Equivariant Cyclic Cohomology of H-Algebras

Abstract

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.

Constant and Equivariant Cyclic Cohomology

“…One can see that H is a unital subalgebra of A ⋊ H, and hence H-constant cyclic cohomology of A ⋊ H is well-defined. Our main goal in this paper is to compute this cohomology in terms of equivariant cyclic cohomology defined in [1,11] which we recall it in the next section.…”

“…Equivariant cyclic cohomology of an algebra under the action of a discrete group is studied in [3,2,8,9,12,13]. Its generalization for the action of a Hopf algebra on an algebra is dealt with in [1,11,10]. In the following we recall this cohomology theory.…”

“…A natural entry into this page of the 'commutativenoncommutative' dictionary is the counterpart of equivariant de Rahm homology, which evidently is the equivariant cyclic cohomology. Equivariant cyclic cohomology has been studied by various authors, in particular we refer the reader to [3,2,8,9,12,13], for the case of groups, and [1,11,10], for Hopf algebras. In homological algebra, it is often important to replace a complex with a quasisomorphic subcomplex.…”

“…It is shown that the equivariant cyclic cohomology and cyclic cohomology of crossed product algebra coincide if H is semisimple [1]. where the left hand side is the cyclic cohomology of subcomplex of Bconstant equivariant co-chains; i.e, ϕ ∈ Hom H (H ⊗ A ⊗(n+1) , C), such that ϕ(h, a 0 , .…”

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

“…In the general case the pertinent formulas are less obvious. Several approaches have been proposed in [AK1,AK2]. In the previous version of this paper we introduced equivariant cyclic cohomology in the case when the Hopf algebra admits a modular element, by which we mean a group-like element implementing the square of the antipode.…”