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Leszek Pysiak IMPRIMITIVITY THEOREM for GROUPOID

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Leszek Pysiak IMPRIMITIVITY THEOREM for GROUPOID DEMONSTRATIO MATHEMATICA Vol.XLIV No1 2011 Leszek Pysiak IMPRIMITIVITY THEOREM FOR GROUPOID REPRESENTATIONS Abstract. We define and investigate the concept of the groupoid representation in- duced by a representation of the isotropy subgroupoid. Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity the- orem for such representations which is a generalization of the classical Mackey’s theorem known from the theory of group representations. 1. Introduction The present paper, devoted to the study of the theory of groupoid rep- resentations, is a continuation of my previous work [20] in which one can find a presentation of the groupoid concept and of the groupoid representa- tion 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 non- commutative geometry, and algebraic topology, and also in numerous ap- plications, 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 non- commutative convolutive algebra on this groupoid in the bundle of Hilbert spaces. In paper ([6]) we have applied this groupoid representation to inves- 2000 Mathematics Subject Classification: 22A22, 22A30, 22D30. Key words and phrases: grupoids, induced representations, imprimitivity systems. 30 L. Pysiak tigate spacetime singularities, and in [10] the representation of the funda- mental 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 Hausdorff space (Section 4). This concept, framed “in the spirit of Mackey” is a natural generalization of induced representation 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 im- primitivity system, is a representation induced by a representation of the isotropy subgroupoid. In Section 6, I investigate induced representations of the transforma- tion gropoid over a homogeneous space of the group G and show that there exists a strict connection between these representations and induced repre- sentations of the group G (in the sense of Mackey). In Section 7, I give a physical interpretation of concepts analyzed in Sec- tion 6. I describe the 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 system for the unitary represen- tation of this groupoid. 2. Preliminaries Let be a groupoid over a set X (the base of ). We recall (cf. [4], [18]) that a groupoidG is a set with a partially definedG multiplication " " on a subset 2 of G, and an inverse map g g−1 defined for every g ◦ . The multiplicationG G ×G is associative when defined.→ One has an embedding ǫ∈: X G called the identity section and two structure maps d, r : X such that→ G G → ǫ(d(g)) = g−1 g ◦ ǫ(r(g)) = g g−1 ◦ for g . ∈ G Imprimitivity theorem for groupoid representations 31 Let us introduce the following fibrations in the set : G = g : d(g) = x Gx { ∈ G } x = g : r(g) = x G { ∈ G } y x x for x X. Let us also denote x = y, and consider the set x = x ∈ G G G G x for x X. It has the group structure and is called the isotropy group G G ∈ T x of the point x. It is clear that the set Γ = x∈X x has the structure of a subgroupoidT of over the base X (all the structureG maps are the restrictions of the structureG maps of to Γ). S G We call a transitive groupoid, if for each pair of elements x1,x2 X there exists Gg such that d(g) = x and r(g) = x . ∈ ∈ G 1 2 A groupoid is a topological groupoid if and X are topological spaces and all structureG maps are continuous (in particular,G the embedding ǫ is a homeomorphism of X onto its image). In the following we assume that (and thus X) is a locally compact Hausdorff space. G Example 1. A pair groupoid. Let X be a locally compact Hausdorff space. Take = X X. We define the set 2 of composable elements as 2 = ((x,G y), (y,z×)) : x,y,z X G and a multiplication, for ((x,G y), (y,z{ )) 2, by ∈ }⊂G×G ∈ G (x, y) (y,z)=(x,z). ◦ Moreover, we have: (x, y)−1 =(y,x), d(x, y) = y, r(x, y) = x, ǫ(x)=(x,x). With such defined structure maps is a groupoid, called pair groupoid. G 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× =→X G by defining the7→ following structure maps. The set of composable elementsG × 2 = ((xg,h), (x,g): x X,g,h G , and the multiplication for ((Gxg,h{), (x,g)) 2 is given∈ by ∈ }⊂G×G ∈ G (xg,h) (x,g)=(x,gh). ◦ −1 −1 And also (x,g) =(xg,g ), d(x,g) = x, r(x,g) = xg, ǫ(x)=(x,eG). 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 is a family λ ∈ G { x}x X of regular Borel measures defined on the sets x (which are locally compact Hausdorff spaces) such that the following threeG conditions are satisfied: 1. the support of each λ is the set , x Gx 32 L. Pysiak 0 2. (continuity) for any f Cc( ) the function f , where ∈ G \ 0 f (x) = fdλx, Gx belongs to Cc(X), 3. (right invariance) for any g and any f Cc( ), \ \ ∈ G ∈ G f(h g)dλ (h) = f(u)dλ (u). ◦ r(g) d(g) Gr(g) Gd(g) x x One can also consider the family λ x∈X of left-invariant measures, each λ x { } x −1 being defined on the set by the formula λ (E) = λx(E ) for any Borel subset E of x (where EG−1 = g : g−1 E ). Then the invariance condition assumesG the form: { ∈ G ∈ } \ \ f(g h)dλd(g)(h) = f(u)dλr(g)(u). ◦ Gd(g) Gr(g) Now, let µ be a regular Borel measure on X. We can consider the following Ì measures which will be called measures associated with µ: ν = λxdµ(x) on Ì Ì , ν−1 = λxdµ(x) and ν2 = λ λxdµ(x) on 2. G x × G If ν = ν−1 we say that the measure µ is a -invariant measure on X. G Definition 2. A topological groupoid on X is called locally trivial if G there exist a point x X, an open cover Ui of X and continuous maps s : U such that∈r s = id for all i{. } i i → Gx ◦ i Ui Proposition 1. Assume that is a locally trivial groupoid on X and X is second countable space. Let µ beG a regular Borel measure on X. Then 1. is transitive, 2. allG isotropy groups of are isomorphic with each other, G 3. for every y X there exist an open cover Vj of X and continuous maps s : V ∈ such that r s = id , { } y,j j → Gy ◦ y,j vj 4. for every x X there exists a section sx : X x which is µ-measurable, ∈ −1 → G i.e., for every Borel set B in x, sx (B) is µ-measurable subset of X, 5. if the measure µ has the propertyG that µ(A) = µ(A) for every µ-measurable subset A of X, then the section sx is µ-a.e. continuous on X. Proof. 1. Let y1, y2 X. Suppose that y1 U1 and y2 U2. Then r(s1(y1)) = y1 and r(s (y∈)) = y . But g = s (y∈) s (y )−1 ∈has the property d(g) = y 2 2 2 2 2 ◦ 1 1 1 and r(g) = y , and this means that is transitive. 2 G 2. For x, y X let gyx be an element of such that d(gyx) = x and r(gyx) = ∈ G x y y.
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