We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a character. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usual Chern-Weil theory. Keywords: Cyclic homology, Hopf algebras, Weil complex 1 Introduction In their computation of the cyclic cocycles involved in the non-commutative index formula in the context of the transverse index theorem, A. Connes and H. Moscovici discovered that the action of the operators involved can be organized in a Hopf algebra action, and that the computation takes place on the cyclic cohomology of their Hopf algebra ([6]). This led them to a definition of the cyclic cohomology HC ffi (H) of a Hopf algebra H, endowed with a character satisfying certain conditions. In their con...

It known from the work of Feigin-Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite group G acting on X the same procedure applied to G-equivariant sheaves gives the orbifold cohomology of X/G. As an application, in some cases we are able to obtain simple proofs of an additive isomorphism between the orbifold cohomology of X/G and the usual cohomology of its crepant resolution (the equality of Euler and Hodge numbers was obtained earlier by various authors). We also state some conjectures on the product structures, as well as the singular case; and a connection with a recent work by Kawamata. 1

2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9

Abstract. In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P → M along with a G-invariant fiberwise Dirac-type operator D on P. The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory group, with a closed graded trace on a certain noncommutative de Rham algebra Ω ∗ B associated to G. The proof is by means of superconnections in the framework of noncommutative geometry.

Abstract. In this paper we define and study the ghost loop orbifold LsX of an orbifold X consisting of those loops that remain constant in the coarse moduli space of X. We construct a configuration space model for LsX using an idea of G. Segal. From this we exhibit the relation between the Hochschild and cyclic homologies of the inertia orbifold of X (that generate the so-called twisted sectors in string theory) and the ordinary and equivariant homologies of LsX. We also show how this clarifies the relation between orbifold K-theory, Chen-Ruan orbifold cohomology, Hochschild homology, and periodic cyclic homology. 1.

this paper is to describe how sheaf theory extends to etale groupoids. More specically, we discuss the construction of the \derived category" of an etale groupoid, and show how the six operations of Grothendieck (namely, tensor, hom, f

Abstract. In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid

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Abstract. It is well-known that an effective orbifold M (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold P by a locally free action of a compact lie group K. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes. 1.