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We introduce the notion of groupoid grading, give some nontrivial examples and prove that groupoid gradings on simple commutative or anti-commutative algebras are necessarily group gradings. We also take advantage of the structure of groupoids to prove some results about groupoid gradings and certain coarsenings of these which turn out to be group gradings. We also study set gradings on arbitrary algebras, by characterizing their homogeneous semisimplicity and their homogeneous simplicity in terms of a property satisfied by the supports of the gradings, and also relate set gradings with groupoid gradings via coarsenings. Finally we study a class of set gradings on M n ( C ) , the orthogonal gradings, and show that all of them which are fine are necessarily groupoid gradings.

In the framework of Banach differential geometry we investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W ∗ -algebra (von Neumann algebra) M . The main role in this theory is played by the complex Banach–Lie groupoid G ( M ) ⇉ L ( M ) of partially invertible elements of M over the lattice L ( M ) of orthogonal projections of M . The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson V B -groupoids with G ( M ) ⇉ L ( M ) as the side groupoid.

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C ∗ -algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C ∗ -algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.

A Poisson structure on a homogeneous space of a Poisson groupoid is homogeneous if the action of the Lie groupoid on the homogeneous space is compatible with the Poisson structures. According to a result of Liu, Weinstein and Xu, Poisson homogeneous spaces of a Poisson groupoid are in correspondence with suitable Dirac structures in the Courant algebroid defined by the Lie bialgebroid of the Poisson groupoid. We show that this correspondence result fits into a more natural context: the one of Dirac groupoids, which are objects generalizing Poisson groupoids and multiplicative closed 2-forms on groupoids.

In the paper we study the algebroid A(M) of the groupoid G(M)⇉L(M) of partially invertible elements over the lattice L(M) of orthogonal projections of a W∗-algebra M. In particular the complex Banach manifold structure of these objects is investigated. The expressions on the algebroid Lie brackets for A(M) and related algebroids are given in noncommutative operator coordinates in the explicit way. We also describe structure of the groupoid of partial isometries U(M)⇉L(M) and the frame groupoid GlinAp0(M)⇉Lp0(M) as well as their algebroids.

Let G be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and K a compact subgroupoid of G with a Haar system too. (G, K) is a Gelfand pair if the algebra of bi-K-invariant functions is commutative under convolution. In this paper, we study spherical functions on groupoids associated to compact Gelfand pairs.

A groupoid is called power-commutative if every mono-generated subgroupoid is commutative. The class Pc of power-commutative groupoids is a variety. A description of free objects in this variety and their characterization by means of injective groupoids in Pc are obtained.

We decompose the full and reducedC∗C^*-algebras of an extension of a groupoid by the circle into a direct sum of twisted groupoidC∗C^*-algebras.

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid C*-algebras. We show that the C*-algebra is generated by an inverse semigroup T of partial isometries. We show that the groupoid 𝓖tight associated with the inverse semigroup T by Exel is exactly the same as the groupoid considered by Sheu.

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks faithful representations., Comment: 20 pages

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of etale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.

For a locally compact groupoid with a fixed Haar system and quasi-invariant measure , we introduce the notion of -measurability and construct the space 1(, , ) of absolutely integrable functions on and show that it is a Banach -algebra and a two-sided ideal in the algebra () of complex Radon measures on . We find correspondences between representations of on Hilbert bundles and certain class of nondegenerate representations of 1(, , ).

In the context of non-abelian gerbes, we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We address their two-dimensional parallel transport, study its properties, and construct non-abelian Wilson surface functionals.

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid G ⋉ X with G compact.

Some aspects of the algebraic structure of TCI-groupoids (totally commutative idempotent groupoids) are presented. Among other things, a full description of the corresponding binary clones is given. To investigate the algebraic structure of TCI-groupoids and their clones the families of two-generated subgroupoids are used. Next the algebraic structure of these families is studied in detail. We present a special construction which allows us to build each finite TCI-groupoid using the two-element semilattice as a kind of elementary "brick". Furthermore it is proved that the class of all regular TCI-groupoids satisfying binary nonbalanced identities is a nonaxiomatizable proper subclass of the class of all regular TCI-groupoids.

The mapping class group of a surface with one boundary component admits numerous interesting representations including 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 Teichm\"uller 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., 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., Comment: 24 pages, 4 figures Theorem 3.6 has been strengthened, and Theorems 8.1 and 8.2 have been added

We present a higher index theorem for a certain class of etale one-dimensional complex-analytic groupoids. The novelty is the use of the local anomaly formula established in a previous paper, which represents the bivariant Chern character of a quasihomomorphism as the chiral anomaly associated to a renormalized non-commutative chiral field theory. In the present situation the geometry is non-metric and the corresponding field theory can be renormalized in a purely conformal way, by exploiting the complex-analytic structure of the groupoid only. The index formula is automatically localized at the automorphism subset of the groupoid and involves a cap-product with the sum of two different cyclic cocycles over the groupoid algebra. The first cocycle is a trace involving a generalization of the Lefschetz numbers to higher-order fixed points. The second cocycle is a non-commutative Todd class, constructed from the modular automorphism group of the algebra., Comment: 38 pages. v2: some inconsistencies with the use of pseudogroups have been fixed

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.

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an action of a bundle of compact topological groups., Comment: 15 pages

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.

A multiplicatively closed, horizontal $n$-plane field $D$ on a Lie groupoid $G$ over $M$ generalizes to intransitive geometry the classical notion of a Cartan connection. The infinitesimalization of the connection $D$ is a Cartan connection $\nabla $ on the Lie algebroid of $G$, a notion already studied elsewhere by the author. It is shown that $\nabla $ may be regarded as infinitesimal parallel translation in the groupoid $G$ along $D$. From this follows a proof that $D$ defines a pseudoaction generating a pseudogroup of transformations on $M$ precisely when the curvature of $\nabla $ vanishes. A byproduct of this analysis is a detailed description of multiplication in the groupoid $J^1 G$ of one-jets of bisections of $G$.

We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of $A$-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart., Comment: Statement of Theorem 4.7 and notation in Section 4.3 corrected

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka–Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representations have finite dimensional fibres. We prove Schur's lemma and Peter–Weyl theorem for compact groupoids.

In the framework of C ∗ -algebraic deformation quantization we propose a notion of a deformation groupoid which could apply to known examples such as Connes’s tangent groupoid of a manifold, its generalization by Landsman and Ramazan, Rieffel’s noncommutative torus, and even Landi’s noncommutative 4-sphere. We construct such a groupoid for a wide class of T n -spaces, that generalizes the one given for C n by Bellissard and Vittot. In particular, using the geometric properties of the moment map discovered in the 1980’s by Atiyah, Delzant, Guillemin and Sternberg, it provides a C ∗ -algebraic deformation quantization for all toric manifolds, including the 2-sphere and all complex projective spaces.

This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson.

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 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.

As a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, including for those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes? tangent groupoid of a manifold. As is well known for a Heisenberg manifold (M,H) the relevant notion of tangent bundle is rather that of a Lie group bundle of graded 2-step nilpotent Lie groups GM. We define the tangent groupoid of (M,H) as a differentiable groupoid gHM encoding the smooth deformation of M × M to GM. In particular, this construction makes a crucial use of a refined notion of privileged coordinates and of a tangent-approximation result for Heisenberg diffeomorphisms.

The notion of H-covariant strong Morita equivalence is introduced for *-algebras over C = R(i) with an ordered ring R which are equipped with a *-action of a Hopf *-algebra H. This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant *-Morita equivalence with its H-covariant *-Picard groupoid. We discuss various groupoid morphisms between the corresponding notions of the Picard groupoids. Moreover, we realize several Morita invariants in this context as arising from actions of the H-covariant strong Picard groupoid. Crossed products and their Morita theory are investigated using a groupoid morphism from the H-covariant strong Picard groupoid into the strong Picard groupoid of the crossed products., LaTeX 2e, 50 pages. Revised version with additional examples and references. To appear in JPAA

We show that Haefliger’s cohomology for étale groupoids, Moore’s cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Čech) cohomology for topological simplicial spaces.

Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-degenerate solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-degenerate solution is also defined. Non-degenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.

To any groupoid, equipped with a Haar system, Jean-Michel Vallin had associated several objects (pseudo-multiplicative unitary, Hopf-bimodule) in order to generalize, up to the groupoid case, the classical notions of multiplicative unitary and Hopf–von Neumann algebra, which were intensely used to construct quantum groups in the operator algebra setting. In two former articles (one in collaboration with Jean-Michel Vallin), starting from a depth-2 inclusion of von Neumann algebras, we have constructed such objects, which allowed us to study two ‘quantum groupoids’ dual to each other. We are now investigating in greater details the notion of pseudo-multiplicative unitary, following the general strategy developed by Baaj and Skandalis for multiplicative unitaries. AMS 2000 Mathematics subject classification: Primary 46L89; 22A22; 81R50

We construct a symplectic groupoid of triangular bilinear forms and establish a relation with the flag variety. We also study the induced Poisson structure and the centre of the corresponding algebroid.

We prove that, as in the case of global actions, any partial action gives rise to a groupoid provided with a Haar system, whose C ∗ C^* -algebra agrees with the crossed product by the partial action.

In these lectures I discuss the Linearization Theorem for Lie groupoids, and its relation to the various classical linearization theorems for submersions, foliations and group actions. In particular, I explain in some detail the recent metric approach to this problem proposed in [6].

It is shown that a central extension of a Lie groupoid by an Abelian Lie group A has a principal A-bundle structure and the extended Lie groupoid is classified by an Euler es-class. Then we prove that for a symplectic α-connected, αβ-transversal or α-simply connected groupoid, there exists at most one central S 1 -extension, the Euler es-class of which corresponds to the Poisson cohomology class of the Poisson manifold of units.

Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a "Lie groupoid" is a subgroupoid of Aut(X) defined by a system of partial differential equations. To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called "the Galois groupoid of the foliation". Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.

In this paper, the monodromy groupoids of internal groupoids in the topological groups with operations are studied and a monodromy principle for internal groupoids in groups with operations is obtained.

The idea of monodromy groupoid we deal here is due to Pradines [1]. An application of the monodromy groupoid on principal bundles was earlier given in [2]. In this paper, a topological version of this application is given.

In this note, we characterise completely the ideals of the groupoid C*-algebra arising from the asymptotic equivalence relation on the points of a Smale space and show that the related Ruelle algebra is simple when the Smale space is topologically transitive.

Groupoids with unit that admit embeddings into quasi-Boolean powers of elementary 2-groups are characterized.

It is well known that a measured groupoid G defines a von Neumann algebra W*(G), and that a Lie groupoid G canonically defines both a C*-algebra C*(G) and a Poisson manifold A*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W*(G), G↦C*(G), and G↦A*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence.

This paper classifies all groupoids that have the property that every map from the groupoid to itself is either an endomorphism or a multiplication map.

We build a longitudinally smooth differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small) b-calculus., 12 pages, LaTeX. This is a slight update from funct-an/9707008 (a citation was missing)

Groupoid actions on C*-bundles and inverse semigroup actions on C*-algebras are closely related when the groupoid is r-discrete.