The Connes–Moscovici Approach to Cyclic Cohomology for Compact Quantum Groups

Abstract

Consider the Hopf algebra (A ,� ) of regular functions on a compact quantum group. Let (A o ,� ) denote its maximal dual Hopf algebra. We show that the tensor product Hopf algebra (H2 ,� 2) of (A o ,� ) and its opposite Hopf algebra is endowed with a modular pair (δ, σ ) in invo- lution; a notion introduced by A. Connes and J. Moscovici, who associate canonically a cocyclic object to such Hopf algebras. Denote the Hopf cyclic cohomology thus obtained by HC ∗ ) (H2). Next we define an action of (H2 ,� 2) on A, and show that the Haar state of (A ,� ) is a δ-invariant σ -trace on A with respect to this action. This gives us a canonical map γ from HC ∗ (δ,σ ) (H2) to the ordinary cyclic cohomology of A. We develop a method for computing HC ∗ ) (H2), including a Kunneth exact sequence for the cyclic cohomology of general cocyclic objects. We recognize the Hopf Hochschild cohomology of any Hopf algebra with a modular pair in involution as a derived functor in the category of bicomodules over the Hopf algebra. Furthermore, we prove a duality theorem to the effect that HC ∗ (δ,σ ) (H2) can be computed from a finitely generated free resolution of A as an A-bimodule, provided a certain assumption on (A ,� ) holds. Unfortunately, we are unable to check this assumption in the case of A = SUq (2). Assuming that it holds, we demonstrate how an explicit resolution of A given by T. Masuda, Y. Nakagami and J. Watanabe can be used to compute both HC ∗ (δ,σ ) (H2) and the map γ for SUq (2). To our surprise γ = 0 in this case, both on the level of Hochschild cohomology, cyclic cohomology and periodic cyclic cohomology. Finally, we dispense with this assumption in the case of SUq (2) by replacing (A o ,� ) with the more tractable Hopf subalgebra (Uq (sl2), �) . Noticing that the γ -map is well-defined also in this case, we combine our computations with those of M. Crainic to prove that the restricted γ -map is again zero, but this time without any extra assumption on SUq (2).