Groupoid cohomology and extensions

Tu, Jean-Louis

doi:10.1090/s0002-9947-06-03982-1

Cited by 39 publications

(39 citation statements)

References 21 publications

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“…When a Cartan subalgebra B of a continuous-trace C * -algebra A exists, the cohomology class [Σ(B)] of its twist is essentially the Dixmier-Douady invariant of A. Indeed, just as in the group case, the groupoid extension Σ(B) defines an element of the cohomology group H 2 (G(B), T) (see [46] for a complete account of groupoid cohomology). Since G(B) is equivalent to B/G(B) = Â, this can be viewed as an element of H 2 ( Â, T ), where T is the sheaf of germs of T-valued continuous functions.…”

Section: Proof Let Us Show (I) Ifmentioning

confidence: 99%

Cartan Subalgebras in $C^*$-Algebras

Renault¹

2008

Irish Math. Soc. Bull.

253

359

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According to J. Feldman and C. Moore's wellknown theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence relations and Cartan pairs, i.e., a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C * -algebraic analogue of this theorem in the early eighties. After a short survey of maximal abelian self-adjoint subalgebras in operator algebras, I present a natural definition of a Cartan subalgebra in a C * -algebra and an extension of Kumjian's theorem which covers graph algebras and some foliation algebras.

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Section: Proof Let Us Show (I) Ifmentioning

confidence: 99%

Cartan Subalgebras in $C^*$-Algebras

Renault¹

2008

Irish Math. Soc. Bull.

253

359

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“…Example 3.2. Given G a topological groupoid, a G-module in [19] is a topological groupoid A with domain and terminus maps equal to p : A → G (0) such that A x…”

Section: Groupoid Actions and Similaritymentioning

confidence: 99%

On groupoids and $C^*$-algebras from self-similar actions

Deaconu¹

2020

Preprint

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Given a self-similar groupoid action (G, E) on the path space of a finite graph, we study the associated Exel-Pardo étale groupoid G(G, E) and its C * -algebra C * (G, E). We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the Ktheory of C * (G, E) and the homology of G(G, E) in certain cases and illustrate with an example.

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“…Many formulations of groupoid cohomology can be found in literature, and relations of the following formulation to previously known theories are not clear. See [20], for example.…”

Section: It Is Directly Checked On All the Generatorsmentioning

confidence: 99%

Cohomology of the adjoint of Hopf algebras

Carter

Crans

Elhamdadi

et al. 2008

J. Gen. Lie Theory Appl.

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A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.in the case of a group algebra, the function algebra on a group, and a calculation of the 1-and 2-dimensional cohomology of the bosonization of the superline are presented in Section 5. Interestingly, the group algebra and the function algebra on a group are cohomologically different. Moreover, the conditions that result when a function on the group algebra satisfies the cocycle condition coincide with the definition of groupoid cohomology. This relationship is given in Section 6, along with a construction of quandle 3-cocycles from groupoid 3-cocycles. In Section 7, we use the deformation cocycles to construct solutions to the Yang-Baxter equation.

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Groupoid cohomology and extensions

Abstract: Abstract. We show that Haefliger's cohomology forétale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (orČech) cohomology for topological simplicial spaces.

Cited by 39 publications

References 21 publications

Cartan Subalgebras in $C^*$-Algebras

Cartan Subalgebras in $C^*$-Algebras

On groupoids and $C^*$-algebras from self-similar actions

Cohomology of the adjoint of Hopf algebras

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