
 Preprint: Institute of Mathematics of the Polish Academy of Sciences:
IM PAN 643
 Preprint: XXX preprint archive:
math.KT/0306288
 Journal: C. R. Acad. Sci., Paris, Sér I, Math. 338 (2004), 667672
Abstract
Following the idea of an invariant differential complex, we construct generaltype cyclic modules that provide the common denominator of the 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 YetterDrinfeld compatibility condition leading to the concept of a stable antiYetterDrinfeld 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 0cocycles 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 0cocycles of a module coalgebra and the cyclic cohomology of a module algebra.



Introduction
Ever since its invention, among the main applications of cyclic cohomology was the computation of Ktheoretical invariants. While enhancing the feasibility of such computations, Connes and Moscovici discovered a new type of cyclic cohomology, notably the cyclic cohomology of Hopf algebras [4]. Invariant cyclic homology, introduced in [7], generalizes the ConnesMoscovici theory and its dual version [8]. It shows that passage from the cyclic homology of algebras to the cyclic cohomology of Hopf algebras is remarkably similar to passage from de Rham cohomology to the cohomology of Lie algebras via invariant de Rham cohomology [2]. The idea of employing invariant complexes proved to be a key in resolving by significantly more effective means the technical challenge of showing that the (n+1)power of the cyclic operator τ_{n} is the identity [5, p. 102], and allowed the introduction of higherdimensional coefficients.
We continue this strategy herein. Our motivation is to obtain and understand computable invariants of Ktheory. The aim of this paper is to provide a general framework for cyclic theories whose cyclicity is based on Hopfalgebraic structures. We refer to such homology and cohomology as Hopfcyclic. The definition and sources of examples of stable antiYetterDrinfeld modules that play the role of coefficients for Hopfcyclic theory are provided in the preceding article [6]. (Note that modular pairs in involution are precisely 1dimensional stable antiYetterDrinfeld modules.) Here we construct cyclic module structures on invariant complexes for module coalgebras and module algebras, respectively. It turns out that the cyclic cohomology of Hopf algebras is a special case of the former, whereas both twisted [9] and usual cyclic cohomology are special cases of the latter. As a result of this generality, we obtain now a very short proof of ConnesMoscovici key result [5, Theorem 1]. Furthermore, as δinvariant σtraces can be viewed as closed 0cocycles on a module algebra, our pairing for Hopfcyclic cohomology generalizes the ConnesMoscovici transfer map [5, Proposition 1] from the cyclic cohomology of Hopf algebras to ordinary cyclic cohomology. Finally, we end this article by deriving Hopfcyclic homology and cohomology of comodule algebras. This extends the formalism for comodule algebras provided in [7].
The coproduct, counit and antipode of H are denoted by Δ, ε and S, respectively. For the coproduct we use the notation
Δ(h) =h^{(1)} ⊗ h^{(2)},
for a left coaction on M we write
_{M}Δ(m) = m^{(1)} ⊗ m^{(0)},
and for a right coaction
Δ_{M}(m) = m^{(0)} ⊗ m^{(1)}.
The summation symbol is suppressed everywhere. We assume all algebras to be associative, unital and over the same ground field k. Partly for the sake of simplicity, we also assume that the antipodes of all Hopf algebras under consideration are bijective.

