[Paper Review] Bivariant Hopf cyclic cohomology

Atabey Kaygun, Masoud Khalkhali|ArXiv.org|Jun 14, 2006

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper introduces bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology for $H$-module algebras and $H$-module coalgebras over a bialgebra $H$, extending Connes' cyclic category to an $H$-equivariant setting. It establishes that these theories generalize Hopf cyclic cohomology and its dual, prove Morita invariance, and exhibit a periodicity $S$-operator via external products, unifying cyclic (co)homology and $H$-module (co)homology in a single framework.

ABSTRACT

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category $Λ$. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories.

Motivation & Objective

• To develop a bivariant extension of Hopf cyclic cohomology for $H$-module algebras and coalgebras, generalizing Connes-Moscovici's theory.
• To unify cyclic (co)homology and $H$-module (co)homology into a single framework via an $H$-equivariant extension of the cyclic category $\Lambda$.
• To establish Morita invariance for both bivariant theories, showing invariance under $H$-equivariant Morita equivalence of module coalgebras.
• To define a periodicity $S$-operator and external product structures, showing that $HC^*_H(k,k;k,k)$ acts on all bivariant cohomology groups.

Proposed method

• Define cyclic $H$-modules and their variants using a large algebra $\mathcal{P}(H)$ constructed from the bialgebra $H$, generalizing Connes' cyclic category $\Lambda$ to an $H$-equivariant setting.
• Construct bivariant cohomology groups as $\mathrm{Ext}$ groups in the category of $H$-equivariant (co)cyclic modules, using the $H$-equivariant structure of the cyclic category.
• For module coalgebras, define the dual $H$-module algebra $C^*$ via the $k$-linear dual of $C$, and show that $C^*$ is isomorphic to $\mathrm{Hom}_C(C,C)^{\mathrm{op}}$ as $H$-module algebras.
• Use spectral sequences to relate bivariant equivariant cohomology to ordinary Hopf cyclic cohomology, with Hopf cyclic cohomology appearing in the $E_2$-term.
• Establish a graded action of $HC^*_H(k,k;k,k)$ and $HC^*_{\rm Hopf}(k,k;k,k)$ on all bivariant cohomology modules via external products.
• Prove Morita invariance by showing that $H$-equivariant equivalence of module coalgebra categories $\mathbf{proj}_H\text{--}C$ and $\mathbf{proj}_H\text{--}C'$ implies isomorphism of their Hopf and equivariant cyclic (co)homologies.

Experimental results

Research questions

• RQ1 How can Hopf cyclic cohomology be generalized to a bivariant theory for $H$-module algebras and coalgebras using an $H$-equivariant extension of the cyclic category $\Lambda$?
• RQ2 What is the relationship between the new bivariant theories and the original Hopf cyclic cohomology of Connes and Moscovici, and its dual?
• RQ3 Can the $S$-operator and periodicity in cyclic cohomology be extended to the bivariant $H$-equivariant setting, and how do they act on the cohomology groups?
• RQ4 To what extent do these bivariant theories satisfy Morita invariance, and how does this relate to the $H$-equivariant structure of module coalgebras?
• RQ5 How do the bivariant theories unify cyclic (co)homology and $H$-module (co)homology in a single framework?

Key findings

• The bivariant Hopf cyclic cohomology $HC^*_{\rm Hopf}(C,C';M,M')$ for $H$-module coalgebras specializes to the standard Hopf cyclic cohomology when $C = k$ and $M = k$, and to the dual theory when $C' = k$ and $M' = k$, confirming consistency with known constructions.
• The bivariant equivariant cohomology $HC^*_H(C,C';M,M')$ fits into spectral sequences converging to it, with the $E_2$-term being the ordinary Hopf cyclic cohomology, showing it as a natural extension.
• The algebra $HC^*_{\rm Hopf}(k,k;k,k)$ acts on all bivariant Hopf cyclic cohomology modules via an external product, and this action is generated by the $S$-operator, generalizing Connes' periodicity.
• The algebra $HC^*_H(k,k;k,k)$ is isomorphic to a graded $k$-algebra extension of $\mathrm{Ext}^*_H(k,k)$ and the $S$-operator algebra, via a spectral sequence of graded $k$-algebras.
• Morita invariance holds for module coalgebras: if $\mathbf{proj}_H\text{--}C$ and $\mathbf{proj}_H\text{--}C'$ are $H$-equivariantly equivalent, then $C^*$ and $C'^*$ have isomorphic Hopf and equivariant cyclic (co)homologies.
• The $k$-linear dual $C^*$ of an $H$-module coalgebra $C$ is isomorphic to $\mathrm{Hom}_C(C,C)^{\mathrm{op}}$ as $H$-module algebras, providing a key duality link between coalgebras and algebras in the theory.