Memorial University of Newfoundland

Department of Mathematics
and Statistics

Atlantic Algebra





Faculty of Science


Hopf-cyclic Homology and Cohomology with Coefficients

Piotr Hajac           Masoud Khalkhali           Bahram Rangipour           Yorck Sommerhäuser

  • 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), 667-672


Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of the known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic 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 a module algebra.


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 τn 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 ΔM(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.