Imprimitivity theorem for groupoid representations

We define and investigate the concept of the groupoid representation induced by a representation of the isotropy subgroupoid. Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity theorem for such representations which is a generalization of the classical Mackey's theorem known from the theory of group representations.


Introduction
The present paper, devoted to the study of the theory of groupoid representations, is a continuation of my previous work [20] in which one can find a presentation of the groupoid concept and of the groupoid representation concept, important examples as well as relationships between groupoid representations and induced group representations (see also [18], [21], [23], [11]).
Groupoids have now found a permanent place in manifold domains of mathematics, such as: algebra, differential geometry, in particular noncommutative geometry, and algebraic topology, and also in numerous applications, notably in physics. It is a natural tool to deal with symmetries of more complex natura than those described by groups (see [23], [11], [2]). Groupoid representations were investigated by many authors and in many ways (see [24], [21], [18], [1], [3]).
In a series of works ( [7], [9], [8], [19]) we have developed a model unify-ing gravity theory with quantum mechanics in which it is a groupoid that describes symmetries of the model, namely the transformation groupoid of the pricipal bundle of Lorentz frames over the spacetime. To construct the quantum sector of the model we have used a regular representation of a noncommutative convolutive algebra on this groupoid in the bundle of Hilbert spaces. In paper ( [6]) we have applaied this groupoid representation to investigate spacetime singulariies, and in [10] the representation of the fundamental groupoid to the gravitational Aharonov-Bohm effect.
The present work is aimed at introducing the concept of the groupoid representation induced by a representation of the isotropy subgroupoid. We assume that the groupoid in question is a locally trivial topological groupoid and as a topological space it is a locally compact Husdorff space (Section 4).
This concept, framed "in the spirit of Mackey" is a natural generalization of induced represetation of locally compact groups, created and investigated by him [15]. Representations, investigated in the present work, are unitary and are realized in Hilbert bundles over the unit spaces of a given groupoid [18]. Section 5 contains the formulation and the proof of the imprimitivity theorem for groupoids which is a generalization of the classical Mackey's imprimitvity theorem for group representations [14], [15]. The theorem says that every unitary groupoid representation, for which there exists the imprimitivity system, is a representation induced by a representation of the isotropy subgroupoid.
In Section 6, I investigate induced representations of the transformation gropoid over a homogeneous space of the group G and show that there exists a strict connection between these representations and induced representations of the group G (in the sense of Mackey. In Section 7, I give a physical interpretation of concepts analyzed in Section 6. I describthe a representation that has been used in the mentioned above model unifying gravity and quanta when this model is reduced (as the result of the act of measurement) to the usual quantum mechanics. And then I consider the energy-momentum space for a massive particle (it is a homogeneous space of the Lorentz group) and the transformation groupoid corresponding to this space. I also give a definition (in the sense of Mackey [14], [13]) of a particle as an imprimitivity sestem for the unitary representation of this groupoid.

Preliminaries
Let G be a groupoid over a set X (the base of G). We recall (cf. [4], [18] ) that a groupoid G is a set with a partially defined multiplication "•" on a subset G 2 of G × G, and an inverse map g → g −1 defined for every g ∈ G. The multiplication is associative when defined. One has an embedding ǫ : X → G called the identity section and two structure maps d, r : G → X such that Let us introduce the following fibrations in the set G: It has the group structure and is called the isotropy group of the point x. It is clear that the set Γ = x∈X G x x has the structure of a subgroupoid of G over the base X (all the structure maps are the restrictions of the structure maps of G to Γ).
We call G a transitive groupoid, if for each pair of elements x 1 , x 2 ∈ X there exists g ∈ G such that d(g) = x 1 and r(g) = x 2 .
A groupoid G is a topological groupoid if G and X are topological spaces and all structure maps are continuous (in particular, the embedding ǫ is a homeomorphism of X onto its image).
In the following we assume that G (and thus X) is a locally compact Hausdorff space.
With such defined structure maps G is a groupoid, called pair groupoid.
Example 2 A transformation groupoid. Let X be a locally compact Hausdorff space, and G a locally compact group. Let G act continuously on X to the right, X × G → X. Denote (x, g) → xg. We introduce the groupoid structure on the set G = X ×G by defining the following structure maps. The set of composable elements G 2 = {((xg, h), (x, g) : x ∈ X, g, h ∈ G} ⊂ G × G, and the multiplication for ((xg, h), (x, g)) ∈ G 2 is given by And also (x, . This groupoid is called the transformation groupoid. Let us recall (cf. [18]) the concept of right Haar System.
Definition 1 A right Haar system for the groupoid G is a family {λ x } x∈X of regular Borel measures defined on the sets G x (which are locally compact Hausdorff spaces) such that the following three conditions are satisfied: 1. the support of each λ x is the set G x , 2. (continuity) for any f ∈ C c (G) the function f 0 , where 3. (right invariance) for any g ∈ G and any f ∈ C c (G), One can also consider the family {λ x } x∈X of left-invariant measures, each λ x being defined on the set G x by the formula λ x (E) = λ x (E −1 ) for any Borel . Then the invariance condition assumes the form: Now, let µ be a regular Borel measure on X. We can consider the following measures which will be called measures associated with µ: If ν = ν −1 we say that the measure µ is a G-invariant measure on X.
Proposition 1 Assume that G is a locally trivial groupoid on X and X is second countable space. Let µ be a regular Borel measure on X. Then 1. G is transitive, 2. all isotropy groups of G are isomorphic with each other, 3. for every y ∈ X there exist an open cover {V j } of X and continuous maps s y,j : V j → G y such that r • s j = id v j , 4. for every x ∈ X there exists a section s x : X → G x which is µ- 5. if the measure µ has the property that µ(A) = µ(A) for every µmeasurable subset A of X, then the section s x is µ − a.e. continuous on
2. For x, y ∈ X let g yx be an element of G such that d(g yx ) = x and r(g yx ) = y. Then we have an isomorphism of the isotropy groups G x x and G y y given by the formula G x x ∋ γ → g yx • γ • g −1 yx ∈ G y y .
3. Let g xy be an element of G such that d(g xy ) = y and r(g xy ) = x. Then in the fiber G y we can simply define s y,i (z) = s x,i (z) • g x,y for z ∈ U i . 4. Since X is second countable space, we can take a countable cover It is easily seen that s x (z) is measurable.
5. The set of discontinuity of s x is contained in the union of sets ∂U i = This ends the proof.
From now on we assume that considered groupoids are locally compact and satisfy the assumptions of Proposition 1. It is known that in the case of any locally trivial groupoid there exists a right Haar system (see [18] Definition 3 A right Haar system {λ x } x∈X on the groupoid G is called consistent with a Borel regular measure µ on the base space X if, for every x ∈ X and any f ∈ C c (G x ), The above formula gives an explicit construction of the right Haar system for many classes of groupoids (see, Section 3 for pair groupoid, and Section 6 for transformation groupoid).
Let us recall the concept of groupoid representation [18], [21]. It involves a Hilbert bundle H over X, H = (X, {H x } x∈X , µ) (Dixmier [5] uses for it the name of µ-measurable field of Hilbert spaces over X). Here all Hilbert spaces H x are assumed to be separable.
Let µ be a G-invariant measure on X, and ν and ν 2 the associated measures on G.

Definition 4 A unitary representation of a groupoid G is the pair (U, H)
where H is a Hilbert bundle over X and U = {U(g)} g∈G is a family of unitary maps U(g) : H d(g) → H r(g) such that: 4. For every φ, ψ ∈ L 2 (X, H, µ), is ν-measurable on G. (Here L 2 (X, H, µ) denotes the space of squareintegrable sections of the bundle H, and (·, ·) x denotes the scalar product in the Hilbert space H x .) 3 Elementary properties of representations of groupoids.
x ∈ X such that for every x, y ∈ X and for ν−a.e. g ∈ G y x the following diagram commutes Definition 6 Let (U, H) be an unitary representation of G and let H 1 be a Hilbert subbundle of H. We say that for every x, y ∈ X and for ν−a.e. g ∈ G y x . Then the representation (U, H 1 ) is called a subrepresentation of (U, H For g ∈ G y x , x, y ∈ X define an operator of the representation A representation (U, H) is called regular representation of the groupoid Now let us consider the regular representation of a pair groupoid G 0 = X ×X.
Let µ be a regular Borel measure on X. Let us define a right Haar system of Then we have a simple invariance condition: For each x ∈ X the Hilbert space L 2 (G 0,x , dµ x ) is obviously isomorphic to L 2 (X).

Example 5
The regular representation of a pair groupoid G 0 in the Hilbert bundle {L 2 (G 0,x )} x∈X over X is given by the following family of operators where z ∈ X.
Let us observe that the regular representation of pair groupoid is equivalent to trivial representation in the trivial Hilbert bundle X × L 2 (X). Now, we shall introduce the quotient groupoid G/Γ (cf. [12]) and consider its representations.
Let Γ be the isotropy groupoid of a groupoid G. Let us define an equivalence relation ∼ on G, for g, h ∈ G, Denote the equivalence class of g ∈ G by [g], and the set of such equivalence classes by G/Γ. Then we can introduce the groupoid structure on G/Γ. The structure mapsd andr, the multiplication, the inverse and the identity sectionǫ are given byd It easy to see that the canonical projection p : G → G/Γ is a homomorphism of (topological) groupoids (in G/Γ we choose the quotient topology).
Let us denote by G 0 the pair groupoid G 0 = X × X over the base X. Recall We observe that the quotient groupoid G/Γ coincides with G 0 .
is an isomorphism of groupoids over X.
for an element x ∈ X, then d(g) = r(g) = x, i.e., g ∈ Γ and [g] in a unit in G/Γ. This means that Φ is an isomorphism. ⋄ Now, let us assume that a representation (U, H) of the groupoid G is Then it is easily seen that one has a unitary representation (U 0 , H) = In such a manner we obtain a Γ-invariant unitary representation of the groupoid G which is called a quasi-regular representation. Let us observe that the corresponding representation U 0 of the quotient groupoid G/Γ coincides with the regular representation of the pair groupoid 4 Induced representations of the groupoid G.
In this section, we define the representation of G induced by a representation of the isotropy subgroupoid Γ. From now on we assume that on the groupoid G there exists a right Haar system {λ x } x∈X consistent with Borel regular measure µ on X.
First, we have to construct an appropriate Hilbert bundle.
Assume that there is given a unitary representation (τ, W) of the subgroupoid Γ. Here W is a Hilbert bundle over X. Let W x denote a fiber over x ∈ X which is a Hilbert space with the scalar product ·, · x , and let W = ∪ x∈X W x denote the total space of the bundle W.
Let us define, for every x ∈ X, the space W x of W -valued functions F defined on the set G x satisfying the following four conditions: 2. for every µ-Borel measurable r-section s x : X → G x (see Proposition 1) the composition F • s x is a µ-measurable section of the bundle W, . F (s x (y)), F (s x (y)) y dµ(y) < ∞.
If we identify two functions F, F ′ ∈ W x satisfying we can introduce the scalar product (·, ·) x in the space W where s x is the section determined by Proposition 1, part 3.
The spaces W x , x ∈ X, with these scalar products are Hilbert spaces. It is easily seen that they are isomorphic to the Hilbert space L 2 (X, W) of squareintegrables sections of the bundle W. Now, let us denote It is a Hilbert bundle over X. We define a unitary representation of the groupoid G in the Hilbert bundle W in the following way Definition 8 The representation of the groupoid G induced by the represen- It is clear that (U τ , W) is a unitary groupoid representation.
Sometimes we shall use the notation U τ = Ind G Γ (τ ).

Systems of imprimitivity.
For a given Hilbert space W 0 we can consider the Hilbert space L 2 (X, W 0 ) of square integrable W 0 -valued functions on X. In the space L 2 (X, W 0 ) one has a representation of the commutative algebra L ∞ (X) given by the multiplication operators by the function: where, for z ∈ X, [π 0 (f )ψ](z) = f (z)ψ(z).
We shall call π 0 the natural representation of L ∞ (X) in L 2 (X, W 0 ). Now, let us consider a representation U of the groupoid G in a Hilbert bundle H over X. We assume that for µ -a.e. x ∈ X there exists a Hilbert space W x with a scalar product ·, · x such that the spaces W x are isomorphic with each other. Let us assume that, for µ -a.e. x, the fiber H x of the bundle H is isomorphic to L 2 (X, W x ). We shall simply write H x = L 2 (X, W x ), and U(g) : L 2 (X, W x ) → L 2 (X, W y ) for g ∈ G y x . It is clear that the collection of the spaces {L 2 (X, W x )} x∈X forms a Hilbert bundle over X which is isomorphic to the bundle H.

Definition 9
We say that there exists a system of imprimitivity (U, π) for the representation (U, H) of the groupoid G if 1. the representation (U, H) satisfies the above assumption (H x = 2. π = (π x ) x∈X is the family of natural representations of the algebra 3. for every f ∈ L ∞ (X), and for µ -a.e. x, y ∈ X, and ν -a.e. g ∈ G y x U(g)π x (f )U(g −1 ) = π y (f ).
Example 7 Let (U, H) be the quasi-regular representation of the groupoid G, defined in Example 6. Then there exists a system of imprimitivity (U, π) for U. Indeed, for f ∈ L ∞ (X), ψ ∈ H y , g ∈ G y x , h ∈ G y , we have The quasi-regular representation can be understood as induced by a trivial one-dimensional representation of the subgroupoid Γ.
We are now in a position to state our main theorem (the imprimitivity theorem for groupoids): Theorem 1 If, for a representation (U, H), there exists a system of imprimitivity (U, π) then the representation U is equivalent to the representation U τ induced by some representation (τ, W) of the subgroupoid Γ.
We can prove even more.
3. if the system of imprimitivity (U, π) is irreducible then (τ, W ) is an irreducible representation of Γ, i.e., for µ -a.e. x ∈ X, the representations (τ x , W x ) of the groups Γ x are irreducible.
Proof: Notice that the Hilbert space L 2 (X, W x ) is isomorphic to the tensor product of Hilbert spaces L 2 (X) W x . A decomposable operator in such a space has the form [A(ψ ⊗ h)](y) = ψ(y) ⊗ A y h. We have to show an orthonormal basis of the space L 2 (X). Consider the unitary operators

decomposable then
A commutes with all operators of the form U ij ⊗ id Hx . Then it follows that for γ 1 , γ 2 ∈ Γ. Thus τ x is a unitary representation of the group Γ x in the Hilbert space W x . This ends the proof of part 1.

Now the assertion 2 of the Lemma is obvious.
To obtain part 3 it is sufficient to see that the condition of irreducibility of the imprimitivity system implies that only operators of the form λid Wx , (λ ∈ C) commute with all τ x (γ), γ ∈ Γ x . But by Schur's lemma it follows that the The next lemma gives us more properties of the representation (τ, W) of the groupoid Γ as well as of the representation (U, H) that has a system of imprimitivity.

Lemma 2
1. The representations τ x , x ∈ X are equivalent to each other, as representations of isomorphic groups Γ x .

The operators U(g)
: x , are decomposable, i.e., there exist unitary operators U 0 (g) : W x → W y such that for ψ ∈ L 2 (X, W x ) and, for z ∈ X, Moreover, the operator U 0 (g) : W x → W y does not depend of z ∈ X.
Proof: First we shall prove part 2. First of all, let us notice that all spaces W x , for µ -a.e. x ∈ X, are isomorphic to each other as Hilbert spaces.
Denote by i y x : W x → W y the isomorphism and define the unitary map R y x : Consider the composition of unitary maps U(g) • (R y x ) −1 : L 2 (X, W y ) → L 2 (X, W y ) where g ∈ G y x . By using the property of the imprimitivity system for U(g), we obtain This means that the operator U(g)•(R y x ) −1 is decomposable in L 2 (X, W y ).
But (R y x ) is a decomposable map by definition, therefore U(g) is decomposable as the composition of decomposable maps. As in the proof of Lemma 1 we conclude that U 0 (g) does not depend of z ∈ X and is unitary.
To prove part 1 let us first observe that the isotropy groups Γ x are isomorphic to each other x ∈ X. Indeed, taking an element g ∈ G y x we define . Therefore, we have τ y (i(γ)) = U 0 (g) • τ x (γ) • U 0 (g) −1 , but this means that the representations τ y and τ x are equivalent. Now, we are in a position to give proofs of Theorems 1 and 2.
Proof. Let us consider the spaces {W x } x∈X , introduced in Section 1, connected to the representation τ of Lemma 1 and the corresponding induced representation U τ . We shall show that the representation (U, H) is equivalent to (U τ , W). We define a family of isomorphisms of Hilbert spaces J x : H x → W x for µ -a.e. x ∈ X. Since H x = L 2 (X, W x ), for ψ ∈ H x , g ∈ G x , and r(g) = y, we put F (g) = (J x ψ)(g) = (U(g)(ψ))(y). The definition is correct since by Lemma 2 we have (U(g)ψ)(y) = U 0 (g)(ψ(y)), and U 0 (g) does not depend of y ∈ X. Since U(g)ψ ∈ L 2 (X, W y ), therefore [U(g)(ψ)](y) ∈ W y .
Also it is clear that F (γ • g) = τ (γ)(F (g)) for γ ∈ Γ y . To see the squareintegrability let us write This also shows that J x are unitary maps and are injective. To see that J x map onto W x , we can give the formula for J −1 x : (J −1 x F )(y) = (U 0 (g)) −1 (F (g)) where F ∈ W x and g ∈ G y x . Then the right-hand side does not change if we take other element g 1 ∈ G y x . Indeed, since g 1 = γ • g, for an element γ ∈ Γ y , therefore we have (U 0 (γ • g)) −1 (F (γ • g)) = ((U 0 (g)) −1 (τ (γ)) −1 (τ (γ))(F (g)) = (U 0 (g)) −1 (F (g)). This shows that J x , x ∈ X, are isomorphisms of Hilbert spaces. Now we can see that J x are intertwining maps for the representations U and U τ , i.e., that the following diagram commutes for µ-a.e. x, z ∈ X and ν -a.e. g ∈ G z x . Let ψ ∈ H x . Then, for h ∈ . This ends the proof of Theorem 1.
The theorem 2 is now a simple consequence of Theorem 1 and Lemma 1, part 3.
6 Representations of the transformation groupoid G = X × G, X = K\G As an introduction to this section we recall the concept of induced representation in Mackey sense (cf [15], [13], [22]) of a Lie group G by a unitary representation (L, V ) of its closed subgroup K defined in a Hilbert space V .
We assume, for simplicity, that X = K\G has a G-invariant measure µ.
We consider H L , a Hilbert space consisting of measurable functions φ on G with values in V , such that We introduce the inner product Then we define the representation U L of G on H L given by the formula It is easily seen that U L is unitary. The representation (U L , H L ) is called induced by the representation L of K.
Let G be a noncompact Lie group and K its compact subgroup. We assume that G is unimodular. Then the homogeneous space X = K\G is a G−manifold with right action of the group G: X × G ∋ (x, g) → xg ∈ X.
As above, we assume that there exists a G-invariant measure µ on the space X, i.e., for f ∈ C c (G) we have We shall consider the structure of transformation groupoid on G = X × G (cf. Example 2) and construct a right Haar system on G consistent with the measure µ.
Let us denote Let us also denote the isotropy group G Lemma 3 Let s 0 be a Borel section of the principal bundle G → K\G = X, i. e., [s 0 (x)] = Ks 0 (x) = x. Then 1. for every x ∈ X there exists a section s x : X → G x with respect to the map r, i.e., r(s x (y)) = y, 2. every element g ∈ G x can be represented as g = k • s x (y) where k ∈ Γ y = G y y . Proof: 1. Let o ∈ X denotes the origin point, i.e., o = [k], k ∈ K. Then we have [ks 0 (x)] = x or, equivalently, os 0 (x) = x. Analogously, os 0 (y) = y for y ∈ X. Thus we can define the section s x : X → G x by the formula s x (y) = (x, s 0 (x) −1 s 0 (y)). It is clear that r(s x (y)) = y.
But the isotropy group of the origin point o is equal to K, what means s 0 (x)gs 0 (y) −1 ∈ K. ⋄ Now, for a function f ∈ C c (G x ), let us define f x (y, k) = f (x, s 0 (x) −1 ks 0 (y)),

Proposition 3
The collection {λ x } x∈X is a right Haar system on the groupoid G consistent with the measure µ on X.
Let us observe that os 0 (x)g 0 s 0 (z) −1 = zs 0 (z) −1 = o, thus s 0 (x)g 0 s 0 (z) −1 ∈ K and s 0 (x)g 0 s 0 (z) −1 k = k 1 ∈ K. But this means that Thus, continuing the computation, we have Now, we shall consider representations of the isotropy subgroupoid Γ. As we have seen, Γ = x∈X {x} × K x with K x = g −1 Kg and g ∈ G such that its coset in X is equal to x ([g] = x). We can use g = s 0 (x).
Let (τ, W) be a unitary representation of the groupoid Γ in a Hilbert Definition 11 A representation (τ, W) is called X-consistent if there exist a unitary representation (τ 0 , W 0 ) of the group K and a family of Hilbert space

⋄
In the sequel we shall consider the representation of the groupoid G = X × G induced by X -consistent representation (τ, W) of the subgroupoid Γ, and we shall establish its connection with the induced representation in the Mackey sense of the group G. We use the notation of section 3. Now condition 3 of the definition of the space W x assumes the form where x, y ∈ X, y = xg, g ∈ G, γ ∈ Γ y = {y} × K y . Thus we have γ = (y, s 0 (y) −1 ks 0 (y)) for an element k ∈ K . Then, by the definition of X-consistent representation, we can write Let introduce a function φ : G → W 0 defined by the formula φ(ks 0 (y)) = A −1 y (F (x, s 0 (x) −1 ks 0 (y))). Then the function φ has the property φ(kg) = τ 0 (k)φ(g).
It is sufficient to check the above formula for g = s 0 (y). If γ = (y, s 0 (y) −1 ks 0 (y)) then we have φ(kg) = A −1 y (F (γ • (x, s 0 (x) −1 s 0 (y)))) = A −1 y (A y τ 0 (k)A −1 y )F (x, s 0 (y)) = τ 0 (k)φ(g). We shall use the notation (L, W 0 ) for the unitary representation of the group K in the space W 0 , L = τ 0 . Thus we have φ(kg) = L(k)φ(g) and we can consider the Hilbert space H L introduced above as well as the representation (U L , H L ) of the group G induced in the sense of Mackey by L from the subgroup K.
The following theorem establishes a connection of the induced represen-tation (U τ , W) of the groupoid G with the representation (U L , H L ) of the group G.
Denote by R g , g ∈ G, the following operator acting in the space W x , x ∈ X, y = xg, , hg)).
Then we have the family of unitary G-representations (R, W x ), x ∈ X.
(The unitarity follows from the fact that the measure µ is G-invariant and the operators A xh , A xhg are Hilbert space isomorphisms.)

Theorem 3
1. For every x ∈ X the G-representation (R, W x ) is unitarily equivalent to the induced representation (U L , H L ).
2. All representations (R, W x ), x ∈ X, are unitarily equivalent to each other. The equivalence is given by the operators I y x : W x → W y , x, y ∈ X. Proof.
1. We define the linear map J x : W x → H L by (J x F )(g) = φ(ks 0 (y)) = A −1 y (F (x, s 0 (x) −1 ks 0 (y))) where g = ks 0 (y). J x is a linear isomorphism since A y is an isomorphism and it is easily seen that J x preserves scalar products of W x and H L and so it is a Hilbert space isomorphism. To see that it defines an equivalence of representations, we have to show that, for g ∈ G, the following diagram is commutative It is sufficient to take h = s 0 (y) and to notice that each g ∈ G can be written in the form g = s 0 (y) −1 ks 0 (z), for z ∈ X, z = yg and an element k ∈ K.
On the other hand 2. Now it is a simple observation that I y x = J −1 y J x . ⋄ 7 A physical picture. A concept of particle in the representation theory framework .
In papers ( [7], [9], [8], [19]) we have studied a model unifying general relativity and quantum mechanics based on noncommutative geometry. The principal structure of the model is provided by a transformation groupoid G = E × G where G is the Lorentz group and E is the principal G-bundle over the spacetime M (the total space of he bundle is formed by all Lorentz frames at all points of M). We have defined a right action of G on E, and the multiplication of elements of the groupoid is introduced as follows (pg, g 1 ) • (p, g) = (p, gg 1 ), p ∈ E, g, g 1 ∈ G.
The model is reduced to the usual quantum mechanics when an act of measurement is performed. Then we choose a frame p ∈ E which represents a reference frame in which the measurement is done. In the sequel we consider the situation when we want to observe a particle from a different reference frame situated at a fixed point x ∈ M. In such a case, our groupoid reduces to the groupoid G = E x ×G where E x is the fiber of the bundle E over x. The groupoid G is transitive and its isotropy subgroupoid is trivial Γ = E x × {e}, where e ∈ G is the neutral element of the group G.
Let us notice that this regular representation can serve to define random operators on the groupoid and then to define the von Neumann algebra of the groupoid (cf. [8], [19], [9]). Now we pass to quantum mechanical momentum representation of a particle with the mass m. Having fixed (by an act of measurement) p ∈ E, we have reduced our initial space to {p} × G ∼ = G. But we want to consider the energy-momentum space H of the particle, H = {(p 0 , p 1 , p 2 , p 3 ) ∈ R 4 : We have an action of the group G = SL 2 (C) on the hyperboloid H (see [22]).
To describe the action we identify H with the set H of hermitian 2 × 2matrices with determinant equal to m, and we let to act g ∈ G on H to the right in the following way, H ∋ A → g * Ag ∈ H. (It is clear that det(g * Ag) = detA = m). Next, we see that the isotropy group of the element (p 0 , 0, 0, 0), p 0 = √ m is equal to K = SU(2). Thus we deduce that the homogoneus space K\G is diffeomorphic to H. We can take the phase space of a particle of the mass m as the space G = K\G × G = H × G and consider the algebraic structure of transformation groupoid on it.
Let (U, W) be a unitary representation of the groupoid G in a Hilbert bundle W. Assume that there exists an imprimitivity system (U, π) for (U, W). We say that a particle of mass m is represented by the pair (U, π).
We say that it is an elementary particle if the imprimitivity system (U, π) is irreducible [13], [14]. Equivalently (on the strength of Theorem 1), we can say that the particle is an induced representation (U τ , W) where τ is a unitary representation of the isotropy subgroupoid Γ. In the same way, we can say that the particle is elementary if the inducing representation τ is irreducible and, in turn, this means that the representation (L, W 0 ), L = τ 0 , of the group K = SU(2) is irreducible. Then the representation (L, W 0 ) is called the spin of the particle.