Cyclic Cohomology of Crossed Coproduct Coalgebras

TL;DR

This paper extends cyclic cohomology theory to crossed coproduct coalgebras involving Hopf comodule coalgebras, introducing new cocylindrical modules and spectral sequence tools for computation.

Contribution

It introduces the cocylindrical module $C atural^{} ext{H}$ for Hopf comodule coalgebras and establishes an isomorphism with the cocyclic module of the crossed coproduct coalgebra.

Findings

Established an isomorphism between cocyclic modules

Developed a spectral sequence for cyclic cohomology approximation

Provided interpretations for spectral sequence terms

Abstract

We extend our work in~\cite{rm01} to the case of Hopf comodule coalgebras. We introduce the cocylindrical module $C \natural^{} \mathcal{H}$, 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 > \blacktriangleleft \mathcal{H} $ and $\Delta(C \natural^{}\mathcal{H}) $, the cocyclic module related to the diagonal of $C \natural^{} \mathcal{H}$. We approximate $HC^{\bullet}(C > \blacktriangleleft \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.