An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ? ? for an ?-comodule algebra A, where the antipode of the Hopf algebra ? is bijective. We show that the cyclic module associated to the diagonal of A ? ? is isomorphic with the cyclic module of the crossed product algebra A ? ?. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Hochschild (Co)Homology of Hopf Crossed Products*

For a general crossed product E = A#_f H, of an algebra A by a Hopf algebra H, we obtain complexes smaller than the canonical ones, giving the Hochschild homology and cohomology of E. These complexes are equipped with natural filtrations. The spectral sequences associated to them coincide with the ones obtained using a natural generalization of the direct method introduced in Trans. Amer. Math. Soc. 74 (1953) 110–134. We also get that if the 2-cocycle f takes its values in a separable subalgebra of A, then the Hochschild (co)homology of E with coefficients in M is the (co)homology of H with coefficients in a (co)chain complex.     

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field or let M be the space k ⁿ and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H _·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C _·(A,A),b) and (^·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.     

The Connes–Moscovici Approach to Cyclic Cohomology for Compact Quantum Groups

Consider the Hopf algebra (A, ) of regular functions on a compact quantum group. Let (A ^o ,) denote its maximal dual Hopf algebra. We show that the tensor product Hopf algebra (H ₂,₂) of (A ^o ,) and its opposite Hopf algebra is endowed with a modular pair (,) in involution; a notion introduced by A. Connes and J. Moscovici, who associate canonically a cocyclic object to such Hopf algebras. Denote the Hopf cyclic cohomology thus obtained by HC ^* _(,)(H ₂). Next we define an action of H ₂),₂ on A and show that the Haar state of (A, ) is a -invariant -trace on A with respect to this action. This gives us a canonical map from HC ^* _(,)(H ₂) to the ordinary cyclic cohomology of A.     

Bialgebra Cyclic Homology with Coefficients

We show that one can extend the definition of Hopf cyclic homology with coefficients such that one can use bialgebras and a larger class of coefficient module/co-modules. With the help of this extension, we calculate the bialgebra cyclic homology of U_q() the quantum deformation of an arbitrary semi-simple Lie algebra and (N) the Hopf algebra of foliations of codimension N, with several coefficient modules.     

Hopf-Cyclic Homology and Relative Cyclic Homology of Hopf-Galois Extensions

Let H be a Hopf algebra and let M_s (H) be the category of allleft H-modules and right H-comodules satisfying appropriatecompatibility relations. An object in M_s (H) will be calleda stable anti-Yetter–Drinfeld module (over H) or a SAYDmodule, for short. To each M M_s (H) we associate, in a functorialway, a cyclic object Z_* (H, M). We show that our constructioncan be used to compute the cyclic homology of the underlyingalgebra structure of H and the relative cyclic homology of H-Galoisextensions. Let K be a Hopf subalgebra of H. For an arbitrary M M_s (K)we define a right H-comodule structure on so that becomes an object in M_s (H). Under some assumptions on K and M we computethe cyclic homology of . As a direct application of this result, we describe the relativecyclic homology of strongly graded algebras. In particular,we calculate the cyclic homology of group algebras and quantumtori. Finally, when H is the enveloping algebra of a Lie algebra g,we construct a spectral sequence that converges to the cyclichomology of H with coefficients in a given SAYD module M. Wealso show that the cyclic homology of almost symmetric algebrasis isomorphic to the cyclic homology of H with coefficientsin a certain SAYD module. 2000 Mathematics Subject Classification16E40 (primary), 16W30 (secondary).     

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.     

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

Dolbeault cohomology of compact nilmanifolds

LetM=G/ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification of the Lie algebra ofG. to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.Research partially supported by MURST and CNR of Italy.Research partially supported by MURST and CNR of Italy.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

On the negative cyclic homology of <Emphasis Type="Italic">shc</Emphasis>-algebras

Let be a field of characteristic and S ¹ the unit circle. We prove that the shc-structure on a cochain algebra (A,d _A ) induces an associative product on the negative cyclic homology HC _*⁻ A. When the cochain algebra (A,d _A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in , then HC _*⁻ A is isomorphic as a graded algebra to the S ¹-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S ¹-equivariant cohomology algebras of the free loop space of the complex projective space when n + 1 = 0 [p] and of the even spheres S ²ⁿ when p = 2.      

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

Quasi-Piecewise-Koszul Modules

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids.     

Coxeter cochain complexes

We define the Coxeter cochain complex of a Coxeter group (G, S) with coefficients in a ?[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.     

Some algebraic properties of cyclic homology groups

In [10], see also [8], a cyclic homology theory HC _* ^– was introduced. The purpose of this paper is to study algebraically the properties of this version of cyclic homology. First we study its relation to Connes cyclic cohomology theory HC ^* and to the usual cyclic homology theory HC _* studied by Loday and Quillen in [15]. We explain the precise sense in which HC _* ^– is dual to HC ^*. Next we study products and describe a general method for constructing product operations in cyclic homology and cohomology theories. Finally we examine the relation between the theory HC _* ^– and algebraic K-theory.