On Hopf-Cyclic Cohomology and Cuntz Algebra

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 𝑘 be a field of characteristic zero and 𝐴 an algebra over 𝑘. In [1] the construction of cyclic cocycles over 𝐴 was related to the construction of traces over some ideals in the Cuntz algebra extension 𝑞𝐴. Let us briefly remind the basic construction.

Definition 1. Let 𝑞𝐴 be an algebra generated by 𝐴 and symbols 𝑞(𝑎) subject to the relation 𝑞(𝑎𝑏)=𝑞(𝑎)𝑏+𝑎𝑞(𝑏)−𝑞(𝑎)𝑞(𝑏), (1) for all 𝑎,𝑏∈𝒜. Equivalently one may identify 𝑞𝐴 with an ideal within a free product algebra 𝐴∗𝐴.

Further, define 𝐽^𝑛 as an ideal of 𝑞𝐴 generated by 𝑎₀𝑞(𝑎₁)⋯𝑞(𝑎_𝑚) and 𝑞(𝑎₁)⋯𝑞(𝑎_𝑚)𝑚≥𝑛. The main result of Connes and Cuntz [1] states as following.

Theorem 2 (see [1, Proposition 3]). If  𝑇 is a trace on 𝐽^𝑛, 𝑛 even, that is a linear functional such that 𝑇𝑥𝑦=𝑇𝑦𝑥,∀𝑥∈𝐽^𝑘,𝑦∈𝐽^𝑙,𝑘+𝑙=𝑛, (2) then 𝜏𝑎₀,𝑎₁,…,𝑎_𝑛=𝑇𝑞𝑎₀𝑞𝑎₁⋯𝑞𝑎_𝑛 (3) defines an even cyclic cocycle on 𝐴.

Odd cocycles arise from graded 𝜌-traces on 𝐽^𝑛+1: 𝑇𝑥𝑦=𝑇𝑦𝜌(𝑥),∀𝑥∈𝐽^𝑘,𝑦∈𝐽^𝑙,𝑘+𝑙=𝑛+1, (4) where 𝜌 is a ℤ₂ action on 𝑞𝐴: 𝜌𝑎₀𝑞𝑎₁⋯𝑞𝑎_𝑚=(−1)^𝑚𝑎₀−𝑞𝑎₀𝑞𝑎₁⋯𝑞𝑎_𝑚,𝜌𝑞𝑎₁⋯𝑞𝑎_𝑚=(−1)^𝑚𝑞𝑎₁⋯𝑞𝑎_𝑚. (5)

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. 𝐻-Module and Comodule Algebras and Hopf-Cyclic Cohomology

Let 𝐻 be a Hopf algebra with an invertible antipode and 𝐴 a left 𝐻-module algebra. Throughout the paper we use the Sweedler notation for coproduct: Δ(ℎ)=ℎ₍₁₎⊗ℎ₍₂₎, (6) and coaction. The action of ℎ∈𝐻 on 𝑎∈𝐴 (from the left) we denote simply by ℎ𝑎.

We begin with the basic lemma, which follows directly from the definition of 𝑞𝐴.

Lemma 3. If 𝐴 is a left 𝐻-module algebra then so is 𝑞𝐴, with the action of 𝐻 extended through: ℎ𝑞(𝑎)=𝑞(ℎ𝑎). (7)
Similarly, if 𝐵 is a left 𝐻-comodule algebra then so is 𝑞𝐵, with the coaction of 𝐻 extended through: Δ_𝐿𝑞(𝑏)=𝑏₍₋₁₎⊗𝑞𝑏₍₀₎. (8)

Let us recall the following.

Definition 4. A left-right stable anti-Yetter-Drinfeld module 𝑀, over 𝐻, is a right 𝐻-module and left 𝐻-comodule, such that 𝑚₍₀₎𝑚₍₋₁₎=𝑚,∀𝑚∈𝑀,𝑚ℎ₍₋₁₎⊗𝑚ℎ₍₀₎=𝑆ℎ₍₃₎𝑚₍₋₁₎ℎ₍₁₎⊗𝑚₍₀₎ℎ₍₂₎,∀𝑚∈𝑀,ℎ∈𝐻. (9)
Let Ω(𝐴) be a differential graded algebra with an injective map 𝑖∶𝐴→Ω⁰(𝐴). Let us assume that the Ω(𝐴) has an 𝐻-module structure compatible with that of 𝐴 and with the exterior derivative 𝑑: ℎ(𝑖(𝑎()(𝜔)(),(𝐴)))=𝑖ℎ𝑎,ℎ𝑑=𝑑ℎ𝜔∀𝑎∈𝐴,𝜔∈Ω,ℎ∈𝐻. (10) Now we are ready to define the following.

Definition 5. We say that ∫_𝑀 is an 𝐻-invariant twisted closed graded trace on 𝑀⊗Ω^𝑛(𝐴) if _𝑀𝑚ℎ₍₁₎⊗𝑆ℎ₍₂₎𝜔=𝜖(ℎ)(𝑚⊗𝜔),∀𝑚∈𝑀,𝜔∈Ω^𝑛(𝐴),ℎ∈𝐻,_𝑀(𝑚⊗𝜔𝑎)=_𝑀𝑚₍₀₎⊗𝑆⁻¹𝑚₍₋₁₎𝑎𝜔,∀𝑚∈𝑀,𝜔∈Ω^𝑛(𝐴),𝑎∈𝐴,ℎ∈𝐻,_𝑀(𝑚⊗𝑑𝜔)=0,∀𝑚∈𝑀,𝜔∈Ω^𝑛−1(𝐴). (11)

Then the following is true.

Proposition 6. If Ω(𝐴) is a differential graded algebra over 𝐴, with an action of 𝐻, and ∫_𝑀 is an 𝐻-invariant closed graded trace as defined above, then the following map: 𝜙𝑚,𝑎₀,𝑎₁,…,𝑎_𝑛=_𝑀𝑚⊗𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑑𝑎_𝑛, (12) defines a Hopf cyclic-cocycle.

Proof. First, let us check the cyclicity: 𝜙𝑚,𝑎₀,𝑎₁,…,𝑎_𝑛=_𝑀𝑚⊗𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑑𝑎_𝑛,=(−1)^𝑛−1_𝑀𝑚⊗𝑑𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑑𝑎_𝑛−1𝑎_𝑛−𝑚⊗𝑑𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑎_𝑛=(−1)^𝑛_𝑀𝑚⊗𝑑𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑎_𝑛=(−1)^𝑛_𝑀𝑚₍₀₎⊗𝑆⁻¹𝑚₍₋₁₎𝑎_𝑛𝑑𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑑𝑎_𝑛−1=(−1)^𝑛𝜙𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑛,𝑎₀,𝑎₁,…,𝑎_𝑛−1. (13)
Similarly, one proves that the Hochschild coboundary of 𝜙 vanishes: 𝑏𝜙𝑚,𝑎₀,𝑎₁,…,𝑎_𝑛+1∶=^𝑛_𝑖=0(−1)^𝑖𝜙𝑚,𝑎₀,…,𝑎_𝑖𝑎_𝑖+1,…,𝑎_𝑛+1+(−1)^𝑛+1𝜙𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑛+1𝑎₀,…,𝑎_𝑛=_𝑀𝑚⊗𝑎₀𝑎₁𝑑𝑎₂⋯𝑑𝑎_𝑛+1,+^𝑛_𝑖=1(−1)^𝑖_𝑀𝑚⊗𝑎₀𝑑𝑎₁⋯𝑑𝑎_𝑖𝑎_𝑖+1⋯𝑑𝑎_𝑛+1,+(−1)^𝑛+1_𝑀𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑛+1𝑎₀𝑑𝑎₁⋯𝑑𝑎_𝑛=(−1)^𝑛_𝑀𝑚⊗𝑎₀𝑑𝑎₁𝑑𝑎₂⋯𝑑𝑎_𝑛𝑎_𝑛+1+(−1)^𝑛+1_𝑀𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑛+1𝑎₀𝑑𝑎₁⋯𝑑𝑎_𝑛=0. (14)

In a trivial way we can also prove the inverse of that theorem, by taking as Ω(𝐴) the universal differential graded algebra over 𝐴 and setting the 𝐻-invariant trace on the bimodule of 𝑛-forms as the given cocycle on all elements 𝑚⊗𝑎₀𝑑𝑎₁⋯𝑑𝑎_𝑛, and as 0 on all elements 𝑚⊗𝑑𝑎₁⋯𝑑𝑎_𝑛.

In the following, we define an 𝐻-invariant trace on 𝑀⊗𝐴.

Definition 7. An 𝐻-invariant trace on 𝑀⊗𝐴 is a bilinear functional 𝜙∶𝑀⊗𝐴→𝑘, which satisfies 𝜙𝑚ℎ₍₁₎⊗𝑆ℎ₍₂₎𝑎(ℎ)𝜙()()()𝑚=𝜖𝑚,𝑎,𝑏𝜙𝑚,𝑎,𝑏=𝜙𝑚,𝑎𝑏−𝜙₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑏𝑎=0. (15)(16)
The main result is as follows.

Proposition 8. If 𝜙 is an 𝐻-invariant trace on 𝑀⊗𝐽^𝑛⊂𝑀⊗𝑞𝐴, 𝑛 even, then 𝜉𝑚,𝑎₀,𝑎₁,…,𝑎_𝑛=𝜙𝑚,𝑞𝑎₀𝑞𝑎₁⋯𝑞𝑎_𝑛 (17) defines a Hopf-cyclic cocycle on 𝑀⊗𝐴.

Proof. Clearly, since 𝜙 is 𝐻-invariant, so is 𝜉. What remains to be checked is the cyclicity and the condition that 𝜉 is a Hochschild cycle. This, however, will be taken care of by the extension of the map 𝜂 from [1].
Using the result of [1] we know that the maps 𝜂(𝑎)=(𝑎)()=(𝑎)𝑞(𝑎)0𝑎00𝑎−𝑞,𝜂𝑑𝑎0−𝑞 (18) define a morphism of differential graded algebras from Ω_𝑢(𝐴) to 𝑀₂(𝑞𝐴). Hence, the image is a differential graded algebra, which we will call Ω(𝐴). Observe that the bimodule of 𝑛-forms is contained in 𝑀₂(𝐽^𝑛+1).
If 𝜏 is an 𝐻-invariant trace on 𝑀⊗𝐽^𝑛 as defined in Definition 7, then the following defines a closed, graded 𝐻-invariant trace on 𝑀⊗Ω^𝑛(𝐴): _𝑀(𝑚⊗𝜔)=𝜏𝑚⊗𝜂(𝜔)₁₁−𝜏𝑚⊗𝜂(𝜔)₂₂. (19)
That ∫_𝑀 is closed follows immediately from the fact that a product of even number of elements 𝜂(𝑑𝑎) is proportional to identity matrix in 𝑀₂(𝑞𝐴). It is clear that the map 𝜂 is 𝐻-linear. Therefore it remains only to check the 𝐻-cyclicity of ∫_𝑀. But again, since 𝜂(𝜔) is diagonal for any even 𝑛-form 𝜔 this follows directly from the fact that 𝜂(𝑎) is diagonal and 𝜏 is an 𝐻-twisted trace.

In a similar way, odd Hopf-cyclic cocycles can be associated with 𝜌-twisted 𝐻-invariant traces on 𝑀⊗𝐽^𝑛.

Consider now the space of 𝐻-invariant (15) linear functionals on 𝑀⊗𝐽^𝑛 and let us split them into odd and even, with respect to the action of 𝜌. For any 𝑘≥𝑛 and any such functional we define 𝑇^(𝑘)𝑚,𝑎₀,𝑎₁,…,𝑎_𝑘=𝑇𝑚⊗𝑎₀𝑞𝑎₁⋯𝑞𝑎_𝑘. (20)

We have the following.

Proposition 9. An even 𝐻-invariant functional 𝑇₊ is a trace if and only if 𝑏𝑇₊^(𝑘)=𝑇₊^(𝑘+1) and 𝑇₊^(𝑘)𝑚,𝑎₀,𝑎₁,…𝑎_𝑘=−𝑇₊^(𝑘)𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑘𝑎₀,𝑎₁,…𝑎_𝑘−1 (21) of all odd 𝑘≥𝑛.
An odd 𝐻-invariant functional 𝑇₋ is a trace if and only if 𝑇₋(𝑚⊗𝑞(𝑎₀)𝑞(𝑎₁)⋯𝑞(𝑎_𝑘)) is a Hopf-cyclic cocycle and 𝑏𝑇₋^(𝑘)=𝑇₋^(𝑘+1) and 𝑇₊^(𝑘)𝑚,𝑎₀,𝑎₁,…𝑎_𝑘=𝑇₊^(𝑘)𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑘,𝑎₀,𝑎₁,…𝑎_𝑘−1+2𝑇₋^(𝑘−1)𝑚₍₀₎,𝑆⁻¹𝑚₍₋₁₎𝑎_𝑘𝑎₀,𝑎₁,…𝑎_𝑘−1. (22)

Since the proof is purely algebraic and follows [1, Proposition 5], the only difference being in the application of cyclicity and 𝐻-invariance, we skip it. In the conclusion we have the following.

Corollary 10. For any even 𝑛 the Hopf-cyclic cohomology 𝐻𝐶^𝑛_𝐻(𝑀⊗𝐴) is isomorphic to the quotient 𝑇𝑟𝑎𝑐𝑒𝑠𝑜𝑛𝐽^𝑛+1𝑇𝑟𝑎𝑐𝑒𝑠𝑜𝑛𝐽^𝑛−1𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑇^(𝑛−2)=0. (23)
The full quotient 𝑇𝑟𝑎𝑐𝑒𝑠𝑜𝑛𝐽^𝑛+1𝑇𝑟𝑎𝑐𝑒𝑠𝑜𝑛𝐽^𝑛−1 (24) is isomorphic with the quotient of the Hopf-cyclic cohomology group 𝐻𝐶^𝑛_𝐻(𝑀⊗𝐴) by the image of 𝐻𝐶_𝐻^𝑛−2(𝑀⊗𝐴) through the periodicity operator 𝒮.

Similar statement for 𝜌-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 SU_𝑞(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, 𝐴 is an algebra (not necessarily unital) over 𝑘 and 𝜎 is an automorphism of 𝐴. Consider 𝐻=ℂℤ, group algebra of ℤ with the action on 𝐴 through the automorphism 𝜎. As an easy corollary of Lemma 3 we have the following.

Corollary 11. The automorphism 𝜎 extends naturally as an automorphism on 𝑞𝐴 through 𝜎𝑞(𝑎)=𝑞(𝜎(𝑎)). (25)
Moreover, the ideals 𝐽^𝑛 are 𝜎-invariant, 𝜎(𝐽^𝑛)⊂𝐽^𝑛.

Consider now stable anti-Yetter-Drinfeld modules over 𝐻. The simplest example comes from one-dimensional vector space 𝑀₁ with the right action and left coaction given by Δ𝑣=𝑒⊗𝑣,𝑣⊲𝑒=𝑣, (26) where 𝑒 denotes the generator of ℤ and 𝑣 a vector from 𝑀₁.

We have the following.

Lemma 12. Let 𝐴,𝜎 be an algebra and its automorphism. Then, any ℤ-invariant, cyclic trace on 𝑀₁⊗𝐴 corresponds to a 𝜎-twisted trace 𝑇 on 𝐴: 𝑇𝑥𝑦=𝑇𝑦𝜎(𝑥),∀𝑥∈𝐽^𝑘,𝑦∈𝐽^𝑙,𝑘+𝑙=𝑛+1 (27)

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 𝑇 is a 𝜎-twisted trace on 𝑞𝐴 then the functional 𝜏_𝜎𝑎₀,𝑎₁,…,𝑎_𝑛=𝑇𝑞𝑎₀𝑞𝑎₁⋯𝑞𝑎_𝑛 (28) defines a 𝜎-twisted 𝑛-cyclic cocycle on 𝐴 for even 𝑛.

Similarly, by composing 𝜎 with the map 𝜌 (5), we obtain another automorphism of 𝑞𝐴: 𝜎=𝜎∘𝜌=𝜌∘𝜎. (29)

Then, we can define odd 𝜎-traces on 𝐽^𝑛, which satisfy 𝑇𝑥𝑦=𝑇𝑦𝜎(𝑥),∀𝑥∈𝐽^𝑘,𝑦∈𝐽^𝑙,𝑘+𝑙=𝑛+1. (30)

The respective functionals, which arise from 𝜎-traces, give 𝜎-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 𝐻-comodule algebra. In this section, 𝐴 is a right 𝐻-comodule algebra and 𝑀 is a right-right stable anti-Yetter-Drinfeld module.

First, we observe the following.

Remark 14. The coaction of 𝐻 extends to 𝑞𝐴 through Δ𝑞(𝑎)=𝑞𝑎₍₀₎⊗𝑎₍₁₎,∀𝑎∈𝐴. (31)

An 𝑛 Hopf-cyclic cocycle with values in 𝑀 is a multilinear map 𝜓 from 𝐴^𝑛+1 to 𝑀, which is cyclic: 𝜓𝑎₀,𝑎₁,…,𝑎_𝑛=(−1)^𝑛𝜓𝑎_𝑛(0),𝑎₀,…,𝑎_𝑛−1𝑎_𝑛(1), (32)

𝐻-colinear: Δ𝜓𝑎₀,𝑎₁,…,𝑎_𝑛=𝜓𝑎₀₍₀₎,𝑎₁₍₀₎,…,𝑎_𝑛(0)⊗𝑎₀₍₁₎𝑎₁₍₁₎⋯𝑎_𝑛(1), (33) and that its coboundary vanishes: 𝑏𝜓𝑎₀,𝑎₁,…,𝑎_𝑛,𝑎_𝑛+1=^𝑛_𝑖=0(−1)^𝑖𝜓𝑎₀,…,𝑎_𝑖𝑎_𝑖+1,…,𝑎_𝑛+1+(−1)^𝑛+1𝜓𝑎_𝑛(0)𝑎₀,𝑎₁,…,𝑎_𝑛−1𝑎_𝑛(1). (34)

We have the following.

Proposition 15. Each 𝐻-colinear, 𝑀-valued trace on 𝐽^𝑛⊂𝐴, 𝑛 even, gives rise to a Hopf-cyclic cocycle on 𝐴 with values in 𝑀.

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 𝐴=𝐻 and one takes the coproduct as the coalgebra structure, and the anti-Yetter-Drinfeld module 𝑀=𝑘 is determined through a modular pair in involution: 𝛾 is a grouplike element, 𝜒 is a character of 𝐴, such that 𝜒(𝛾)=1 and the right coaction and action are Δ(𝑣)=𝑣⊗𝛾,𝑣⊲ℎ=𝑣𝜒(ℎ), (35) for any 𝑣∈𝑘 (for details see [9]).
The compatibility condition 𝛾 and 𝜒 is 𝑆_𝛾,𝜒(ℎ)=𝛾𝜒ℎ₍₁₎𝑆ℎ₍₂₎,𝑆²_𝛾,𝜒=𝑖𝑑. (36)
Then, since 𝑞𝐴 is a comodule algebra over 𝐴 and 𝑘 remains an anti-Yetter-Drinfeld module, one can construct even Hopf-cyclic cocycles over 𝐴 with values in 𝑘 from 𝑘-valued linear maps on 𝐽^𝑛, 𝑛-even, that satisfy 𝑇(𝑥)⊗𝛾=𝑇𝑥₍₀₎⊗𝑥₍₁₎,𝑇𝑥𝑦=𝑇𝑦₍₀₎𝑥𝜒𝑦₍₁₎, (37) for each 𝑥∈𝐽^𝑛,𝑥𝑦∈𝐽^𝑘.

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.