Cup Coproducts in Hopf Cyclic Cohomology

We define cup coproducts for Hopf cyclic cohomology of Hopf algebras and for its dual theory. We show that for universal enveloping algebras and group algebras our coproduct recovers the standard coproducts on Lie algebra homology and group homology, respectively.

💡 Research Summary

The paper introduces a novel “cup coproduct” operation for Hopf cyclic cohomology and its dual theory, Hopf cyclic homology. The authors start by recalling the standard Hopf cyclic cohomology complex ((C^\bullet(H),b,B)) associated with a Hopf algebra (H) and point out that, while cup products have been extensively studied, a systematic coproduct structure has been missing. To fill this gap, they construct a map (\Delta_{\smile}) on the tensor product complex (C^\bullet(H)\otimes C^\bullet(H)) using the Hopf algebra’s comultiplication (\Delta) and antipode (S). This map respects the cyclic coboundary operators (b) and (B) (i.e., it commutes with them) and preserves total degree, thereby defining a genuine coproduct on Hopf cyclic cohomology groups.

The dual construction is carried out for Hopf cyclic homology. By working with the homological complex ((C_\bullet(H),d,\delta)) and defining a corresponding map (\nabla_{\smile}) on the tensor product of homological chains, the authors obtain a coproduct that is dual to (\Delta_{\smile}). Both constructions are shown to be compatible with the Hopf algebra’s antipode and coassociative diagonal, guaranteeing coassociativity of the resulting coproducts.

A central part of the paper is the verification of these abstract definitions in two fundamental examples. For the universal enveloping algebra (U(\mathfrak g)) of a Lie algebra (\mathfrak g), the Hopf cyclic cohomology is identified with the Chevalley–Eilenberg Lie algebra cohomology. The authors prove that (\Delta_{\smile}) coincides exactly with the classical coproduct on Lie algebra homology, using explicit chain maps and the Künneth theorem. In the case of a group algebra (k