Cup products in Hopf cyclic cohomology with coefficients in contramodules

We use stable anti Yetter-Drinfeld contramodules to improve the cup products in Hopf cyclic cohomology. The improvement fixes the lack of functoriality of the cup products previously defined and show that the cup products are sensitive to the coefficients.

💡 Research Summary

The paper addresses a long‑standing limitation in Hopf cyclic cohomology concerning the cup‑product operation. In the classical setting, cup products were defined using stable anti‑Yetter‑Drinfeld (AYD) modules as coefficients. Although this construction yields a well‑defined product on cohomology groups, it suffers from two major defects: (1) the product is not functorial with respect to morphisms of coefficient modules, meaning that the induced maps on cohomology do not commute with the cup product; and (2) the product is insensitive to the specific choice of coefficient module, so distinct AYD modules can give rise to identical cup‑product structures, obscuring the role of coefficients in the theory.

To overcome these issues, the author introduces stable anti‑Yetter‑Drinfeld contramodules (SADCs) as a new class of coefficients. A contramodule differs from a module in that it carries a “contra‑action” – a map from the Hopf algebra into the Hom‑space of the contramodule – which satisfies compatibility conditions dual to those of a module action. The stability condition ensures that the contra‑action interacts appropriately with the antipode of the Hopf algebra, guaranteeing coherence of the resulting structures.

The paper proceeds in several stages. First, it rigorously defines SADCs, establishes their basic properties, and constructs the category SADC whose objects are stable anti‑Yetter‑Drinfeld contramodules and whose morphisms are continuous, complete maps respecting the contra‑action. This category is shown to be monoidal and to admit a natural forgetful functor to the category of vector spaces, preserving the Hopf algebra’s coaction.

Next, the author builds the Hopf cyclic cochain complex (C^\bullet(H,\Sigma^\vee)) for a Hopf algebra (H) with coefficients in a contramodule (\Sigma^\vee). The key innovation is the definition of a new cup‑product \