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.


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.
for all , ∈ A. Equivalently one may identify with an ideal within a free product algebra * .
Odd cocycles arise from graded -traces on +1 : ( ) = ( ( )) , ∀ ∈ , ∈ , + = + 1, (4) where is a Z 2 action on : In the paper we will extend this result to a version of Hopf-cyclic cohomology (see [2][3][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.

-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 ℎ ∈ on ∈ (from the left) we denote simply by ℎ .

ISRN Algebra
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 Ω( ) be a differential graded algebra with an injective map : → Ω 0 ( ). Let us assume that the Ω( ) has an -module structure compatible with that of and with the exterior derivative : Now we are ready to define the following.

Proposition 6.
If Ω( ) 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 coboundary of vanishes: In a trivial way we can also prove the inverse of that theorem, by taking as Ω( ) the universal differential graded algebra over and setting the -invariant trace on the bimodule of -forms as the given cocycle on all elements ⊗ 0 1 ⋅ ⋅ ⋅ , and as 0 on all elements ⊗ 1 ⋅ ⋅ ⋅ . In the following, we define an -invariant trace on ⊗ .

Definition 7. An -invariant trace on
⊗ is a bilinear functional : ⊗ → , which satisfies The main result is as follows.
Proof. Clearly, since is -invariant, so is . What remains to be checked is the cyclicity and the condition that is a Hochschild cycle. This, however, will be taken care of by the extension of the map from [1].
Using the result of [1] we know that the maps define a morphism of differential graded algebras from Ω ( ) to 2 ( ). Hence, the image is a differential graded algebra, which we will call Ω( ). Observe that the bimodule offorms is contained in 2 ( +1 ).
If is an -invariant trace on ⊗ as defined in Definition 7, then the following defines a closed, gradedinvariant trace on ⊗ Ω ( ): That ∫ is closed follows immediately from the fact that a product of even number of elements ( ) is proportional to identity matrix in 2 ( ). It is clear that the map islinear. Therefore it remains only to check the -cyclicity of ∫ . But again, since ( ) is diagonal for any even -form this follows directly from the fact that ( ) is diagonal and is an -twisted trace.
In a similar way, odd Hopf-cyclic cocycles can be associated with -twisted -invariant traces on ⊗ .
Since the proof is purely algebraic and follows [1, Proposition 5], the only difference being in the application of cyclicity and -invariance, we skip it. In the conclusion we have the following.

Corollary 10. For any even the Hopf-cyclic cohomology ( ⊗ ) is isomorphic to the quotient
The full quotient is isomorphic with the quotient of the Hopf-cyclic cohomology group ( ⊗ ) by the image of −2 ( ⊗ ) through the periodicity operator S.
Similar statement for -traces gives the correspondence to odd Hopf-cyclic cohomology.

Example: Twisted Cyclic Cocycles
Twisted cyclic cocycles appeared first in a context of quantum deformations [6], where they appeared to be a good replacement of the usual cyclic cocycles. In particular, for the quantum SU (2) and the family of quantum spheres, certain automorphisms lead to a similar behavior of twisted cyclic theory as in the classical nondeformed case, without the dimension drop, that appears in the standard cyclic homology [7]. A detailed study of the twisted case, including the geometric realization through modular Fredholm modules, was presented in [5]; here we recall the basic facts to illustrate the above general case.
The notation used in this section is as follows: again, is an algebra (not necessarily unital) over and is an automorphism of . Consider = CZ, group algebra of Z with the action on through the automorphism . As an easy corollary of Lemma 3 we have the following.

ISRN Algebra
Consider now stable anti-Yetter-Drinfeld modules over . The simplest example comes from one-dimensional vector space 1 with the right action and left coaction given by where denotes the generator of Z and V a vector from 1 . We have the following.
Lemma 12. Let , be an algebra and its automorphism. Then, any Z-invariant, cyclic trace on 1 ⊗ corresponds to a -twisted trace on : ( ) = ( ( )) , ∀ ∈ , ∈ , + = + 1 We skip the proof as it follows directly from the properties of Hopf-cyclic traces applied to this particular example. As a corollary, we obtain the following.

Proposition 13. If is a -twisted trace on then the functional
defines a -twisted -cyclic cocycle on for even .
The detailed presentation of the construction of twisted cyclic cocycles from finitely summable modular Fredholm modules is in [5].

Example: Hopf Algebras
A different set of examples of Hopf-cyclic cohomology originated from studies of Hopf algebras. Let us begin with an example of the Hopf-cyclic homology of an -comodule algebra. In this section, is a right -comodule algebra and is a right-right stable anti-Yetter-Drinfeld module. First, we observe the following.
Remark 14. The coaction of extends to through An Hopf-cyclic cocycle with values in is a multilinear map from +1 to , which is cyclic: and that its coboundary vanishes: We have the following.
Proposition 15. Each -colinear, -valued trace on ⊂ , even, gives rise to a Hopf-cyclic cocycle on with values in .
The proof follows exactly the same lines as in the previous section and therefore we skip it. What is interesting, however, is the application, which was discussed in [8].

Lemma 16. If
= and one takes the coproduct as the coalgebra structure, and the anti-Yetter-Drinfeld module = is determined through a modular pair in involution: is a grouplike element, is a character of , such that ( ) = 1 and the right coaction and action are for any V ∈ (for details see [9]). The compatibility condition and is Then, since is a comodule algebra over and remains an anti-Yetter-Drinfeld module, one can construct even Hopfcyclic cocycles over with values in from -valued linear maps on , -even, that satisfy ( ) ⊗ = ( (0) ) ⊗ (1) , ( ) = ( (0) ) ( (1) ) , for each ∈ , ∈ .
Again, the proof is a direct consequence of Proposition 9 and Corollary 10.

Conclusions
We have shown that the results of [1] extend to the case of Hopf-cyclic cohomology with coefficients. This is, in itself, an anticipated result. Its value, however, is that such presentation offers a possibility for a geometric presentation of Hopf-cyclic cocycles thus opening a new insight in the theory. Similarly as in the standard or twisted case it is conceivable that Hopfcyclic cocycles might be constructed from certain type of objects like Fredholm modules. While the general theory is still not available yet, the above construction shows a path, which could be followed, at least in some particular cases, like for the modular pair in involution. The work in this direction is already in progress.