Bialgebra Cyclic Homology with Coefficients Part I

Cyclic cohomology of Hopf algebras admitting a modular pair was first defined in [2] and further developed in [3] and [4] in the context of transverse geometry. Their results are followed by several papers computing Hopf cyclic (co)homology of certain Hopf algebras such as [15], [5] and [13]. In a series of papers [1], [10], [11], and [6] the authors developed a theory of cyclic (co)homology which works with Hopf module coalgebras or Hopf comodule algebras with coefficients in stable Hopf module/comodules satisfying anti-Yetter-Drinfeld condition (SaYD.) In this paper, We show that one can extend the Hopf cyclic homology non-trivially by using just stable module/comodules, dropping the aYD condition. This also allows us to extend the definition of the cyclic homology to bialgebras.