On Hopf-Cyclic Cohomology and Cuntz Algebra

We demonstrate that Hopf cyclic cocycles, that is, cyclic cocycles with coefficients in stable anti-Yetter-Drinfeld modules, arise from invariant traces on certain ideals of Cuntz-type extension of the algebra. 1. Introduction Let be a field of characteristic zero and an algebra over . In [1] the construction of cyclic cocycles over was related to the construction of traces over some ideals in the Cuntz algebra extension . Let us briefly remind the basic construction. Definition 1. Let be an algebra generated by and symbols subject to the relation for all . Equivalently one may identify with an ideal within a free product algebra . Further, define as an ideal of generated by and . The main result of Connes and Cuntz [1] states as following. Theorem 2 (see [1, Proposition 3]). If？？ is a trace on , even, that is a linear functional such that then defines an even cyclic cocycle on . Odd cocycles arise from graded -traces on : where is a action on : In the paper we will extend this result to a version of Hopf-cyclic cohomology (see [2–4]) for review and details) with coefficients in a stable anti-Yetter-Drinfeld module and present, as a particular example, the case of a twisted cyclic cohomology. The latter was already studied in [5], with the view to geometric construction of modular Fredholm modules. 2. -Module and Comodule Algebras and Hopf-Cyclic Cohomology Let be a Hopf algebra with an invertible antipode and a left -module algebra. Throughout the paper we use the Sweedler notation for coproduct: and coaction. The action of on (from the left) we denote simply by . We begin with the basic lemma, which follows directly from the definition of . Lemma 3. If is a left -module algebra then so is , with the action of extended through: Similarly, if is a left -comodule algebra then so is , with the coaction of extended through: Let us recall the following. Definition 4. A left-right stable anti-Yetter-Drinfeld module , over , is a right -module and left -comodule, such that Let be a differential graded algebra with an injective map . Let us assume that the has an -module structure compatible with that of and with the exterior derivative : Now we are ready to define the following. Definition 5. We say that is an -invariant twisted closed graded trace on if Then the following is true. Proposition 6. If is a differential graded algebra over , with an action of , and is an -invariant closed graded trace as defined above, then the following map: defines a Hopf cyclic-cocycle. Proof. First, let us check the cyclicity: Similarly, one proves that the Hochschild