[Paper Review] Pairings in Hopf-cyclic cohomology of algebras and coalgebras with coefficients

Igor Nikonov, G. Sharygin|ArXiv.org|Oct 20, 2006

Advanced Operator Algebra Research4 references4 citations

TL;DR

This paper establishes a unified framework for cup-products in Hopf-cyclic cohomology of algebras and coalgebras with coefficients in stable anti-Yetter-Drinfeld modules. It constructs a pairing via the noncommutative Weil algebra and proves that the cup product defined here coincides with the Crainic-type pairing and the construction in [15], with a key identity showing the compatibility of the pairing with $S$-operations and periodicity.

ABSTRACT

This paper is concerned with the theory of cup-products in Hopf-type cyclic cohomology of algebras and coalgebras. Here we give detailed proofs of the statements, announced in our previous paper. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving noncommutative Weil algebra. Then we use a generalization of Quillen and Crainic's construction to define the cup-product. We discuss the relation of the introduced cup-product and $S$-operations on cyclic cohomology. After this we describe the relation of this type of product and bivariant cyclic cohomology. In the last section we briefly discuss the relation of our constructions with that of Khalkhali and Rangipour.

Motivation & Objective

• To provide a comprehensive construction of cup-products in Hopf-cyclic cohomology for algebras and coalgebras with coefficients in stable anti-Yetter-Drinfeld modules.
• To establish a connection between the cyclic cohomology of a coalgebra and the cohomology of a noncommutative Weil algebra.
• To generalize Quillen and Crainic’s construction to define a cup-product in the context of Hopf-cyclic cohomology.
• To clarify the relationship between the introduced cup-product and $S$-operations on cyclic cohomology.
• To compare the proposed pairing with existing constructions, particularly those in [15], and prove their equivalence.

Proposed method

• Construct the noncommutative Weil algebra $W^{oldsymbol{ ext{H}}}(C,oldsymbol{ ext{M}})$ for a coalgebra $C$ and a stable anti-Yetter-Drinfeld module $\mathcal{M}$, and compute its cohomology.
• Define the paracocyclic module $C^{\ast}(C,\mathcal{M}) = \mathcal{M} \otimes C^{\otimes n+1}$ with cyclic operations $\delta_i$, $\sigma_i$, and $\tau_n$ satisfying all cocyclic relations except $\tau_n^{n+1} = 1$.
• Pass to the $\mathcal{H}$-invariant quotient $C^{\ast}_{\mathcal{H}}(C,\mathcal{M}) = \mathcal{M} \otimes_{\mathcal{H}} C^{\otimes n+1}$, which becomes a cocyclic module when $\mathcal{M}$ is stable and anti-Yetter-Drinfeld.
• Define the cup-product via a generalized Crainic-type construction using the Weil algebra and the map $\rho: W_{\natural}/\operatorname{Im} d \to \operatorname{Hom}(BA^{\natural}, \Bbbk)$.
• Use the element $cs_{m+n} = i w^{m+n} \in W$ as a cocycle in $W_{\natural}/\operatorname{Im} d$ to define the pairing as a composition: $\int \circ \mathbf{ev}_{\xi} \circ \tau \circ \rho(cs_{m+n})$.
• Prove the key identity $\alpha_n[\sigma(cs_{m+n})(\xi)] = \frac{m+1}{m+n+1}[\xi]$ in $HC^{m}_{\mathcal{H}}(C,\mathcal{M})$ by induction on $n$, using boundary maps and the action of $b$ and $\delta$ on $W$.

Experimental results

Research questions

• RQ1 How can a cup-product structure be consistently defined in Hopf-cyclic cohomology of algebras and coalgebras with coefficients?
• RQ2 What is the role of the noncommutative Weil algebra in realizing the cyclic cohomology of a coalgebra?
• RQ3 How does the introduced cup-product relate to $S$-operations on cyclic cohomology?
• RQ4 In what way does the Crainic-type pairing coincide with the cup-product construction in this framework?
• RQ5 Is the pairing defined here equivalent to the construction in [15], and if so, under what conditions?

Key findings

• The cyclic cohomology of a coalgebra $C$ with coefficients in a stable anti-Yetter-Drinfeld module $\mathcal{M}$ is realized as the cohomology of the noncommutative Weil algebra $W^{\mathcal{H}}(C,\mathcal{M})$.
• The cup-product in Hopf-cyclic cohomology is constructed via a generalized Crainic-type pairing using the Weil algebra and the map $\rho$, which identifies closed traces with cyclic cocycles.
• The key identity $\alpha_n[\sigma(cs_{m+n})(\xi)] = \frac{m+1}{m+n+1}[\xi]$ is proven by induction, establishing the compatibility of the pairing with the $S$-operation.
• The cup product defined here coincides with the construction in [15], as shown by the composition $\int \circ \mathbf{ev}_{\xi} \circ \tau \circ \rho(cs_{m+n})$ being equal to the pairing $\sigma(cs_{m+n})(\xi)$.
• The boundary map $\varphi$ satisfies $\varphi[cs_n] = \frac{n+1}{n+2}[cs_{n+1}]$ in $H^*(W^{n+1}_{\natural}/\operatorname{Im} d, \delta)$, which is essential for the inductive proof.
• The construction establishes a natural isomorphism between the pairing and the $S$-operation, showing that the cup-product is compatible with periodicity and the Connes-Tonneli map.