Bialgebra Cyclic Homology with Coefficients, Part II

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This is the second part of the article [math.KT/0408094]. In the first paper, we used the underlying coalgebra structure to develop a cyclic theory. In this paper we define a dual theory by using the algebra structure. We define a cyclic homology theory for triples (X,B,Y) where B is a bialgebra, X is a B--comodule algebra and Y is just a stable B--module/comodule. We recover the main result of [math.KT/0310088] that these homology theories are dual to each other in the appropriate sense when the bialgebra is a Hopf algebra and the stable coefficient module satisfies anti-Yetter-Drinfeld condition. We also compute this particular homology for the quantum deformation of an arbitrary semi-simple Lie algebra and the Hopf algebra of foliations of codimension N with stable but non-anti-Yetter-Drinfeld coefficients.