Cyclic Cohomology of Crossed Coproduct Coalgebras

Abstract

We extend our work in~\cite{rm01} to the case of Hopf comodule coalgebras. We introduce the cocylindrical module $C{♮}^{}\mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra with bijective antipode and $C$ is a Hopf comodule coalgebra over $\mathcal{H}$. We show that there exists an isomorphism between the cocyclic module of the crossed coproduct coalgebra $C>◀\mathcal{H}$ and $\Delta (C{♮}^{}\mathcal{H})$, the cocyclic module related to the diagonal of $C{♮}^{}\mathcal{H}$. We approximate $H{C}^{\bullet }(C>◀\mathcal{H})$ by a spectral sequence and we give an interpretation for ${\mathsf{E}}^0,{\mathsf{E}}^1$ and ${\mathsf{E}}^2$ terms of this spectral sequence.

18 pages, The paper is totaly revised. The main result is extended to all Hopf algebras with bijective antipode