Cyclic structures in algebraic (co)homology theories

TL;DR

This paper extends cyclic cohomology to Hopf algebroids with new coefficients, generalizes cyclic duality, and explores dual homology theories, including applications to Lie-Rinehart homology and twisted cyclic homology.

Contribution

It introduces a generalized cyclic cohomology framework for Hopf algebroids and develops a duality theory applicable to para-cyclic objects, broadening the scope of cyclic (co)homology.

Findings

Lie-Rinehart homology as a special case

A dual homology theory via generalized cyclic duality

Examples include twisted cyclic homology with non-stable coefficients

Abstract

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-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.