Representation Of Groupoid Research Articles

Partial groupoid representations and a relation with the Birget–Rhodes expansion

We introduce partial groupoid representations of a finite groupoid \mathcal{G} on an algebra \mathcal{A} . We also show that the partial groupoid representations of \mathcal{G} are in one-to-one correspondence with the representations of the algebra generated by the Birget–Rhodes expansion \mathcal{G}^{\mathrm{BR}} of \mathcal{G} .

Ideals of étale groupoid algebras and Exel’s Effros–Hahn conjecture

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author’s Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.

Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions.

On the Structure of Finite Groupoids and Their Representations

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self maps. Then representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and smooth representations. It is known that in the topological category, this correspondence can be reversed for certain topological groupoids. We extend this result to the smooth category under weaker assumptions on the groupoids.

Some Groupoids and Their Representations by Means of Integer Sequences

In some previous works, we have discussed the groupoids related to the integer sequences of Mersenne, Fermat, Cullen, Woodall and other numbers. These groupoids possess different binary operators. As we can easily see, other integer sequences can have the same binary operators, and therefore can be used to represent the related groupoids. Using the On-Line Encyclopedia of Integer Sequences (OEIS), we are able to identify the properties of these representations of groupoids. At the same time, we can also find integer sequences not given in OEIS and probably not yet studied.

A universal property for groupoid C*-algebras. I

We describe representations of groupoid C*-algebras on Hilbert modules over arbitrary C*-algebras by a universal property. For Hilbert space representations, our universal property is equivalent to Renault's Integration-Disintegration Theorem. For a locally compact group, it is related to the automatic continuity of measurable group representations. It implies known descriptions of groupoid C*-algebras as crossed products for \'etale groupoids and transformation groupoids of group actions on spaces.

Multiplicity formulas for representations of transformation groupoids

Abstract We study the representations of transitive transformation groupoids with the aim of generalizing the Mackey theory. Using the Mackey theory and a bijective correspondence between the imprimitivity systems and the representations of a transformation groupoid we derive the irreducibility theory. Then we derive the direct sum decomposition for representations of a groupoid together with the formula for the multiplicity of subrepresentations. We discuss a physical interpretation of this formula. Finally, we prove the claim analogous to the Peter-Weyl theorem for a noncompact transformation groupoid. We show that the representation theory of a transitive transformation groupoids is closely related to the representation theory of a compact groups.

Topological construction of $C^*$-correspondences for groupoid $C^*$-algebras

Let G and H be locally compact, Hausdor groupoids with Haar systems. We de fine a topological correspondence from G to H to be a G-H bispace X carrying a G-quasi invariant and H-invariant family of measures. We show that such a correspondence gives a C*-correspondence from C *(G) to C* (H). If the groupoids and the spaces are second countable, then this construction is functorial. We show that under a certain amenability assumption, similar results hold for the reduced C *-algebras. We apply this theory of correspondences to study induction techniques for groupoid representations, construct morphisms of Brauer groups and produce some odd unbounded KK-cycles.

On spectra of Koopman, groupoid and quasi-regular representations

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.

Modular Classes of Lie Groupoid Representations up to Homotopy

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's "The volume of a differentiable stack".

A short proof of the localic groupoid representation of Grothendieck toposes

It is known that each Grothendieck topos is the category of G \mathbb {G} -equivariant sheaves for some localic groupoid G \mathbb {G} . A simple proof of this is given which relies on the recently observed fact that the pullback adjunction between locales induced by any geometric morphism satisfies Frobenius reciprocity.

Topological representation of geometric theories

AbstractUsing Butz and Moerdijk's topological groupoid representation of a topos with enough points, a ‘syntax‐semantics’ duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantic topological groupoid of models and isomorphisms of a theory. It is then shown how to extract a theory from equivariant sheaves on a topological groupoid in such a way that the result is a contravariant adjunction between theories and groupoids, the restriction of which is a duality between theories with enough models and semantic groupoids. Technically a variant of the syntax‐semantics duality constructed in 1 for first‐order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids—reflecting the use of ‘indexed’ models in the representation theorem—which in several respects simplifies the construction and the characterization of semantic groupoids.

Imprimitivity theorem for groupoid representations

Abstract 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.

Groupoid extensions of mapping class representations for bordered surfaces

The mapping class group of a surface with one boundary component admits numerous interesting representations including a representation as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., to construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representations on the moduli spaces of flat connections.

Irreducible representations of groupoid 𝐶*-algebras

If G G is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.

REPRESENTATIONS OF INTERNAL CATEGORIES

Adams gave the notion of a Hopf algebroid generalizing the notion of a Hopf algebra and showed that certain generalized homology theories take values in the category of comodules over the Hopf algebroid associated with each homology theory. A Hopf algebra represents an affine group scheme which is a group in the category of a scheme and the notion of comodules over a Hopf algebra is equivalent to the notion of representations of the affine group scheme represented by a Hopf algebra. On the other hand, a Hopf algebroid represents a groupoid in the category of schemes. Therefore, it is natural to consider the notion of comodules over a Hopf algebroid as representations of the groupoid represented by a Hopf algebroid. This motivates the study of representations of groupoids, and more generally categories, for topologists. The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category, using the notion of a fibered category.

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur’s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle ℋ = {Hu}u∈¸G0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle (G0, {Hu}, μ), then dimHu = 1 (u ∈¸ G0). Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.

Regular representation of groupoid C* -algebras and applications to inverse semigroups

The analogue of the left regular representation of a locally compact groupoid is constructed in the Hausdoras well as in the non-Hausdorcase. A necessary and su‰cient condition for a locally compact groupoid with a cocycle to be Morita equi- valent to a group action is obtained. As an application, the C � -algebras of a class of inverse semigroups is shown to be Morita-equivalent to crossed products of groups by abelian C � -

An addendum to “Path-lifting for Grothendieck toposes"

The purpose of this note is to prove the conjecture in [M], viz. that if F → E is a connected and locally connected morphism between (Grothendieck) toposes then F → E is an open surjection. In [M], the following partial results were proved (cf. also Remark (ii) below): (1) If F → E is a connected and locally connected, then F → E is a stable surjection (between toposes). (2) If Y → X is a connected and locally connected map between locales, then Y I → X is an open surjection (between locales). Here, as in [M], I denotes the unit interval [0, 1], viewed as a locale or as a topos (viz. the topos of sheaves of sets on [0, 1]). I would like to gratefully acknowledge that my attention was drawn again to this issue in relation to my work with Bunge on the paths-fundamental group of a topos [BM], and with Kock on etendues. Indeed, it is the stronger result to be proved here which is needed in these applications. (Without going into details, the point is that this result implies that for a localic groupoid G with connected and locally connected source and target maps, one can now immediately obtain a groupoid representation for the path-space of the classifying topos of G, by the equivalence B(G) ' B(G); I assume here that G is etale complete, of course.) Consider for a topos E its “canonical” connected and locally connected cover XE → E by a locale, where XE = E [En(G)] is the topos of E-internal sheaves on the locale En(G) of infinite-to-one partial enumerations of a generating object G in E (see [JM]). Recall that this construction is “natural in E” in the sense that it is the pullback of the generic case where E is the object classifier S[U ] and G is the universal object U . I will show: