joined space. In addition, a sequence of elements f(x n ; y n ; n ) j (x n ; y n ; n ) G 1 g converges, on any local chart, to an element of the tangent bundle (z; X z ) if and only if it veries: x n! z ; yn! z x n? yn ;! Xz n

15 Lie groupoids and algebroids in Classical and Quantum Mechanics 15 The condition above does not depend on the choice of the local chart. What does it serve for in quantization theory? The main idea is that the choice of the dierential structure on the whole groupoid as a dierentiable manifold with boundary denes a setting where the functions on the groupoid have two pieces, one corresponding to G 1 and other to G, and the topology condition gives a continuity condition for the pair. The piece in the interior of the manifold will play the role of a quantum kernel corresponding to an operator which acts on L (M) (M is a Riemannian manifold and the measure is dened canonically), while the function on the boundary (T M) corresponds, via the Fourier transform, to a classical observable (a function on the cotangent bundle T M). The continuity condition corresponds precisely to the quantization mapping. Thus, the tangent groupoid provides a very nice geometrical description of the quantization procedure, at least for the cotangent bundle case. The details of this method of introducing the strict quantization can be found in [5]. Acknowledgments This work has been partially supported DGES (PB96{0717). References [1] Weinstein A., Groupoids: Unifying Internal and External Symmetries, Notices Amer. Math. Soc {5 (1996) [] Libermann P., Lie algebroids and Mechanics, Archivum Mathematicorum (Brno) 3 147{6 (1996) [3] Weinstein A., Lagrangian mechanics and groupoids, in the book: Mechanics Day, Shadwick W.F., Krishnaprasad P.S. and Ratiu T.S. eds., Fields Institute Communications, American Mathematical Society, [4] Connes A., Noncommutative Geometry, Academy Press, San Diego, 1994 [5] Cari~nena J. F., Clemente-Gallardo J., Follana E., Gracia{Bonda J. M., Rivero A. and Varilly J. C., Connes' Tangent Groupoid and Strict Quantization, J. Geom. Phys. (1998) (to appear) [6] Cari~nena J. F. and Clemente-Gallardo J., Some examples and applications of Fedosov's star products and Deformation Quantization, Czech. J. Phys. 48 (1998) (to appear) [7] Courant T., Tangent Lie algebroids, J. Phys. A: Math. Gen {36 (1994)