Cyclic structures in algebraic (co)homology theories

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.

💡 Research Summary

The paper presents a systematic extension of Connes‑Moscovici’s cyclic cohomology from Hopf algebras to the broader setting of left Hopf algebroids (also called ×ₐ‑Hopf algebras). The authors begin by recalling the original framework, where a Hopf algebra H acts on a stable anti‑Yetter‑Drinfel’d (SAYD) module M, and the cyclic operators b, B, and τ are defined on the complex Cₙ = M ⊗ H^{⊗ n}. They observe that the SAYD condition is rather restrictive and that many natural coefficient objects, especially those arising from non‑commutative geometry and Lie‑Rinehart theory, do not satisfy it.

To overcome this limitation, the paper introduces a more flexible coefficient: a right H‑module that is simultaneously a left H‑comodule, with compatibility conditions that are weaker than the SAYD axioms. This structure is natural for Hopf algebroids because the base algebra A introduces two distinct A‑module structures (source and target), and the comodule side must respect the left A‑action while the module side respects the right A‑action. The authors verify that with these coefficients the usual cyclic operators can still be defined: the face maps d_i, degeneracy maps s_i, and the cyclic operator τ_n. Although τ_n does not necessarily satisfy τ_n^{n+1}=Id, the authors show that τ_n^{n+1} is an A‑linear automorphism, which is sufficient to obtain a para‑cyclic object.

A major conceptual contribution is the generalisation of cyclic duality to arbitrary para‑cyclic objects. Classical cyclic duality exchanges a cyclic object with its cocyclic dual, interchanging the roles of b and B and yielding a correspondence between homology and cohomology. In the para‑cyclic context the lack of strict cyclicity obstructs this exchange. The authors construct a duality functor D that sends a para‑cyclic object C to its A‑linear dual Hom_A(C, A). They prove that D(C) inherits a natural para‑cyclic structure, and that the resulting dual homology theory is genuinely new, not merely a re‑interpretation of the original cohomology.

The paper then illustrates the theory with two important examples. First, Lie‑Rinehart algebras (A, L) are considered. Their universal enveloping algebra U(L) carries a Hopf algebroid structure, and the exterior algebra Λ_A^· L, viewed as a right‑module/left‑comodule over U(L), fits precisely into the new coefficient framework. Consequently, Lie‑Rinehart homology appears as a special case of the generalized cyclic homology, confirming that the authors’ construction recovers known results while providing a unified perspective.

Second, the authors treat twisted cyclic homology of an associative algebra A. Given an algebra automorphism φ, one can form the φ‑twisted bimodule A_φ, which is generally not a SAYD module. Nevertheless, A_φ satisfies the right‑module/left‑comodule compatibility required in the new setting. The resulting para‑cyclic complex reproduces the twisted cyclic homology introduced by Kustermans, Murphy, and others, but now without invoking SAYD conditions. This demonstrates that the generalized framework can accommodate coefficients that were previously inaccessible.

In the concluding section, the authors emphasize that the combination of Hopf algebroid theory, para‑cyclic structures, and the broadened notion of coefficients dramatically expands the applicability of cyclic (co)homology. The generalized cyclic duality opens the door to new homological invariants for non‑commutative spaces, quantum groups, and higher‑algebraic structures. Potential future directions include exploring connections with Hopf‑cyclic cohomology of groupoids, developing index theorems in the algebroid context, and investigating categorical extensions where para‑cyclic objects arise naturally in higher‑dimensional algebra.