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41
A short survey of noncommutative geometry
 J. Math. Physics
, 2000
"... We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the RiemannHilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geom ..."
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Cited by 49 (4 self)
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We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the RiemannHilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It expresses the gamma five matrix as the pairing between the operator theoretic chern characters of e and D. It is of degree five in the idempotent and four in the Dirac operator which only appears through its commutant with the idempotent. It determines both the sphere and all its metrics with fixed volume form. We also show using the noncommutative analogue of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude on some questions related to string theory. I
The cyclic homology of crossed products algebras
 J. Reine Angew. Math
, 1993
"... In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a cros ..."
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Cited by 39 (0 self)
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In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a crossed product algebra, see Brylinski [2], [3].
A Homology Theory for Étale Groupoids
"... Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haeflig ..."
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Cited by 38 (5 self)
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Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.
Cyclic Cohomology of Étale Groupoids; The General Case
 Ktheory
, 1999
"... We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution ..."
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Cited by 24 (1 self)
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We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the nonHausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...
Hopf algebra equivariant cyclic homology and cyclic homology of crossed product algebras
, 2008
"... We introduce the cylindrical module A♮H, where H is a Hopf algebra and A is a Hopf module algebra over H. We show that there exists an isomorphism between C•(A op ⋊H cop) the cyclic module of the crossed product algebra A op ⋊ H cop, and ∆(A♮H), the cyclic module related to the diagonal of A♮H. If S ..."
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Cited by 21 (10 self)
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We introduce the cylindrical module A♮H, where H is a Hopf algebra and A is a Hopf module algebra over H. We show that there exists an isomorphism between C•(A op ⋊H cop) the cyclic module of the crossed product algebra A op ⋊ H cop, and ∆(A♮H), the cyclic module related to the diagonal of A♮H. If S, the antipode of H, is invertible it follows that C•(A ⋊ H) ≃ ∆(A op ♮H cop). When S is invertible, we approximate HC•(A ⋊ H) by a spectral sequence and give an interpretation of E 0,E 1 and E 2 terms of this spectral sequence.
(CO)CYCLIC (CO)HOMOLOGY OF BIALGEBROIDS: AN APPROACH VIA (CO)MONADS
, 2008
"... ... there is a (co)simplex Z ∗: = ΠTl ∗+1 X in C. The aim of this paper is to find criteria for para(co)cyclicity of Z ∗. Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i: ΠTl → ΠTr, and a morphism w: TrX → TlX in M. The (symmetrical) ..."
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Cited by 17 (3 self)
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... there is a (co)simplex Z ∗: = ΠTl ∗+1 X in C. The aim of this paper is to find criteria for para(co)cyclicity of Z ∗. Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i: ΠTl → ΠTr, and a morphism w: TrX → TlX in M. The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads Tl = T ⊗R (−) and Tr = (−) ⊗R T on the category of Rbimodules. The functor Π can be chosen such that Z n = T b⊗R... b⊗RT b⊗RX is the cyclic Rmodule tensor product. A natural transformation i: T b⊗R(−) → (−)b⊗RT is given by the flip map and a morphism w: X ⊗R T → T ⊗R X is constructed whenever T is a (co)module algebra or coring of an Rbialgebroid. The notion of a stable anti YetterDrinfel’d module over certain bialgebroids, so called ×RHopf algebras, is introduced. In the particular example when T is a module coring of a ×RHopf algebra B and X is a stable anti YetterDrinfel’d Bmodule, the paracyclic object Z ∗ is shown to project to a cyclic structure on T ⊗ R ∗+1 ⊗B X. For a BGalois extension S ⊆ T, a stable anti YetterDrinfel’d Bmodule TS is constructed, such that
On higher etainvariants and metrics of positive scalar curvature
 K Theory 24:4 (2001), 341–359. MR 2002k:58051 Zbl 1010.58019
"... Abstract. Let N be a closed connected spin manifold admitting one metric of positive scalar curvature. In this paper we use the higher etainvariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, ..."
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Abstract. Let N be a closed connected spin manifold admitting one metric of positive scalar curvature. In this paper we use the higher etainvariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, involving π1(N) and dim N,forNto admit an infinite number of metrics of positive scalar curvature that are nonbordant. Mathematics Subject Classifications (2000): 55N22, 19L41. Key words: bordism groups, positive scalar curvature metrics, Galois coverings, higher etainvariants,