[Paper Review] Cyclic Cohomology of Crossed Coproduct Coalgebras

R. Akbarpour, Masoud Khalkhali|ArXiv.org|Jul 23, 2001

Algebraic structures and combinatorial models6 references5 citations

TL;DR

This paper extends cyclic cohomology theory to crossed coproduct coalgebras $C \!>\!\blacktriangleleft\!\mathcal{H}$ using a cocylindrical module $C\natural\mathcal{H}$, establishing an isomorphism between the cyclic module of the crossed coproduct and the diagonal of $C\natural\mathcal{H}$. It constructs a spectral sequence approximating $HC^\bullet(C \!>\!\blacktriangleleft\!\mathcal{H})$, with explicit interpretations of the $\mathsf{E}^0$, $\mathsf{E}^1$, and $\mathsf{E}^2$ terms, and proves that for cosemisimple $\mathcal{H}$, the cyclic cohomology of the crossed coproduct is isomorphic to that of the coinvariant module $\mathsf{C}^\bullet_{\mathcal{H}}(C)$.

ABSTRACT

We extend our work in~\cite{rm01} to the case of Hopf comodule coalgebras. We introduce the cocylindrical module $C atural^{} \mathcal{H}$, 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 $Δ(C atural^{}\mathcal{H}) $, the cocyclic module related to the diagonal of $C atural^{} \mathcal{H}$. 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.

Motivation & Objective

• To generalize cyclic cohomology methods from crossed product algebras to crossed coproduct coalgebras.
• To construct a cocylindrical module $C\natural\mathcal{H}$ for a Hopf comodule coalgebra $C$ over a Hopf algebra $\mathcal{H}$ with bijective antipode.
• To establish an isomorphism between the cyclic module of the crossed coproduct $C \!>\!\blacktriangleleft\!\mathcal{H}$ and the diagonal of $C\natural\mathcal{H}$.
• To approximate the cyclic cohomology of $C \!>\!\blacktriangleleft\!\mathcal{H}$ using a spectral sequence and interpret its initial terms.
• To provide a cohomological interpretation of the coinvariant module $\mathsf{C}^\bullet_{\mathcal{H}}(C)$ as dual to the coinvariant homology found in prior work.

Proposed method

• Construct the cocylindrical module $C\natural\mathcal{H}$ from a Hopf comodule coalgebra $C$ and a Hopf algebra $\mathcal{H}$ with bijective antipode.
• Define the diagonal $\Delta(C\natural\mathcal{H})$ as the paracocyclic object on $A(n,n) = (C\otimes\mathcal{H})^{igotimes(n+1)}$, with operators induced by horizontal and vertical structures.
• Apply the Eilenberg-Zilber theorem for cocylindrical modules to obtain a quasi-isomorphism between $\Delta(C\natural\mathcal{H})$ and the total complex $Tot^\bullet(C\natural\mathcal{H})$, inducing a spectral sequence converging to $HC^\bullet(C \!>\!\blacktriangleleft\!\mathcal{H})$.
• Filter the total complex $Tot^\bullet$ by subcomplexes $\mathsf{F}^i_{pq}$ to define the spectral sequence, with $\mathsf{E}^0$-term given by $\mathsf{C}^p(\mathcal{H}, \mathsf{C}^q(C^\natural_{\mathcal{H}}) \boxtimes \mathsf{W})$.
• Identify the $\mathsf{E}^1$-term as the cohomology of the $\mathcal{H}$-cohomology of $\mathsf{C}^q(C^\natural_{\mathcal{H}}) \boxtimes \mathsf{W}$, and the $\mathsf{E}^2$-term as the cyclic cohomology of the $\mathcal{H}$-cohomology modules with coefficients in $\mathsf{W}$.
• Use the existence of a homotopy operator for cosemisimple $\mathcal{H}$ to show that $H^p(\mathcal{H}, M) = 0$ for $p > 0$, leading to collapse of the spectral sequence and isomorphism of cyclic cohomologies.

Experimental results

Research questions

• RQ1 How can the cyclic cohomology of a crossed coproduct coalgebra $C \!>\!\blacktriangleleft\!\mathcal{H}$ be computed using a cocylindrical module structure?
• RQ2 What is the relationship between the cyclic module of $C \!>\!\blacktriangleleft\!\mathcal{H}$ and the diagonal of $C\natural\mathcal{H}$?
• RQ3 What is the interpretation of the $\mathsf{E}^0$, $\mathsf{E}^1$, and $\mathsf{E}^2$ terms in the spectral sequence approximating $HC^\bullet(C \!>\!\blacktriangleleft\!\mathcal{H})$?
• RQ4 How does the cohomology of the coinvariant module $\mathsf{C}^\bullet_{\mathcal{H}}(C)$ relate to the cyclic cohomology of the crossed coproduct?
• RQ5 Under what conditions does the spectral sequence collapse, and what are the consequences for the cyclic cohomology of $C \!>\!\blacktriangleleft\!\mathcal{H}$?

Key findings

• The cyclic cohomology of the crossed coproduct coalgebra $C \!>\!\blacktriangleleft\!\mathcal{H}$ is isomorphic to the cyclic cohomology of the coinvariant module $\mathsf{C}^\bullet_{\mathcal{H}}(C)$ when $\mathcal{H}$ is cosemisimple.
• The $\mathsf{E}^0$-term of the spectral sequence is isomorphic to the complex $\mathsf{C}^p(\mathcal{H}, \mathsf{C}^q(C^\natural_{\mathcal{H}}) \boxtimes \mathsf{W})$, with differential $\boldsymbol{\delta}$.
• The $\mathsf{E}^1$-term is given by $H^p(\mathcal{H}, \mathsf{C}^q(C^\natural_{\mathcal{H}}) \boxtimes \mathsf{W})$ with the differential $\mathfrak{b} + \mathbf{u}\mathfrak{B}$.
• The $\mathsf{E}^2$-term is $HC^q(H^p(\mathcal{H}, \mathsf{C}^q(C^\natural_{\mathcal{H}})); \mathsf{W})$, the cyclic cohomology of the $\mathcal{H}$-cohomology modules with coefficients in $\mathsf{W}$.
• For a cosemisimple Hopf algebra $\mathcal{H}$, the spectral sequence collapses at $\mathsf{E}^2$, and $HC^\bullet(C \!>\!\blacktriangleleft\!\mathcal{H}) \simeq HC^\bullet(\mathsf{C}^\bullet_{\mathcal{H}}(C))$.
• The Hochschild cohomology of $C \!>\!\blacktriangleleft\!\mathcal{H}$ is also isomorphic to that of $\mathsf{C}^\bullet_{\mathcal{H}}(C)$ under the same cosemisimple condition.