# On Hopf-Cyclic Cohomology and Cuntz Algebra

Authors: Andrzej Sitarz
Journal: ISRN Algebra
 (2013)

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## 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.


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## Body

## 1. Introduction

Letk 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.
LetqA be an algebra generated by A and symbols q(a) subject to the relation
(1)q(ab)=q(a)b+aq(b)-q(a)q(b),
for all a,b∈𝒜. Equivalently one may identify qA with an ideal within a free product algebra A*A.Further, defineJn as an ideal of qA generated by a0q(a1)⋯q(am) and q(a1)⋯q(am)m≥n. The main result of Connes and Cuntz [1] states as following.Theorem 2 (see [1, Proposition 3]).
IfT is a trace on Jn, n even, that is a linear functional such that
(2)T(xy)=T(yx),∀x∈Jk,y∈Jl,k+l=n,
then
(3)τ(a0,a1,…,an)=T(q(a0)q(a1)⋯q(an))
defines an even cyclic cocycle on A.Odd cocycles arise from gradedρ-traces on Jn+1:
(4)T(xy)=T(yρ(x)),∀x∈Jk,y∈Jl,k+l=n+1,
where ρ is a ℤ2 action on qA:
(5)ρ(a0q(a1)⋯q(am))=(-1)m(a0-q(a0))q(a1)⋯q(am),ρ(q(a1)⋯q(am))=(-1)mq(a1)⋯q(am).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

LetH 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:
(6)Δ(h)=h(1)⊗h(2),
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 ofqA.Lemma 3.
IfA is a left H-module algebra then so is qA, with the action of H extended through:
(7)hq(a)=q(ha).
Similarly, ifB is a left H-comodule algebra then so is qB, with the coaction of H extended through:
(8)ΔLq(b)=b(-1)⊗q(b(0)).Let us recall the following.Definition 4.
A left-right stable anti-Yetter-Drinfeld moduleM, over H, is a right H-module and left H-comodule, such that
(9)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.
LetΩ(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:
(10)h(i(a))=i(ha),hd(ω)=d(hω),∀a∈A,ω∈Ω(A),h∈H.
Now we are ready to define the following.Definition 5.
We say that∫M‍ is an H-invariant twisted closed graded trace on M⊗Ωn(A) if
(11)∫M‍(mh(1)⊗S(h(2))ω)=ϵ(h)∫‍(m⊗ω),∀m∈M,ω∈Ωn(A),h∈H,∫M(m⊗ωa)=∫M‍(m(0)⊗S-1(m(-1))aω),∀m∈M,ω∈Ωn(A),a∈A,h∈H,∫M‍(m⊗dω)=0,∀m∈M,ω∈Ωn-1(A).Then the following is true.Proposition 6.
IfΩ(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:
(12)ϕ(m,a0,a1,…,an)=∫M‍(m⊗a0da1da2⋯dan),
defines a Hopf cyclic-cocycle.Proof.
First, let us check the cyclicity:(13)ϕ(m,a0,a1,…,an)=∫M‍(m⊗a0da1da2⋯dan),=(-1)n-1∫M‍(m⊗d(a0da1da2⋯dan-1an)-m⊗da0da1da2⋯an)=(-1)n∫M‍(m⊗da0da1da2⋯an)=(-1)n∫M‍(m(0)⊗(S-1m(-1)an)da0da1da2⋯dan-1)=(-1)nϕ(m(0),(S-1m(-1)an),a0,a1,…,an-1).
Similarly, one proves that the Hochschild coboundary ofϕ vanishes:
(14)bϕ(m,a0,a1,…,an+1):=∑i=0n‍(-1)iϕ(m,a0,…,aiai+1,…,an+1)+(-1)n+1ϕ(m(0),S-1(m(-1))an+1a0,…,an)=∫M‍(m⊗a0a1da2⋯dan+1),+∑i=1n‍(-1)i∫M(m⊗a0da1⋯d(aiai+1)⋯dan+1),+(-1)n+1∫M(m(0),S-1(m(-1))an+1a0da1⋯dan)=(-1)n∫M(m⊗a0da1da2⋯danan+1)+(-1)n+1∫M(m(0),S-1(m(-1))an+1a0da1⋯dan)=0.In a trivial way we can also prove the inverse of that theorem, by taking asΩ(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⋯dan, and as 0 on all elements m⊗da1⋯dan.In the following, we define anH-invariant trace on M⊗A.Definition 7.
AnH-invariant trace on M⊗A is a bilinear functional ϕ:M⊗A→k, which satisfies
(15)ϕ(mh(1)⊗Sh(2)a)=ϵ(h)ϕ(m,a),(16)(bϕ)(m,a,b)=ϕ(m,ab)-ϕ(m(0),(S-1m(-1)b)a)=0.
The main result is as follows.Proposition 8.
Ifϕ is an H-invariant trace on M⊗Jn⊂M⊗qA, n even, then
(17)ξ(m,a0,a1,…,an)=ϕ(m,q(a0)q(a1)⋯q(an))
defines a Hopf-cyclic cocycle on M⊗A.Proof.
Clearly, sinceϕ 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
(18)η(a)=(a00a-q(a)),η(da)=(0-q(a)q(a)0)
define a morphism of differential graded algebras from Ωu(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(Jn+1).
Ifτ is an H-invariant trace on M⊗Jn as defined in Definition 7, then the following defines a closed, graded H-invariant trace on M⊗Ωn(A):
(19)∫M‍(m⊗ω)=τ(m⊗η(ω)11)-τ(m⊗η(ω)22).
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 withρ-twisted H-invariant traces on M⊗Jn.Consider now the space ofH-invariant (15) linear functionals on M⊗Jn 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
(20)T(k)(m,a0,a1,…,ak)=T(m⊗a0q(a1)⋯q(ak)).We have the following.Proposition 9.
An evenH-invariant functional T+ is a trace if and only if bT+(k)=T+(k+1) and
(21)T+(k)(m,a0,a1,…ak)=-T+(k)(m(0),S-1m(-1)aka0,a1,…ak-1)
of all odd k≥n.
An oddH-invariant functional T- is a trace if and only if T-(m⊗q(a0)q(a1)⋯q(ak)) is a Hopf-cyclic cocycle and bT-(k)=T-(k+1) and
(22)T+(k)(m,a0,a1,…ak)=T+(k)(m(0),S-1m(-1)ak,a0,a1,…ak-1)+2T-(k-1)(m(0),S-1m(-1)aka0,a1,…ak-1).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 evenn the Hopf-cyclic cohomology HCHn(M⊗A) is isomorphic to the quotient
(23)TracesonJn+1TracesonJn-1suchthatT(n-2)=0.
The full quotient(24)TracesonJn+1TracesonJn-1
is isomorphic with the quotient of the Hopf-cyclic cohomology group HCHn(M⊗A) by the image of HCHn-2(M⊗A) through the periodicity operator 𝒮.Similar statement forρ-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 SUq(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 automorphismσ extends naturally as an automorphism on qA through
(25)σ(q(a))=q(σ(a)).
Moreover, the idealsJn are σ-invariant, σ(Jn)⊂Jn.Consider now stable anti-Yetter-Drinfeld modules overH. The simplest example comes from one-dimensional vector space M1 with the right action and left coaction given by
(26)Δv=e⊗v,v⊲e=v,
where e denotes the generator of ℤ and v a vector from M1.We have the following.Lemma 12.
LetA,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:
(27)T(xy)=T(yσ(x)),∀x∈Jk,y∈Jl,k+l=n+1We 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.
IfT is a σ-twisted trace on qA then the functional
(28)τσ(a0,a1,…,an)=T(q(a0)q(a1)⋯q(an))
defines a σ-twisted n-cyclic cocycle on A for even n.Similarly, by composingσ with the map ρ (5), we obtain another automorphism of qA:
(29)σ~=σ∘ρ=ρ∘σ.Then, we can defineoddσ-traces on Jn, which satisfy
(30)T(xy)=T(yσ~(x)),∀x∈Jk,y∈Jl,k+l=n+1.The respective functionals, which arise fromσ~-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 anH-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 ofH extends to qA through
(31)Δq(a)=q(a(0))⊗a(1),∀a∈A.Ann Hopf-cyclic cocycle with values in M is a multilinear map ψ from An+1 to M, which is cyclic:
(32)ψ(a0,a1,…,an)=(-1)nψ(an(0),a0,…,an-1)an(1),H-colinear:
(33)Δψ(a0,a1,…,an)=ψ(a0(0),a1(0),…,an(0))⊗a0(1)a1(1)⋯an(1),
and that its coboundary vanishes:
(34)bψ(a0,a1,…,an,an+1)=∑i=0n‍(-1)iψ(a0,…,aiai+1,…,an+1)+(-1)n+1ψ(an(0)a0,a1,…,an-1)an(1).We have the following.Proposition 15.
EachH-colinear, M-valued trace on Jn⊂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.
IfA=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
(35)Δ(v)=v⊗γ,v⊲h=vχ(h),
for any v∈k (for details see [9]).
The compatibility conditionγ and χ is
(36)Sγ,χ(h)=γχ(h(1))S(h(2)),Sγ,χ2=id.
Then, sinceqA 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 Jn, n-even, that satisfy
(37)T(x)⊗γ=T(x(0))⊗x(1),T(xy)=T(y(0)x)χ(y(1)),
for each x∈Jn,xy∈Jk.Again, the proof is a direct consequence of Proposition9 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.



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