The Experts below are selected from a list of 14328 Experts worldwide ranked by ideXlab platform
Nicolaas P. Landsman - One of the best experts on this subject based on the ideXlab platform.
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Operator Algebras and Poisson Manifolds Associated to Groupoids
Communications in Mathematical Physics, 2001Co-Authors: Nicolaas P. LandsmanAbstract: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.
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Compact Quantum Groupoids
arXiv: Mathematical Physics, 1999Co-Authors: Nicolaas P. LandsmanAbstract:Quantum Groupoids are a joint generalization of Groupoids and quantum groups. We propose a definition of a compact quantum Groupoid that is based on the theory of C*-algebras and Hilbert bimodules. The essential point is that whenever one has a tensor product over the complex numbers in the theory of quantum groups, one now uses a certain tensor product over the base algebra of the quantum Groupoid.
Pedro Resende - One of the best experts on this subject based on the ideXlab platform.
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Actions of étale-covered Groupoids
Journal of Algebra, 2021Co-Authors: Juan Pablo Quijano, Pedro ResendeAbstract:Abstract By restricting to a class of localic open Groupoids G which, similarly to Lie Groupoids, possess appropriate covers G ˆ → G by etale Groupoids, we extend results about Groupoid actions and quantales that were previously proved for etale Groupoids but do not seem to work for arbitrary open Groupoids. In particular we obtain a characterization of the category of G-actions as a category of quantale modules on O ( G ˆ ) that satisfy a condition related to the quantale O ( G ) . This leads to a simple description of G-sheaves and the classifying topos of G in terms of Hilbert O ( G ˆ ) -modules. The bicategory whose 1-cells are the Groupoid bi-actions is bi-equivalent to a corresponding bicategory of quantales and bimodules.
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Étale Groupoids as germ Groupoids and their base extensions
2010Co-Authors: Dmitry Matsnev, Pedro ResendeAbstract:We introduce the notion of wide representation of an inverse semi- group and prove that with a suitably defined topology there is a space of germs of such a representation which has the structure of anetale Groupoid. This gives an elegant description of Paterson's universal Groupoid and of the translation Groupoid of Skandalis, Tu, and Yu. In addition we characterize the inverse semigroups that arise from Groupoids, leading to a precise bijection between the class ofetale Groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and we explain the sense in which quantales and localic Groupoids carry a generalization of this correspondence.
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Étale Groupoids as germ Groupoids and their base extensions
Proceedings of the Edinburgh Mathematical Society, 2010Co-Authors: Dmitry Matsnev, Pedro ResendeAbstract:AbstractWe introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation that has the structure of an étale Groupoid. This gives an elegant description of Paterson's universal Groupoid and of the translation Groupoid of Skandalis, Tu and Yu. In addition, we characterize the inverse semigroups that arise from Groupoids, leading to a precise bijection between the class of étale Groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and we explain the sense in which quantales and localic Groupoids carry a generalization of this correspondence.
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Étale Groupoids and their quantales
Advances in Mathematics, 2007Co-Authors: Pedro ResendeAbstract:Abstract We establish close and previously unknown relations between quantales and Groupoids. In particular, to each etale Groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic etale Groupoids and their quantales, which are given a rather simple characterization and here are called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids, and as a consequence we obtain a (non-functorial) correspondence between these and localic etale Groupoids that generalizes more classical results concerning inverse semigroups and topological etale Groupoids. This generalization is entirely algebraic and it is valid in an arbitrary topos. As a consequence of these results we see that a localic Groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen, and an analogue of this holds for topological Groupoids. In practice we are provided with new tools for constructing localic and topological etale Groupoids, as well as inverse semigroups, for instance via presentations of quantales by generators and relations. The characterization of inverse quantal frames is to a large extent based on a new quantale operation, here called a support, whose properties are thoroughly investigated, and which may be of independent interest.
Osman Mucuk - One of the best experts on this subject based on the ideXlab platform.
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Monodromy Groupoid of an internal Groupoid in topological groups with operations
Filomat, 2015Co-Authors: Osman Mucuk, H. Fulya AkızAbstract:In this paper, the internal Groupoid structure of the monodromy Groupoid for topological internal Groupoids is developed.
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Group-Groupoids and monodromy Groupoids
Topology and its Applications, 2011Co-Authors: Osman Mucuk, Tunçar Şahan, Berrin Kılıçarslan, Nazmiye AlemdarAbstract:Abstract This paper gives an introduction to some results on monodromy Groupoids and the monodromy principle, and then develops the notion of monodromy Groupoid for group Groupoids.
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Holonomy and monodromy Groupoids
Banach Center Publications, 2001Co-Authors: Ronald Brown, Ilhan Icen, Osman MucukAbstract:We outline the construction of the holonomy Groupoid of a locally Lie Groupoid and the monodromy Groupoid of a Lie Groupoid. These specialise to the well known holonomy and monodromy Groupoids of a foliation, when the Groupoid is just an equivalence relation.
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EXTENDIBILITY, MONODROMY, AND LOCAL TRIVIALITY FOR TOPOLOGICAL GroupoidS
International Journal of Mathematics and Mathematical Sciences, 2001Co-Authors: Osman MucukAbstract:A Groupoid is a small category in which each morphism has an inverse. A topo- logical Groupoid is a Groupoid in which both sets of objects and morphisms have topolo- gies such that all maps of Groupoid structure are continuous. The notion of monodromy Groupoid of a topological Groupoid generalizes those of fundamental Groupoid and uni- versal cover. It was earlier proved that the monodromy Groupoid of a locally sectionable topological Groupoid has the structure of a topological Groupoid satisfying some proper- ties. In this paper a similar problem is studied for compatible locally trivial topological Groupoids.
Florentin Smarandache - One of the best experts on this subject based on the ideXlab platform.
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Theory of Abel Grassmann's Groupoids
2020Co-Authors: Florentin Smarandache, Madad Khan, Saima AnisAbstract:An AG-Groupoid is an algebraic structure that lies in between a Groupoid and a commutative semigroup. It has many characteristics similar to that of a commutative semigroup. If we consider x^2y^2= y^2x^2, which holds for all x, y in a commutative semigroup, on the other hand one can easily see that it holds in an AG-Groupoid with left identity e and in AG**-Groupoids. This simply gives that how an AG-Groupoid has closed connections with commutative algebras. We extend now for the first time the AG-Groupoid to the Neutrosophic AG-Groupoid. A neutrosophic AG-Groupoid is a neutrosophic algebraic structure that lies between a neutrosophic Groupoid and a neutrosophic commutative semigroup.
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A Kind of Variation Symmetry: Tarski Associative Groupoids (TA-Groupoids) and Tarski Associative Neutrosophic Extended Triplet Groupoids (TA-NET-Groupoids)
Symmetry, 2020Co-Authors: Xiaohong Zhang, Wangtao Yuan, Mingming Chen, Florentin SmarandacheAbstract:The associative law reflects symmetry of operation, and other various variation associative laws reflect some generalized symmetries. In this paper, based on numerous literature and related topics such as function equation, non-associative Groupoid and non-associative ring, we have introduced a new concept of Tarski associative Groupoid (or transposition associative Groupoid (TA-Groupoid)), presented extensive examples, obtained basic properties and structural characteristics, and discussed the relationships among few non-associative Groupoids. Moreover, we proposed a new concept of Tarski associative neutrosophic extended triplet Groupoid (TA-NET-Groupoid) and analyzed related properties. Finally, the following important result is proved: every TA-NET-Groupoid is a disjoint union of some groups which are its subgroups.
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Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids
Symmetry, 2020Co-Authors: Xiaohong Zhang, Florentin SmarandacheAbstract:Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative Groupoid (CA-Groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-Groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-Groupoids, quasi-cancellative CA-Groupoids and power cancellative CA-Groupoids are found out. Moreover, the concept of variant CA-Groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-Groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-Groupoids are established.
Mohammad Qasim Mann'a - One of the best experts on this subject based on the ideXlab platform.
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Monoid and Topological Groupoid
Journal of the Indian Mathematical Society, 2018Co-Authors: Mohammad Qasim Mann'aAbstract:Here we introduce some new results which are relative to the concept of topological monoid-Groupoid and prove that the category of topological monoid coverings of X is equivalent to the category covering Groupoids of the monoid-Groupoid π 1 (X). Also, it is shown that the monoid structure of monoid-Groupoid lifts to a universal covering Groupoid.
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A note on topological semigroup-Groupoid
2013Co-Authors: Mohammad Qasim Mann'aAbstract:In this paper we prove that the set of homotopy classes of paths in topological semigroup is a semigroup-Groupoid. Further, we define the category TSGCov/X of topological semigroup coverings of X and prove that its equivalent to the category SGpGpdCov/ of covering Groupoids of the semigroup-Groupoid . We also prove that the topological semigroup structure of a topological semigroup-Groupoid lifts to a universal topological covering Groupoid.
