Hopf-cyclic Homology with Contramodule Coefficients

A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions. 1. Introduction. It has been demonstrated in [8], [9] that the Hopf-cyclic homology developed by Connes and Moscovici [5] admits a class of non-trivial coefficients. These coefficients, termed anti-Yetter-Drinfeld modules are modules and comodules of a Hopf algebra satisfying a compatibility condition reminiscent of that for cross modules. The aim of this note is to show that the Hopf-cyclic (co)homology of module coalgebras and module algebras also admits coeffcients that are modules and contramodules of a Hopf algebra with a compatibility condition. All (associative and unital) algebras, (coassociative and counital) coalgebras in this note are over a field k. The coproduct in a coalgebra C is denoted by ∆ C , and counit by ε C. A Hopf algebra H is assumed to have a bijective antipode S. We use the standard Sweedler notation for coproduct ∆ C (c) = c (1) ⊗c (2) , ∆