Groupoids Research Articles

Dirac groupoids and Dirac bialgebroids

Dirac groupoids and Dirac bialgebroids

On sofic groupoids and their full groups

We prove that the class of sofic groupoids is stable under several measure-theoretic constructions. In particular, we show that virtually sofic groupoids are sofic. We answer a question of Conley, Kechris, and Tucker-Drob by proving that an aperiodic pmp groupoid is sofic if and only if its full group is metrically sofic.

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.

Congruences and decompositions of AG$^{**}$-groupoids

Congruences and decompositions of AG$^{**}$-groupoids

The interplay between Steinberg algebras and skew rings

We study the interplay between Steinberg algebras and skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorff ample groupoid, can be seen as a partial skew inverse semigroup ring.

Some classifiable groupoid $$C^{*}$$ C ∗ -algebras with prescribed K-theory

Given a simple, acyclic dimension group $$G_{0}$$ and countable, torsion-free, abelian group $$G_{1}$$ , we construct a minimal, amenable, etale equivalence relation, R, on a Cantor set whose associated groupoid $$C^{*}$$ -algebra, $$C^{*}(R)$$ , is tracially AF, and hence classifiable in the Elliott classification scheme for simple, amenable, separable $$C^{*}$$ -algebras, and with $$K_{*}(C^{*}(R)) \cong (G_{0}, G_{1})$$ .

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.

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.

Conway Groupoids and Completely Transitive Codes

To each supersimple 2-(n,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M13 which is constructed from P3.

Braided regular crossed modules bifibered over regular groupoids

Braided regular crossed modules bifibered over regular groupoids

Measured quantum transformation groupoids

In this article, when \mathbb G is a locally compact quantum group, we associate, to a braided-commutative \mathbb G -Yetter–Drinfel'd algebra (N, \mathfrak a, \widehat{\mathfrak a}) equipped with a normal faithful semi-finite weight verifying some appropriate condition (in particular if it is invariant with respect to \mathfrak a , or to \widehat{\mathfrak a} ), a structure of a measured quantum groupoid. The dual structure is then given by (N, \widehat{\mathfrak a}, \mathfrak a) . Examples are given, especially the situation of a quotient type co-ideal of a compact quantum group. This construction generalizes the standard construction of a transformation groupoid. Most of the results were announced by the second author in 2011, at a conference in Warsaw.

Partial Transformation Groupoids Attached to Graphs and Semigroups

We introduce the notion of continuous orbit equivalence for partial dynamical systems, and give an equivalent characterization in terms of Cartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras and semigroup C*-algebras can be described as C*-algebras attached to partial dynamical systems. As applications, for graphs, we generalize and explain a result of Matsumoto and Matui relating orbit equivalence and Cartan-isomorphism, and for semigroups, we strengthen several structural results for semigroup C*-algebras concerning amenability, nuclearity as well as simplicity of boundary quotients. We also discuss pure infiniteness for partial transformation groupoids arising from graphs and semigroups.

On Cyclic Associative Abel-Grassman Groupoids

A new subclass of AG-groupoids, so called, cyclic associative Abel-Grassman groupoids or CAAG-groupoid is studied. These have been enumerated up to order 6. A test for the verication of cyclic associativity for an arbitrary AG-groupoid has been introduced. Various properties of CAAG-groupoids have been studied. Relationship among CA-AG-groupoids and other subclasses of AG-groupoids is investigated. It is shown that the subclass of CA-AG-groupoid is dierent from that of the AG*-groupoid as well as AG**-groupoids.

Groupoids corresponding to relational systems

Groupoids corresponding to relational systems

Hyperbolic groupoids and duality

Introduction Technical preliminaries Preliminaries on groupoids and pseudogroups Hyperbolic groupoids Smale quasi-flows and duality Examples of hyperbolic groupoids and their duals Bibliography Index

Quantum Heisenberg manifolds as twisted groupoid [formula omitted]-algebras

The quantum Heisenberg manifolds are noncommutative manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product C⁎-algebras, and they also can be realized as crossed products of C(T2) by Hilbert C⁎-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid C⁎-algebras. In the final section, for a transformation groupoid X×Z with X compact, and (E,p,X) a locally trivial principal G-bundle, where G is a compact abelian group, we form a groupoid Λ that is a locally trivial principal G-bundle over X×Z, which reduces to the twist groupoid of the second author when G=T.

О пополнении группоида до программной алгебры

О пополнении группоида до программной алгебры

Geometry and general relativity in the groupoid model with a finite structure group

In a series of papers (M. Heller et al. J. Math. Phys. 38, 5840 (1997). doi:10.1063/1.532186 ; M. Heller and W. Sasin. Int. J. Theor. Phys. 38, 1619 (1999). doi:10.1023/A:1026617913754 ; M. Heller et al. Int. J. Theor. Phys. 44, 619 (2005). doi:10.1007/s10773-005-3992-7 ) we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra, [Formula: see text], defined on a transformation groupoid Γ determined by the action of the Lorentz group on the frame bundle (E, πM, M) over space–time M. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group, G, and the frame bundle is trivial E = M × G. The model is fully computable. We define the Einstein–Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space–time (giving the “general relativistic limit”), the extra terms that appear due to our generalization can be interpreted as “matter terms”, as in Kaluza–Klein-type models. To illustrate this effect we further simplify the metric matrix to a block diagonal form, compute for it the generalized Einstein equations and find two of their “Friedman-like” solutions for the special case when G = [Formula: see text]. One of them gives the flat Minkowski space–time (which, however, is not static), another, a hyperbolic, linearly expanding universe.

Exact circle maps and KMS states

We describe the KMS states and the ground states for the gauge action on the C*-algebra of the oriented transformation groupoid of a continuous piecewise monotone and exact map of the circle.

On Modulo AG-groupoids

A groupoid G is called an AG-groupoid if it satisfies the left invertive law: (ab)c = (cb)a. An AG-group G, is an AG-groupoid with left identity e 2 G (that is, ea = a for all a 2 G) and for all a 2 G there exists a−1 2 G such that a −1 a = aa−1 = e. In this article we introduce the concept of AG-groupoids (mod n) and AG-group (mod n) using Vasantha’s constructions [1]. This enables us to prove that AG-groupoids (mod n) and AG-groups (mod n) exist for every integer n 3. We also give some nice characterizations of some classes of AG-groupoids in terms of AG-groupoids (mod n).