• Preprint: Mathematisches Institut der polnischen Akademie der Wissenschaften: IM PAN 643
• Preprint: XXX-Preprint-Archiv: math.KT/0306288
• Zeitschrift: C. R. Acad. Sci., Paris, Sér I, Math. 338 (2004), 667-672

Von der Idee eines invarianten Differentialkomplexes geleitet konstruieren wir allgemeine zyklische Moduln, die einen gemeinsamen Nenner der bekannten zyklischen Theorien darstellen. Die Zyklizität dieser Moduln wird durch hopfalgebraische Strukturen beherrscht. Wir zeigen, daß die Existenz eines zyklischen Operators eine Modifikation der Yetter-Drinfel'd-Verträglichkeitsbedingung erzwingt, was zum Konzept eines stabilen Anti-Yetter-Drinfel'd-Moduls führt. Dieser Modul spielt in der so erhaltenen zyklischen Kohomologie von Modulalgebren und Koalgebren, und auch in der zyklischen Homologie und Kohomologie von Komodulalgebren, die Rolle des Koeffizientenmoduls. Wie bei Connes and Moscovici zeigen wir, daß es zwischen der zyklischen Kohomologie einer Modulkoalgebra, die auf eine Modulalgebra wirkt, und den geschlossenen 0-Kozyklen dieser Modulalgebra eine Paarung gibt. Diese Paarung nimmt ihre Werte in der gewöhnlichen zyklischen Kohomologie dieser Algebra an. Weiter argumentieren wir, daß es eine analoge Paarung zwischen den geschlossenen 0-Kozyklen einer Modulkoalgebra und der zyklischen Kohomologie einer Modulalgebra gibt.

Ever since its invention, among the main applications of cyclic cohomology was the computation of K-theoretical 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 Connes-Moscovici 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 higher-dimensional coefficients.

We continue this strategy herein. Our motivation is to obtain and understand computable invariants of K-theory. The aim of this paper is to provide a general framework for cyclic theories whose cyclicity is based on Hopf-algebraic structures. We refer to such homology and cohomology as Hopf-cyclic. The definition and sources of examples of stable anti-Yetter-Drinfeld modules that play the role of coefficients for Hopf-cyclic theory are provided in the preceding article [6]. (Note that modular pairs in involution are precisely 1-dimensional stable anti-Yetter-Drinfeld 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 Connes-Moscovici key result [5, Theorem 1]. Furthermore, as δ-invariant σ-traces can be viewed as closed 0-cocycles on a module algebra, our pairing for Hopf-cyclic cohomology generalizes the Connes-Moscovici transfer map [5, Proposition 1] from the cyclic cohomology of Hopf algebras to ordinary cyclic cohomology. Finally, we end this article by deriving Hopf-cyclic 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 _MΔ(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.

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• Zuletzt geändert: 18.11.2005