This note discusses the cyclic cohomology of a left Hopf algebroid (×A-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd modules.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 1, pp.297-318.
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