Abstract
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 coboundary of vanishes:
In a trivial way we can also prove the inverse of that theorem, by taking as the universal differential graded algebra over and setting the -invariant trace on the bimodule of -forms as the given cocycle on all elements , and as on all elements .
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.
Proposition 8. If is an -invariant trace on , even, then defines a Hopf-cyclic cocycle on .
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 to . Hence, the image is a differential graded algebra, which we will call . Observe that the bimodule of -forms is contained in .
If is an -invariant trace on as defined in Definition 7, then the following defines a closed, graded -invariant trace on :
That is closed follows immediately from the fact that a product of even number of elements is proportional to identity matrix in . It is clear that the map is -linear. 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 .
Consider now the space of -invariant (15) linear functionals on and let us split them into odd and even, with respect to the action of . For any and any such functional we define
We have the following.
Proposition 9. An even -invariant functional is a trace if and only if and
of all odd .
An odd -invariant functional is a trace if and only if is a Hopf-cyclic cocycle and and
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 through the periodicity operator .
Similar statement for -traces gives the correspondence to odd Hopf-cyclic cohomology.
3. 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 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 , group algebra of with the action on through the automorphism . As an easy corollary of Lemma 3 we have the following.
Corollary 11. The automorphism extends naturally as an automorphism on through
Moreover, the ideals are -invariant, .
Consider now stable anti-Yetter-Drinfeld modules over . The simplest example comes from one-dimensional vector space with the right action and left coaction given by where denotes the generator of and a vector from .
We have the following.
Lemma 12. Let be an algebra and its automorphism. Then, any -invariant, cyclic trace on corresponds to a -twisted trace on :
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 .
Similarly, by composing with the map (5), we obtain another automorphism of :
Then, we can define odd -traces on , which satisfy
The respective functionals, which arise from -traces, give -twisted odd cyclic cocycles.
The detailed presentation of the construction of twisted cyclic cocycles from finitely summable modular Fredholm modules is in [5].
4. 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 to , which is cyclic:
-colinear: 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 and the right coaction and action are
for any (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 Hopf-cyclic cocycles over with values in from -valued linear maps on , -even, that satisfy
for each .
Again, the proof is a direct consequence of Proposition 9 and Corollary 10.
5. 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 Hopf-cyclic 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.