Generalized Coefficients for Hopf Cyclic Cohomology

A category of coefficients for Hopf cyclic cohomology is defined. It is shown that this category has two proper subcategories of which the smallest one is the known category of stable anti Yetter-Drinfeld modules. The middle subcategory is comprised of those coefficients which satisfy a generalized SAYD condition depending on both the Hopf algebra and the (co)algebra in question. Some examples are introduced to show that these three categories are different. It is shown that all components of Hopf cyclic cohomology work well with the new coefficients we have defined.


Introduction
A suitable class of coefficients for Hopf cyclic cohomology was first introduced in [6,7]. These coefficients generalize modular pairs in involution discovered by Connes-Moscovici for Hopf cyclic cohomology [4], and are called stable anti Yetter-Drinfeld (SAYD) modules. Hopf cyclic cohomology is defined for a datum consisting of a symmetry, a space, and a space of coefficients. The symmetry is defined by a Hopf algebra, the space is defined by an algebra or a coalgebra on which the Hopf algebra acts or coacts, and the coefficients space is a module and comodule over the Hopf algebra satisfying two conditions defined merely by the Hopf algebra structure. One notes that bialgebra cyclic cohomology introduced in [8] extends the Hopf cyclic cohomology tremendously by relaxing the symmetry to be a bialgebra and the coefficients to be just stable module-comodule. However for computing the latter cohomology one cannot rely on the homological algebra of coalgebras and algebras. What we introduce in this paper is a class of coefficients larger than SAYD modules but still computable in terms of derived category of (co)algebras. We show that the bialgebra cyclic cohomology for the new coefficients coincide with their Hopf cyclic cohomology.
Based on examples, we observe that Hopf cyclic cohomology works well with some coefficients which are not SAYD modules over Hopf algebras. We study these modules and prove that there are at least three noticeable categories of such coefficients. The idea is to show that the coefficients not only depend on the Hopf algebra in question but that they may also be related to the (co)algebra on which the Hopf algebra (co)acts. This observation unties our hands in situations where the Hopf algebra lacks a large class of SAYD modules. Existence of such a Hopf algebra is already shown in [15], where it is proved that any finite-dimensional SAYD module over Connes-Moscovici Hopf algebra is a direct sum of the original modular pair in involution found in [4].
We cover two cases of Hopf cyclic cohomology in this paper. For an H-module coalgebra C we define C H-SAYD modules and for an H-module algebra A we introduce A H-SAYD modules. We show that SAYD modules are both C H-SAYD modules and A H-SAYD modules for any A and C, however we give several examples showing that these inclusions of categories are proper.
Next we show that there are examples of coefficients for H-module (co)algebras (C)A which are not even ( C H) A H-SAYD modules, while the correspondent Hopf cyclic complexes are welldefined. We call these coefficient Hopf cyclic coefficients (HCC). It is not difficult to see that the cup product in Hopf cyclic cohomology works well with these new coefficients.
One notes that after introducing Hopf cyclic cohomology with coefficients in [6,7], there are some generalizations of the theory in [1,8,11]. However in all computable examples the authors had to transform their examples into the original Hopf cyclic complexes of some new Hopf algebras. As an important example of such, in [8] the coefficients are module/comodules over bialgebras and do not have to satisfy any YD like condition. However, the author in [8] proves that if these coefficients are SAYD, when the bialgebra is Hopf algebra, then the Hopf cyclic cohomology and bialgebra cyclic cohomoloy coincide. We prove that the same result holds for new coefficients. The proof is a modification of the proof given in [8] for SAYD modules over Hopf algebras.
Hopf cyclic cohomology generalizes Lie algebra cohomology. One of the most useful features of Lie algebra cohomology is the induction/restriction functor which simply allows us to use a subalebra of the Lie algebra in question to calculate the Lie algebra cohomology of the Lie algebra with coefficients in the induced module in terms of the Lie algebra cohomology of the subalgebra with coefficients in the given module. This feature is successfully adopted for Hopf cyclic cohomology in [7]. Thanks to one of the referees some of our examples can be calculated via the induction procedure.
Notations. In this paper we denote a Hopf algebra by H and its counit by ε. We use the Sweedler summation notation ∆(h) = h (1) ⊗ h (2) for the coproduct of a Hopf algebra. Furthermore (h) = h −1 ⊗ h 0 and (h) = h 0 ⊗ h 1 are used for the left and right coactions of a coalgebra, respectively.

Hopf cyclic cohomology coef f icients
In this section we introduce two generalized versions of SAYD modules for module coalgebras and module algebras. Let a Hopf algebra act on an algebra or a coalgebra, this action being compatible with (co)algebra structure. The main idea here is to show that there are coefficients by which the Hopf cyclic cohomology of the datum is well-defined where these coefficients are not only (co)representations of the Hopf algebras but also represent the algebra or coalgebra in question. Let us recall from [6] that a right-left SAYD module M over a Hopf algebra H is a right module and a left comodule over H such that Since bicrossed product Hopf algebras will be used repeatedly in this paper, let us recall this notion here. Let U and F be Hopf algebras and furthermore let U be a right F-comodule coalgebra. One then forms a cocrossed product coalgebra U < F that has F ⊗ U as underlying vector space and the following coalgebra structure Dually if F is a left U-module algebra, we can endow the underlying vector space F ⊗ U with an algebra structure, to be denoted by F > U, with 1 > 1 as its unit and the product given by The Hopf algebras U and F as above, with the action and coaction as above, form a matched pair of Hopf algebras if they satisfy the following compatibility conditions for any u ∈ U, and any f ∈ F One forms a new Hopf algebra F U, called the bicrossed product of the matched pair (F, U); it has F < U as underlying coalgebra, F > U as underlying algebra and the antipode is defined by Throughout the paper all Hopf algebras are assumed to have invertible antipodes.

The C H-SAYD modules for module coalgebras
For basics of (co)cyclic modules and their correspondence cyclic (co)homology we refer the reader to [3] and [12]. If M is a right module and left comodule over a Hopf algebra H, then for any left H-module coalgebra C one defines the following para-cocyclic structure on where c = c 0 ⊗ · · · ⊗ c n . In addition, when M is SAYD module over H, it is shown in [5] that the above para-cocyclic structure is well-defined on C n H (C, M ) := M ⊗ H ⊗C ⊗n+1 . Here H acts on C ⊗n+1 diagonally.
However the cyclic structure on C * H (C, M ) can be well-defined for some other types of modules M which are not SAYD over H.
Here the action of H on C ⊗• is diagonal. We use the symbol C H-SAYD for the category whose objects are C H-SAYD modules and whose morphisms are H-module and H-comodule morphisms.
Let us introduce some examples of C H-SAYD modules which are not necessarily a SAYD module over Hopf algebras. First we generalize the notion of modular pair in involution [4]. Lemma 2.2. Let C be an H-module coalgebra, δ a character and σ a group-like element for H. Let (δ, σ) be a modular pair, i.e. δ(σ) = 1 and C H-in involution, i.e.
where S δ (h) = δ h (1) S h (2) . Then σ C δ is a C H-SAYD module via the action and coaction induced by δ and σ respectively.
Proof . One checks that S −1 δ (h) = S −1 h (1) δ h (2) . On the other hand we have Using the fact that (δ, σ) is a modular pair, the C H-stability condition is obvious.
For any H-module coalgebra C, the following lemma introduce a one-dimensional D H-SAYD module for some H-module coalgebra D.
Lemma 2.3. Let (δ, σ) be a modular pair for the Hopf algebra H and C be a left H-module coalgebra. We define the following subspace of C Then I is a coideal of C and D := C I is an H-module coalgebra. Furthermore σ C δ is a D H-SAYD module.
Proof . Since is a coalgebra map, for any c ∈ C and h ∈ H we have This shows ∆(I) ⊆ I ⊗ C + C ⊗ I. Also we have ε(y) = 0 for all y ∈ I. Therefore I is a coideal of C and D = C I is coalgebra. Now it is enough to show that I is an H-module. To do this we show that for all h, g ∈ H and c ∈ C we have First we observe that Since S 2 δ is an algebra map, we have On the other hand we have Adding relations (2.3) and (2.4), we obtain (2.2). Therefore I is an H-module coalgebra and using definition of I, (δ, σ) is a D H-modular pair in involution. Thus by Lemma 2.2 we obtain that σ C δ is a D H-SAYD module.
To exemplify, let us recall that H := H cop 1 , the co-opposite Hopf algebra of the Connes-Moscovici Hopf algebra in codimension one, [4] is generated by the elements X, Y and δ k , where k ∈ N, subject to the following relations Its coalgebra structure and antipode are defined as follows Let U be the enveloping algebra of the Lie algebra generated by X and Y . Also suppose F is the co-opposite Hopf algebra of the Hopf algebra generated by all the δ k . It is shown in [13] that H is canonically isomorphic to the bicrossed product Hopf algebra F U where U acts on F via and F coacts on U via A similar decomposition exists for the higher dimensions of this Hopf algebra denoted H cop n . Its is shown in [15,Theorem 3.16] that there is only one-dimensional SAYD module for the co-opposite Hopf algebra of the Connes-Moscovici Hopf algebra H cop 1 , namely 1 C δ . Since for H cop 1 we have S 2 = Id, it follows that δ = ε. But in the following example we show that 1 C ε defines a C H cop 1 -SAYD module.
where the H-module structure is given by the multiplication. The relations show that I is the coideal generated by the elements of the form Using S 2 (X) = X − δ 1 , Xδ i = δ i+1 + δ i X and the fact that S 2 is a morphism of algebras, one obtains the following relation easily by induction where h n,i are certain elements in H. This shows that the coideal I is generated by We define a coalgebra map ϕ : H −→ U, given by One checks that ker(ϕ) = I and therefore H I ∼ = U. As a result the action of H on D ∼ = U which has been defined by multiplication simplifies to

HCC modules for coalgebras
Definition 2.5. Let C be a coalgebra which is also an H-module but not necessarily modulecoalgebra where the action of H on C ⊗• is diagonal. A module-comodule M over H is called a (H, C)-Hopf cyclic coefficients, abbreviated by C H-HCC, if the cosimplicial and cyclic operators on C * H (C, M ) are well-defined and turn it to a cocyclic module. We use the symbol C H-HCC for the category whose objects are C H-HCC modules and whose morphisms are Hmodule and H-comodule morphisms In the following statement we show that for any C H-SAYD module the cocyclic module structure defined in (2.1) is well-defined and therefore we obtain a Hopf cyclic cohomology with these new coefficients: Proposition 2.6. Let C be an H-module coalgebra. Then any C H-SAYD module M is a C H-HCC.
Proof . Since C is assumed to be an H-module coalgebra, all cosimplicial structure operators except the very last coface are well-defined. On the other hand, the last coface is a composition of the first coface and the cyclic operator. Therefore it is enough to show that the cyclic operator is well-defined. Let π : C n (C, M ) → C n H (C, M ) be the natural projection then we see that the same proof as in [5] works here We use the C H-AYD module condition in the second equality. To verify the cocyclicity condition, using the C H-stability condition one has Lemma 2.7. Let C be an H-module coalgebra. If the action of H on C is cocommutative, i.e. Proof . Since C is assumed to be an H-module coalgebra it suffices to show that τ n is welldefined. Indeed, let π : C n (C, M ) → C n H (C, M ) be the natural projection, then We use (2.5) in the second equality.
The following statement is a dual result to the previous lemma. Proof . Using the fact that the action of H on M is via ε and H acts commutatively on C we have The C H-stability condition is obvious by the triviality of the action.
Example 2.9. Let G be a discrete group acting on a set X normally, i.e. ghx = hgx for all g, h ∈ G and x ∈ X. Then consider the group algebra H = CG and the obvious coalgebra structure C X on X where each element is considered to be a group-like element. Obviously C X is a module coalgebra over CG by the action of G on X. The action of H on C X is commutative. This introduces a source of C H-SAYDs which are not ordinary SAYD modules over H = CG.
One notes that all types of SAYD modules M on CG are known, see [6], which in all cases Ggraded vector spaces M = g M g where for the (non-trivial) action we have gm = m for all m ∈ M g and hm ∈ M hgh −1 .
We present another example. Since F is a Hopf subalgebra of H = F U, the Hopf algebra H acts on the coalgebra C := H ⊗ F C via left multiplication. It is easy to see that as a coalgebra C is isomorphic to U via u −→ (1 u) ⊗ F 1. Via this identification, the action of H on C is as follows. Let h = f u ∈ H, and v ∈ C = U. Then One easily checks that for any bicrossed product Hopf algebra F U, the Hopf algebra U is an H-module coalgebra via this action.
Example 2.10. Consider any bicrossed product Hopf algebra H = F U where U is a cocommutative Hopf algebra. We have seen that C = U is an H-module coalgebra by the following action Since U is cocommutative, it is obvious that H acts on C cocommutatively. Therefore any comodule M over H with the trivial action defines a U H-SAYD module.
Example 2.11. Consider any bicrossed product Hopf algebra H = F U where U is a cocommutative Hopf algebra. Consider the module coalgebra U with the action defined in (2.6). We show that H acts on U cocommutatively. First let us recall that Using the cocommutativity of U, for h := f u we get Therefore any module M over H, providing it with the trivial coaction of H, defines a U H-HCC.
As an example the co-opposite Hopf algebra of Connes-Moscovici Hopf algebra H cop 1 ∼ = F U acts cocommutatively on U.
Here we introduce an example of C H-HHC which is not a C H-SAYD module. We have shown that if H acts on C cocommutatively then any module M defines an C H-HHC. Indeed in this example, we introduce a triple (H, C, M ) such that H acts on C cocommutatively but M is not a C H-SAYD module.
We have seen that C = U is a H-module coalgebra by the action (f u)v = ε(f )uv and that this action is a cocommutative action by Example 2.11. Now let M = H be a right H-module via multiplication and left H-comodule by trivial coaction. Since the action is cocommutative, it is a U H-HCC module. We show that M is not a C H-SAYD. Let f u := δ 1 ⊗ X ∈ H, c := X ∈ U, and m := 1 1 ∈ M . First we notice that After substituting into the AYD condition we observe that

Relation to Kaygun's cyclic cohomology of bialgebras
In this subsection we first recall the bialgebra cyclic cohomology developed by Kaygun in [8] and then show that for any C H-SAYD module the result of these two cohomology coincide. Let H be a bialgebra, C be an H-module coalgebra, M be a right module and left comodule over H.
We let H act on C n (C, M ) from left by and defines the following subspace of C n (H, M ) W n (C, M ) := L g , τ i (C n (H, M )) | g ∈ H, i ∈ Z + .
Here [L g , τ ] := L g τ − τ L g . Let us see that W is stable under cyclic structure of C * (C, M ). Indeed for τ we simply see that For ∂ m , where 0 ≤ m ≤ n, we use the identity ∂ m τ = τ ∂ m+1 and the fact that ∂ m L g = L g ∂ m to see that Using the facts that L g σ j = σ j L g and τ σ j = σ j−1 τ , we see that for 1 ≤ j ≤ n, we have Finally for σ 0 = σ n τ we obtain This paracocyclic module is denoted by PCM * (C, M ) in [8]. On the other hand if M is stable then which proves that the paracocyclic module CM * (C, M ) := C⊗ H PCM * (C, M ) is in fact a cocyclic module.
Proposition 2.13. Let H be a Hopf algebra, C be an H-module coalgebra, and M be a C H-SAYD module. Then W ⊆ ker π, where π : C n (C, M ) → C n H (C, M ) is the canonical projection. Proof . Let us fix m ⊗ c 0 ⊗ · · · ⊗ c n ∈ C n (C, M ). Using the C H-SAYD property of M , we see that On the other hand we have To finish the proof one uses the facts that, by Proposition 2.6, τ on C H (C, M ) is well-defined, and that [L g , −] is a derivation. Indeed, Finally we prove the main result of this subsection. Here we assume that the action of H on C n H (C, M ) is trivial. It is easily seen that So π induces a map of cocyclic modules On the other hand since W * is a para-cocyclic submodule of C * (C, M ) we have projections of para-cocyclic modules.
Let us check that π is balanced over H. In the following we use [x] as the class of x ∈ C n (C, M ) in PCM n (C, M ) and 1 ∈ C as the generator of C as trivial H-module.
So π induces a map of cocyclic modules It is obvious that Π and Π are inverse to each other.

The A H-SAYD and HCC modules for module algebras
Let A be a left H-module algebra and M be a right-left SAYD module over H. We notice that M ⊗ A ⊗(n+1) is a right H-module via (m ⊗ a)h := mh (1) ⊗ S(h (2) ) a and the right Hmodule structure of the ground field C is given by rh := ε(h)r for all r ∈ C and h ∈ H. It is shown that the following cocyclic structure on the space of right H-linear homomorphisms (d i f )(m ⊗ a) = f (m ⊗ a 0 ⊗ · · · ⊗ a i a i+1 ⊗ · · · ⊗ a n ), 0 ≤ i < n, (d n f )(m ⊗ a) = f m 0 ⊗ S −1 m −1 a n a 0 ⊗ a 1 ⊗ · · · ⊗ a n−1 , (s n f )(m ⊗ a) = f (m ⊗ a 0 ⊗ · · · ⊗ a i ⊗ 1 ⊗ · · · ⊗ a n ), 0 ≤ i ≤ n, (t n f )(m ⊗ a) = f (m 0 ⊗ S −1 (m −1 )a n ⊗ a 0 ⊗ · · · ⊗ a n−1 ).
Here the action of H on A ⊗(n+1) is diagonal. We use the symbol A H-SAYD for the category whose objects are A H-SAYD modules and whose morphisms are H-module and H-comodule morphisms Proof . Let M be a right-left A H-SAYD module. To show that M is A H-HCC it is enough to check that the cyclic map is well-defined (t n f )((m ⊗ a 0 ⊗ · · · ⊗ a n )h) = (t n f ) mh (1) ⊗ S h (2) (a 0 ⊗ · · · ⊗ a n ) )a n ⊗ a 0 ⊗ · · · ⊗ a n−1 )ε(h) = (t n f )(m ⊗ a 0 ⊗ · · · ⊗ a n )ε(h) = ((t n f )(m ⊗ a 0 ⊗ · · · ⊗ a n )) h.
Lemma 2.19. Let A be a left H-module algebra, and (δ, σ) be a modular pair and A H-in involution, i.e.
Proof . The following computations show the A H-AYD condition The stability condition is obvious from the modular pair condition.
Proposition 2.20. Let (δ, σ) be a modular pair for H and A be a left H-module algebra. We define the following subspace of A Then B is an H-module subalgebra of A and (δ, σ) is a B H-modular pair in involution.
Proof . It is obvious that 1 ∈ B. Since A is an H-module algebra, σ is a group-like element and S −2 δ is a coalgebra map, the following computation shows that B is a subalgebra of A Let h ∈ H and b ∈ B. The following computation shows that hb ∈ B and therefore that B is a left H-module Therefore B is an H-module subalgebra of A and (δ, σ) is a B H-modular pair in involution.
Lemma 2.21. Let A be a left H-module algebra. If the action of H on A is cocommutative, i.e. Proof . The A H-stability condition is obvious by the triviality of the coaction. Since A is assumed to be H-module algebra it suffices to show that the cyclic map τ is well-defined (1) ⊗ S h (n+1) a n ⊗ S h (n) a 0 ⊗ · · · ⊗ S h (2) a n−1 = f (m ⊗ a n ⊗ a 0 ⊗ · · · ⊗ a n−1 )ε(h) = (t n f )(m ⊗ a 0 ⊗ · · · ⊗ a n )ε(h).
We use (2.9) in the third equality. Proof . The A H-stability condition is obvious. The following computation proves the A H-AYD condition We use the commutativity of the action on the second equality.
Here we introduce an example of the preceding lemma.
Example 2.23. Similar to Example 2.9, let G be a group acting normally on a set X from the right and H = CG be the group algebra of G acting on A = Fun(X, C), the commutative algebra of all complex valued functions on X, by (gf )(x) = f (xg). It is easy to check that A is a left H-module algebra and furthermore that this action is commutative.
One notes that for any bicrossed product Hopf algebra H = F U, the Hopf algebra F is an H-module algebra by the following action (2.10) Example 2.24. Let H := F U be any bicrossed product Hopf algebra where U is a cocommutative Hopf algebra. Then the following computation proves that the action in (2.10) is cocommutative Therefore any module M over H, with the trivial coaction of H, defines a F H-HCC.
Here we introduce an example which shows that the categories of A H-HCC and A H-SAYD modules are different. In fact we introduce an example of an A H-HCC module which is not a A H-SAYD. We have seen that the Hopf algebra F is a left H-module algebra by the action defined in (2.10) and this action is a cocommutative action by Example 2.24 and therefore that M is a F H-HCC module. We show that M is not a F H-AYD. First one notices that using the formula Using (2.7) and the triviality of the coaction, by substituting h = δ 1 X, m = 1 1 and a = δ 1 into the F H-AYD condition we have

Cup products in Hopf cyclic cohomology
In this section we show that all features of Hopf cyclic cohomology work well with the new coefficients which we have defined in Section 2.1. One of the most important features of Hopf cyclic cohomology is its cup product [9,10,14] as a generalization of the Connes-Moscovici characteristic map [4]. We show that this cup product works well with the new coefficients in this paper. One may construct an AYD module over a Hopf algebra as the tensor product of a YD module with a AYD module [6,16]. This shows that the category of AYD modules is a C-category over the category of YD-modules. In this section we show the same expectation for generalized modules is satisfied. In other words we prove that the category of C H-SAYD modules is a C-category over the category of C H-YD modules.  The rest of this section is devoted to the cup product in Hopf cyclic cohomology with generalized coefficients.
Similar to [14] one has the following statement. Therefore we obtain a cyclic map χ : C * (B, C) −→ C * (A, C). As a result we have the following cyclic map Ψ = χ • Ψ a,c : C n,n a,c −→ C n (A, C).

Now we define
to be a tensor product of cocyclic modules which has a mixed complex structure [2,12]. The cyclic cohomology of this mixed complex is p,q HC p H (A, M ) ⊗ HC q H (C, M ).
As in [14], one uses the Alexander-Whitney map to obtain the following map AW : HC p H (A, M ) ⊗ HC q H (C, M ) −→ HC p+q (C p,q a,c ) −→ HC n (C n,n a,c ).
Now we obtain the following statement.