On the cyclic Homology of multiplier Hopf algebras (original) (raw)

A New Cyclic Module for Hopf Algebras

K-Theory, 2000

We define a new cyclic module, dual to the Connes-Moscovici cocyclic module, for Hopf algebras, and give a characteristic map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

Cyclic Homology of Hopf Algebras

K-theory, 2001

A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday-Quillen ([LQ]) and Karoubi's work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra ([B], [KV], [L]) in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras.

On the Cyclic Cohomology of Extended Hopf Algebras

2001

We introduce the concept of {\it extended Hopf algebras} and consider their cyclic cohomology in the spirit of Connes-Moscovici. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscovici algebra mathcalHFM\mathcal{H}_{FM}mathcalHFM are extended Hopf algebras. We show that the cyclic cohomology of the extended Hopf algebra U(L,R)U(L,R)U(L,R) naturally associated to a Lie-Rinehart algebra (L,R)(L,R)(L,R) coincides with the homology of (L,R)(L,R)(L,R). We also give some other examples of extended Hopf algebras and their cyclic cohomology.

Cyclic cohomology and Hopf algebras

Letters in Mathematical Physics, 1999

We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the ...

A Note on Cyclic Duality and Hopf Algebras

Communications in Algebra, 2005

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes' duality isomorphism for the cyclic category. * PIMS postdoctoral fellow

Bialgebra Cyclic Homology with Coefficients, Part I

2004

The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will show that one can further extend the cyclic homology of Hopf algebras with coefficients non-trivially. The new homology, called the bialgebra cyclic homology, admits stable coefficient module/comodules, dropping the anti-Yetter-Drinfeld condition. This fact allows the new homology to use bialgebras, not just Hopf algebras. We will also give computations for bialgebra cyclic homology of the Hopf algebra of foliations of codimension n and the quantum deformation of an arbitrary semi-simple Lie algebra with several stable but non-anti-Yetter-Drinfeld coefficients.

Cyclic homology of Hopf Galois extensions and Hopf algebras

Arxiv preprint math/0307099, 2003

Abstract: Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z (H, M). The cyclic homology of Z (H, M) extends ...

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

Proceedings of the London Mathematical Society, 2006

Let H be a Hopf algebra and let CM m (H) be the category of all left H-modules and right H-comodules satisfying the following two compatibility relations: ρ(hm) = h (2) m 0 ⊗ h (3) m 1 Sh (1) , for all m ∈ M and h ∈ H. m 1 m 0 = m, for all m ∈ M. An object in CM m (H) will be called a modular crossed module (over H). For example, if A/B is an H-Galois extension, then the quotient A B := A/[A, B] of A modulo commutators [A, B] is a modular crossed module over H. More particularly, H itself can be regarded as an object ad H ∈ CM m (H). The category CM m (H) has very nice homological properties: it is abelian and contains enough injective objects. Furthermore, if K is a Hopf subalgebra of H, then the categories CM m (K) and CM m (H) can be related by a functor Ind H K (−), where Ind H K M := H ⊗ K M , with appropriate structures. To every M ∈ CM m (H) we associate a cyclic object Z * (H, M). The cyclic homology of Z * (H, M) extends the usual cyclic homology of the algebra structure of H (for M := ad H). The relative cyclic homology of an H-Galois extension A/B can be also regarded as a particular case (M := A B). We compute the cyclic homology of Ind H K M := H ⊗ K M when K is cocommutative, and M decomposes as a direct sum of one dimensional subcomodules such that the associated group-like elements are central. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the (usual) cyclic homology of group algebras and quantum tori. Finally, when H := U (g) is the enveloping algebra of a Lie algebra g, we construct a spectral sequence that converges to the cyclic homology of H with coefficients in an arbitrary modular crossed module M. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of H with coefficients in a certain modular crossed-module.

K T ] 1 J ul 2 00 3 Hopf-cyclic homology and cohomology with coefficients

2003

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopfalgebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of...