process

Equivariant cyclic homology for quantum groups 177between the coinvariant spaces.We may view a linear map H .A/ ! C as a linear map .A/ ! F.H/ whereF.H/ denotes the linear dual space of H . Under this identification an element inHom H . H .A/ ; C/ corresponds to a linear map f W .A/ ! F.H/ satisfying theequivariance conditionf.t !/.x/ D f .!/.S.t .2/ /xt .1/ /for all t 2 H and ! 2 .A/. In a completely analogous fashion to the case of homologydiscussed above, a direct inspection using the canonical isomorphismsHom H . H .A/ ; C/ Š Hom. H .A/ H ; C/ Š Hom. H .A/ H ; C/shows that the cyclic type cohomologies of the cocyclic module CH .A/ agree forevery unital H -algebra A with the ones associated to the mixed complex H .A/ H .Inparticular, the definition in [2] is indeed obtained by dualizing the construction givenin [1].The main difference between the cocyclic module used byAkbarbour and Khalkhaliand the definition in [22] is that Neshveyev and Tuset work with right actions instead ofleft actions. It is explained in [22] that the two approaches lead to isomorphic cocyclicmodules and hence to isomorphic cyclic type cohomologies.Now assume that H is a semisimple Hopf algebra. Then the coinvariant space H .A/ H is naturally isomorphic to the space H .A/ H of invariants. If A is a unital H -algebra then Theorem 6.9 and Proposition 8.6, together with the above considerations,yield a natural isomorphismHP .C H .A// Š HP H .C;A/which identifies the periodic cyclic homology of the cyclic module C H .A/ with theequivariant cyclic homology of A in the sense of Definition 7.1. Similarly, for asemisimple Hopf algebra H one obtains a natural isomorphismHP .C H .A// Š HP H .A; C/for every unital H -algebra A.Both of these isomorphisms fail to hold more generally, even in the classical settingof group actions. Let be a discrete group and consider the group ring H D C. Forthe H -algebra C with the trivial action one easily obtainsHP .C H .C// Š Hom H .H ad ; C/located in degree zero where H ad is the space H D C viewed as an H -module withthe adjoint action. On the other hand, a result in [25] shows that there is a naturalisomorphismHP H .C; C/ Š HP .H /which identifies the H -equivariant theory of the complex numbers with the periodiccyclic cohomology of H D C. This is the result one should expect from equivariant