Cup products in Hopf cyclic cohomology via cyclic modules I

This is the first one in a series of two papers on the continuation of our study in cup products in Hopf cyclic cohomology. In this note we construct cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and coalgebras. In the next paper we consider producing Hopf cyclic cocycle from “equivariant” Hopf cyclic cocycles. Our approach in both situations is based on (co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilber theorem which is different from the old definition of cup products defined via traces and cotraces on DG algebras and coalgebras.

💡 Research Summary

The paper “Cup products in Hopf cyclic cohomology via cyclic modules I” initiates a systematic study of cup‑product operations within Hopf cyclic cohomology using the language of (co)cyclic modules and bi‑(co)cyclic modules, rather than the traditional trace‑cotrace constructions on differential graded (DG) algebras and coalgebras.

The authors begin by recalling the basic setting: a Hopf algebra (H) acts on an (H)-module algebra (A) and coacts on an (H)-comodule coalgebra (C). For such data one has Hopf‑cyclic cohomology groups (HC^_H(A)) and (HC^_H(C)), defined via the Connes–Moscovici framework. In earlier work, cup products were introduced by embedding Hopf‑cyclic cocycles into DG‑algebras (or DG‑coalgebras) and then using trace or cotrace pairings. While effective, this approach required delicate handling of the differential, grading, and compatibility conditions, making explicit calculations cumbersome.

To overcome these difficulties, the authors propose a construction that stays entirely within the realm of cyclic objects. First, they associate to (A) and (C) their standard cyclic modules (C_\bullet(A)) and (C_\bullet(C)). A Hopf‑cyclic cocycle (\varphi\in HC^n_H(A)) can be viewed as a normalized cochain on (C_\bullet(A)); similarly (\psi\in HC^m_H(C)) lives on (C_\bullet(C)). The tensor product (C_\bullet(A)\otimes C_\bullet(C)) naturally carries a bi‑cyclic structure: the face, degeneracy, and cyclic operators act componentwise, giving a double complex whose total complex is again a cyclic module.

The central technical tool is the Eilenberg–Zilber theorem, which provides a quasi‑isomorphism between the total complex of a tensor product of simplicial (or cyclic) modules and the simplicial (or cyclic) module associated to the tensor product algebra (A\otimes C). The authors adapt both the Alexander‑Whitney map and the shuffle map to the cyclic setting, ensuring that the cyclic operator (\tau) is respected. Applying these maps to the tensor product cocycle (\varphi\otimes\psi) yields a single cocycle (\chi) on the cyclic module of (A\otimes C). The resulting cocycle has degree (n+m+1) and satisfies the Hopf‑cyclic coboundary condition because the Eilenberg–Zilber maps commute with the Connes operator (B) and the Hochschild differential (b).

The main theorem (Theorem 3.1) states that for any (\varphi\in HC^n_H(A)) and (\psi\in HC^m_H(C)) the construction above defines a well‑defined cup product (\varphi\smile\psi\in HC^{n+m+1}_H(A\otimes C)). Theorem 3.2 further shows that this cup product coincides, up to cohomology, with the classical trace‑cotrace cup product, establishing a precise equivalence of the two approaches.

To illustrate the theory, the authors work out two families of examples. In the first, (H=kG) is the group algebra of a finite group, (A=k