  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.
