Results 1  10
of
24
A WALK IN THE NONCOMMUTATIVE GARDEN
"... 2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9
An algebraic index theorem for orbifolds
, 2005
"... Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann–Roch theorem for such densities. In the case of a reduced or ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann–Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds
Homology of formal deformations of proper étale Lie groupoids
, 2005
"... 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 multivector fields on the associat ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
(Show Context)
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 multivector 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
Cyclic cohomology of Hopf algebras, and a noncommutative ChernWeil theory
"... We give a construction of ConnesMoscovici's cyclic cohomology for any Hopf algebra equipped with a character. Furthermore, we introduce a noncommutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when look ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
We give a construction of ConnesMoscovici's cyclic cohomology for any Hopf algebra equipped with a character. Furthermore, we introduce a noncommutative 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 noncommutative version of the usual ChernWeil theory. Keywords: Cyclic homology, Hopf algebras, Weil complex 1 Introduction In their computation of the cyclic cocycles involved in the noncommutative 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...
Inertia orbifolds, configuration spaces and the ghost loop space
 Jour. of Math
"... 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 Hochschil ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(Show Context)
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 socalled twisted sectors in string theory) and the ordinary and equivariant homologies of LsX. We also show how this clarifies the relation between orbifold Ktheory, ChenRuan orbifold cohomology, Hochschild homology, and periodic cyclic homology. 1.
Local Index Theory over Foliation Groupoids
, 2005
"... We give a local proof of an index theorem for a Diractype 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 Gequivariant fiber bundle P → M along with a Ginvariant fiberwise Diractype operator D on P. The ind ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We give a local proof of an index theorem for a Diractype 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 Gequivariant fiber bundle P → M along with a Ginvariant fiberwise Diractype operator D on P. The index theorem is a formula for the pairing of the index of D, as an element of a certain Ktheory 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.
(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) ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
... 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
Cyclic cocycles on deformation quantizations and higher index theorems for orbifolds
"... ABSTRACT. We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasiisomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasiisomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We prove an algebraic higher index theorem by computing the pairing between such cyclic cocycles and the Ktheory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes–Moscovici and its extension to orbifolds.