Cyclic cohomology of entwining structures

In this paper, we introduce and study a cyclic cohomology theory H λ ∙ ​ (A,C,ψ) for an entwining structure (A,C,ψ) over a field k. We then provide a complete description of the cocycles and the coboundaries in this theory using entwined traces applied to dg-entwining structures over (A,C,ψ). We then apply these descriptions to construct a pairing H λ m ​ (A,C,ψ)⊗H λ n ​ (A ′ ,C ′ ,ψ ′ )→H λ m+n ​ (A⊗A ′ ,C⊗C ′ ,ψ⊗ψ ′ ), where (A,C,ψ) and (A ′ ,C ′ ,ψ ′ ) are entwining structures.