Cyclic cohomology of entwining structures

Abstract

In this paper, we introduce and study a cyclic cohomology theory $H^\bullet_λ(A,C,ψ)$ 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^m_λ(A,C,ψ) \otimes H^n_λ(A',C',ψ') \longrightarrow H^{m+n}_λ(A \otimes A', C \otimes C', ψ\otimes ψ') $, where $(A,C,ψ)$ and $(A',C',ψ')$ are entwining structures.