Cyclic cohomology of entwining structures

Abstract

In this paper, we introduce and study a cyclic cohomology theory $H_{\lambda }^{\bullet }(A,C,\psi )$ for an entwining structure $(A,C,\psi )$ 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 $(A,C,\psi )$. We then apply these descriptions to construct a pairing $H_{\lambda }^m(A,C,\psi )\otimes H_{\lambda }^n(A^{\prime },C^{\prime },{\psi }^{\prime })\to H_{\lambda }^{m+n}(A\otimes A^{\prime },C\otimes C^{\prime },\psi \otimes {\psi }^{\prime })$, where $(A,C,\psi )$ and $(A^{\prime },C^{\prime },{\psi }^{\prime })$ are entwining structures.