On the anti-Yetter-Drinfeld module-contramodule correspondence

TL;DR

This paper explores the relationship between anti-Yetter-Drinfeld modules and contramodules over Hopf algebras, establishing conditions for their equivalence and revealing new structural insights and examples.

Contribution

It introduces a functor connecting anti-Yetter-Drinfeld modules to contramodules, identifies conditions for equivalence, and provides novel examples and periodicity observations.

Findings

Conditions under which the functor is an equivalence

Equivalence of the center of the opposite category of comodules to anti-Yetter Drinfeld modules

Existence of a symmetric 2-contratrace not arising from anti-Yetter Drinfeld modules

Abstract

We study a functor from anti-Yetter Drinfeld modules to contramodules in the case of a Hopf algebra $H$. Some byproducts of this investigation are the establishment of sufficient conditions for this functor to be an equivalence, verification that the center of the opposite category of $H$-comodules is equivalent to anti-Yetter Drinfeld modules, and the observation of two types of periodicities of the generalized Yetter-Drinfeld modules introduced previously. Finally, we give an example of a symmetric $2$-contratrace on $H$-comodules that does not arise from an anti-Yetter Drinfeld module.

Equations

$$\rho f(w)=f(w_0{)}_0\otimes f(w_0{)}_{1}S(w_1).$$

$$‘‘Id\otimes \Delta "=(Id\otimes \Delta )\circ -$$

$$‘‘\rho \otimes Id"=(I{d}_V\otimes m\otimes I{d}_H)\circ (I{d}_{V\otimes H}\otimes {\sigma }_{H,H})\circ ({\rho }_V\otimes I{d}_{H\otimes H})\circ (f\otimes S)\circ {\rho }_W$$

$$‘‘\rho \otimes Id"(f)(w)=((w_0){}^{{}^{\left(1\right)}}{)}_0\otimes ((w_0){}^{{}^{\left(1\right)}}{)}_{1}S(w_1)\otimes (w_0){}^{{}^{\left(2\right)}}.$$

$$‘‘Id\otimes \Delta "\circ \rho =‘‘\rho \otimes Id"\circ \rho .$$

$$\mathrm{Hom}(W,V)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{Hom}(T\otimes W,V{)}^H\simeq \mathrm{Hom}(T,{\mathrm{Hom}}^l(W,V){)}^H,$$

$$f(t\otimes w{)}_0\otimes f(t\otimes w{)}_1=f(t_0\otimes w_0)\otimes t_1{w}_1.$$

$$\rho {\varphi }_t={\varphi }_{t_0}\otimes t_1$$

$$\rho f_t(w)$$

$$=f(t_0\otimes w_{0,0})\otimes t_1{w}_{0,1}S(w_1)$$

$$=f(t_0\otimes w_0)\otimes t_1{w}_{1}S(w_2)$$

$$=f(t_0\otimes w)\otimes t_1$$

$$=f_{t_0}\otimes t_1.$$

$$\varphi (t_0\otimes w_0)\otimes t_1{w}_1$$

$$=\varphi (t\otimes w_0{)}_0\otimes \varphi (t\otimes w_0{)}_{1}S(w_1)w_2$$

$$=\varphi (t\otimes w{)}_0\otimes \varphi (t\otimes w{)}_1$$

$${\rho }^l:{\mathrm{Hom}}^l(W,V)\to {\mathrm{Hom}}^l(W,V)\otimes H.$$

$$\mathrm{Hom}(W,V{)}^{rat}={\mathrm{Hom}}^l(W,V)$$

$${\rho }^{r}f(w)=f(w_0{)}_0\otimes S^{-1}(w_1)f(w_0{)}_1.$$

$$\mathrm{Hom}(W\otimes T,V{)}^H\simeq \mathrm{Hom}(T,{\mathrm{Hom}}^r(W,V){)}^H.$$

$$V◃W={\mathrm{Hom}}^r(W,V),\text{and}W▹V={\mathrm{Hom}}^l(W,V)$$

$${\mathrm{Hom}}^l(V,W)=W\otimes V^{\ast }$$

$${\mathrm{Hom}}^r(V,W)={}^{\ast }V\otimes W$$

$${}^{\ast \ast }V=V^{S^{-2}}\text{and}{V}^{\ast \ast }=V^{S^2},$$

$$(hm{)}_0\otimes (hm{)}_1=h^2{m}_0\otimes h^3{m}_{1}S(h^1).$$

$$\tau :\mathrm{Hom}(W,M)\to \mathrm{Hom}(W,M)$$

$$\tau f(w)=w_{1}f(w_0).$$

$$‘‘\tau \otimes Id"=(a\otimes Id)\circ {\sigma }_{M,H,H}\circ (f\otimes Id)\circ {\rho }_W$$

$$‘‘\tau \otimes Id"(f)(w)=w_1(w_0){}^{{}^{\left(1\right)}}\otimes (w_0){}^{{}^{\left(2\right)}}.$$

$$\mathrm{Hom}(W,M)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\theta \circ \tau f(w)=\theta (w_{1}f(w_0))=w_1\theta f(w_0)=\tau (\theta \circ f)(w)$$

$$W▹(V▹M)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

2010 Mathematics Subject Classification. Monoidal category (18D10), abelian and additive category (18E05), cyclic homology (19D55), Hopf algebras (16T05).

1 Introduction.

This paper grew out of the author’s attempts to better understand contramodules at least in some simple examples. The simplest case being the Hopf algebra $kG$ where $G$ is a discrete infinite group. Contramodules over a coalgebra were introduced by Eilenberg and Moore in 1965 and can be viewed either as algebraic structures allowing infinite combinations or a better behaved notion than that of modules over the dual algebra (see Remark 4.6). They do not strictly speaking generalize comodules, but do have a non-trivial intersection with them. In our investigations we found [6] to be very helpful, in fact the phenomenon of this underived comodule-contramodule correspondence without the anti-Yetter-Drinfeld enhancement is investigated there as well.

The introduction of anti-Yetter-Drinfeld contramodule coefficients to the Hopf-cyclic cohomology theory in [2] that followed the definition of anti-Yetter-Drinfeld module coefficients in [3] can in retrospect be conceptually understood as being completely natural since they are seen to be exactly corresponding to the representable symmetric $2$-contratraces, see [7] and [5]. The latter form a well behaved class of Hopf-cyclic coefficients explored in [4] and [7], that lead directly to Hopf-cyclic type cohomology theories.

Roughly speaking, the category of stable anti-Yetter-Drinfeld modules consists of $H$-modules and comodules such that the two structures are compatible in a way that ensures that they form the center of a certain bimodule category [4]. A similar statement with contramodule structure replacing the comodule one can be made about anti-Yetter-Drinfeld contramodules. In general understanding objects in these categories is not a simple task, however in the case of $H=kG$ the former category is known to consist of $G$-graded $G$-equivariant vector spaces, i.e., $\oplus_{g\in G}M_{g}$ with $x:M_{g}\to M_{xgx^{-1}}$. Stability, a condition that ensures cyclicity, translates to $x=Id_{M_{x}}$. We could find no similarly simple description of the anti-Yetter-Drinfeld contramodule category in the literature. It turns out, Corollary 4.10, that this category is also equivalent to $G$-graded $G$-equivariant vector spaces but the objects are now $\prod_{g\in G}M_{g}$. The Theorem 4.5 is a more general case of this correspondence.

The above anti-Yetter-Drinfeld module-contramodule correspondence was the motivation for the rest of the results in this paper. Namely, the Proposition 4.7 shows that the equivalence arises from a functor $M\mapsto\widehat{M}$ from comodules to contramodules. This functor can be found in [6] but arose independently from the considerations of [7] which furthermore demonstrate that it works on the anti-Yetter-Drinfeld versions as well. More precisely, for $M$ a stable anti-Yetter-Drinfeld module we consider $\mathcal{F}(-)=\mathop{\rm Hom}\nolimits(-,M)^{H}$ which is a symmetric $2$-contratrace on $H$-comodules, i.e., a contravariant functor from $\mathcal{M}^{H}$ to Vec subject to a trace-like symmetry. Its pullback to the category ${{}_{H}\mathcal{M}}$ of $H$-modules is $\mathop{\rm Hom}\nolimits_{H}(-,\widehat{M})$. The pullback construction reduces in this case to the observation that $H\in\mathcal{M}^{H}$ is an algebra and the category of $H$-bimodules in $\mathcal{M}^{H}$ is equivalent to ${{}_{H}\mathcal{M}}$. The pullback $\mathop{\rm Hom}\nolimits_{H}(-,\widehat{M})$ is obtained as $\mathcal{F}_{H}$, i.e., the equalizer of the action maps $\mathcal{F}(V)\to\mathcal{F}(H\otimes V)$ and $\mathcal{F}(V)\to\mathcal{F}(V\otimes H)$ with the targets identified via the symmetry of $\mathcal{F}$. Though this can be used as the definition of $\widehat{M}$, we give an explicit construction of both the contramodule structure, essentially agreeing with [6], and the $H$-action on $\widehat{M}=\mathop{\rm Hom}\nolimits(H,M)^{H}$.

It turns out that, not surprisingly, $M\mapsto\widehat{M}$ is not always an equivalence, but it does have a left adjoint, that we found in [6] and upgraded to the anti-Yetter-Drinfeld setting here. The key object when studying the question of equivalence is the ideal of left integrals for $H$ as introduced in [8]. This object seems to be the first example of a generalized Yetter-Drinfeld module of charge other than $1$ or $-1$, its charge is $2$. These were introduced in [4] without any hope that anything other than $\pm 1$ would be useful. In fact, the conditions for the comodule-contramodule correspondence are closely related to the presence of a $2$-periodicity of the charges, see Remark 5.7. Furthermore, in studying the question of stability of anti-Yetter-Drinfeld modules/contramodules and the generalization of this concept to more general charges (in a way that was necessarily different from [4]) we observed a second kind of periodicity within a generalized Yetter-Drinfeld category of a fixed charge. The Remark 5.8 describes an action of $\mathbb{Z}/i\mathbb{Z}$ on Yetter-Drinfeld modules of charge $i-1$ and Yetter-Drinfeld contramodules of charge $i+1$. This action is compatible with the generalized $M\mapsto\widehat{M}$ that sends Yetter-Drinfeld modules of charge $i-1$ to Yetter-Drinfeld contramodules of charge $i+1$; the case of $i=0$ is the usual anti-Yetter-Drinfeld situation.

Identifying categories of interest with centers of bimodule categories such as was done in [4] and [5] is carried through this paper as well. We point to the summary Theorem 3.8 that is of the [4] flavor, and the Corollary 2.8 of the [5] flavor as examples. One of the natural questions that arose after [4] was if symmetric $2$-contratraces give a more general class of coefficients for Hopf-cyclic type theories even in the case of Hopf algebras. It turns out [5] that for the case of ${{}_{H}\mathcal{M}}$ the representable symmetric $2$-contratraces are equivalent to the anti-Yetter-Drinfeld contramodules, and similarly (Corollary 2.9) for the case of $\mathcal{M}^{H}$ the representable symmetric $2$-contratraces are equivalent to the anti-Yetter-Drinfeld modules. Thus one needs only find a non-representable example of a contratrace in order to have a new coefficient in the $H$-comodule case. This is explained in Section 4.2.

The paper is arranged as follows: Section 2 is devoted to the establishment of the fact that $\mathcal{M}^{H}$ is biclosed, and thus it makes sense to consider the opposite bimodule category $\mathcal{M}^{Hop}$ with the adjoint action. The center is shown to consist of anti-Yetter-Drinfeld modules; this identifies symmetric representable $2$-contratraces with stable anti-Yetter-Drinfeld modules. Finally we introduce the functor $M\mapsto\widehat{M}$. In Section 3 we demonstrate that the adjoint pair of functors between comodules and contramodules: $N\mapsto N^{\prime}$ and $M\mapsto\widehat{M}$ induce the same between the anti-Yetter-Drinfeld versions and also the stable anti-Yetter-Drinfeld versions. Section 4 deals with the question of equivalence established by $M\mapsto\widehat{M}$ and ends with an example of a new coefficient for the case of $H=kG$. In Section 5 we extend $M\mapsto\widehat{M}$ to the generalized Yetter-Drinfeld modules and discuss two types of periodicities and the compatibility of $M\mapsto\widehat{M}$ with them.

Some things to keep in mind: for a coalgebra $C$ we use the following version of Sweedler notation $\Delta(c)=c^{1}\otimes c^{2}$. For a right comodule $N$ over $C$ we use $\rho(n)=n_{0}\otimes n_{1}$. All Hopf algebras have invertible antipodes $S$ and are over a field $k$ of characteristic [math]. We denote by $\mathop{\rm Hom}\nolimits(-,-)^{H}$ and $\mathop{\rm Hom}\nolimits_{H}(-,-)$ the morphisms in $\mathcal{M}^{H}$ and ${{}_{H}\mathcal{M}}$ respectively, while $\mathop{\rm Hom}\nolimits(-,-)$ stands for $k$-linear maps.

Acknowledgments: The author wishes to thank Masoud Khalkhali and Piotr Hajac for stimulating questions and discussions. This research was supported in part by the NSERC Discovery Grant number 406709.

2 The category of $H$-comodules.

This section is primarily dedicated to the establishment of the fact that the monoidal category $\mathcal{M}^{H}$ of $H$-comodules is biclosed, and the analysis of the center of the bimodule category over $\mathcal{M}^{H}$ resulting from considering $\mathcal{M}^{Hop}$. This establishes an analogue of a result in [5] describing the center as the category of $aYD$-modules for $H$.

2.1 Internal Homs in the category of $H$-comodules.

Motivated by the existence of internal Homs in ${{}_{H}\mathcal{M}}$, and thus the possibility of describing representable contratraces on ${{}_{H}\mathcal{M}}$ as central elements in the opposite category, we will now address the same question in $\mathcal{M}^{H}$, the monoidal category of $H$-comodules.

Since in the finite dimensional $H$ case, we have that $\mathcal{M}^{H}\simeq{{}_{H^{*}}\mathcal{M}}$ so we have a suggestive way of obtaining the required formulas. We note that some modifications do need to be made to account for possible infinite dimensionality of $H$.

For $W,V\in\mathcal{M}^{H}$ consider $\rho:\mathop{\rm Hom}\nolimits(W,V)\to\mathop{\rm Hom}\nolimits(W,V\otimes H)$ given by

[TABLE]

Definition 2.1**.**

Define $\mathop{\rm Hom}\nolimits^{l}(W,V)$ as the subspace of $\mathop{\rm Hom}\nolimits(W,V)$ that consists of $f$ such that $\rho f\in\mathop{\rm Hom}\nolimits(W,V)\otimes H$.

We can define two maps

[TABLE]

and

[TABLE]

from $\mathop{\rm Hom}\nolimits(W,V\otimes H)$ to $\mathop{\rm Hom}\nolimits(W,V\otimes H\otimes H)$. The latter can be written down more manageably as follows: let $f(w)=w~{}^{{}^{\left(1\right)}}\otimes w~{}^{{}^{\left(2\right)}}$ then

[TABLE]

A direct computation shows that

[TABLE]

Note that when restricted to $\mathop{\rm Hom}\nolimits(W,V)\otimes H$ the maps $``Id\otimes\Delta"$ and $``\rho\otimes Id"$ are actually $Id\otimes\Delta$ and $\rho\otimes Id$ respectively. The formula (2.2) has two important and immediate consequences: $\rho:\mathop{\rm Hom}\nolimits^{l}(W,V)\to\mathop{\rm Hom}\nolimits^{l}(W,V)\otimes H$, whereas before we only knew that it lands in $\mathop{\rm Hom}\nolimits(W,V)\otimes H$, and $\rho$ is a coaction.

It is not hard to see that $\mathop{\rm Hom}\nolimits^{l}(W,V)$ is contravariant in $W$ and covariant in $V$. More precisely, let $\phi\in\mathop{\rm Hom}\nolimits(W^{\prime},W)^{H}$ and $\theta\in\mathop{\rm Hom}\nolimits(V,V^{\prime})^{H}$, then the following diagram commutes:

[TABLE]

and so $\theta\circ-\circ\phi:\mathop{\rm Hom}\nolimits^{l}(W,V)\to\mathop{\rm Hom}\nolimits^{l}(W^{\prime},V^{\prime})$ and it is a map of $H$-comodules.

Lemma 2.2**.**

We have natural identifications

[TABLE]

i.e., $\mathop{\rm Hom}\nolimits^{l}(-,-)$ is the left internal Hom in $\mathcal{M}^{H}$.

Proof.

Note that $f\in\mathop{\rm Hom}\nolimits(T\otimes W,V)^{H}$ if and only if

[TABLE]

On the other hand $\phi\in\mathop{\rm Hom}\nolimits(T,\mathop{\rm Hom}\nolimits^{l}(W,V))^{H}$ if and only if

[TABLE]

where $\phi_{t}\in\mathop{\rm Hom}\nolimits^{l}(W,V)$.

Let $f\in\mathop{\rm Hom}\nolimits(T\otimes W,V)^{H}$ then if we define $f_{t}(w)=f(t\otimes w)$ we have

[TABLE]

So that $t\mapsto f_{t}\in\mathop{\rm Hom}\nolimits(T,\mathop{\rm Hom}\nolimits^{l}(W,V))^{H}$. Conversely, if $\phi\in\mathop{\rm Hom}\nolimits(T,\mathop{\rm Hom}\nolimits^{l}(W,V))^{H}$ then define $\phi(t\otimes w)=\phi_{t}(w)$ then $\phi(t_{0}\otimes w)\otimes t_{1}=\phi(t\otimes w_{0})_{0}\otimes\phi(t\otimes w_{0})_{1}S(w_{1})$ so that

[TABLE]

and thus $t\otimes w\mapsto\phi_{t}(w)\in\mathop{\rm Hom}\nolimits(T\otimes W,V)^{H}$.

So the usual bijection $f(t\otimes w)=f_{t}(w)$ establishes a natural identification between $\mathop{\rm Hom}\nolimits(T\otimes W,V)^{H}$ and $\mathop{\rm Hom}\nolimits(T,\mathop{\rm Hom}\nolimits^{l}(W,V))^{H}$ as required. ∎

Remark 2.3**.**

From now on we will denote the coaction $\rho$ of (2.1) by $\rho^{l}$ since

[TABLE]

Remark 2.4**.**

Note that we have a natural fully faithful embedding of $\mathcal{M}^{H}$ into ${{}_{H^{*}}\mathcal{M}}$. The right adjoint to it can be used to define $\mathop{\rm Hom}\nolimits^{l}(W,V)$. Namely, the formula (2.1) defines a left $H^{*}$-module structure on $\mathop{\rm Hom}\nolimits(W,V)$ via $\chi f=(Id_{V}\otimes\chi)(\rho f)$. Then it is easy to see that

[TABLE]

where $(-)^{rat}$is the right adjoint that features prominently in [8].

Repeating the above considerations nearly verbatim, we define the right internal Hom for $\mathcal{M}^{H}$ as follows. Begin by defining

[TABLE]

Definition 2.5**.**

Define $\mathop{\rm Hom}\nolimits^{r}(W,V)$ as the subspace of $\mathop{\rm Hom}\nolimits(W,V)$ that consists of $f$ such that $\rho^{r}f\in\mathop{\rm Hom}\nolimits(W,V)\otimes H$.

We again obtain that $\rho^{r}:\mathop{\rm Hom}\nolimits^{r}(W,V)\to\mathop{\rm Hom}\nolimits^{r}(W,V)\otimes H$ and is a coaction. Furthermore, we have natural adjunctions:

[TABLE]

As usual we now have the opposite category $\mathcal{M}^{Hop}$ with

[TABLE]

and we may examine its center $\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})$.

Remark 2.6**.**

We observe that if $V\in\mathcal{M}^{H}$ is finite dimensional then

[TABLE]

and

[TABLE]

where $V^{*}=\mathop{\rm Hom}\nolimits^{l}(V,k)$ and ${{}^{*}V}=\mathop{\rm Hom}\nolimits^{r}(V,k)$ and both $V^{*}$ and ${{}^{*}V}$ are $\mathop{\rm Hom}\nolimits_{k}(V,k)$ as vector spaces. So that $\mathcal{M}^{H}_{fd}$ is rigid with

[TABLE]

where $V^{S^{2i}}$ denotes the $H$-comodule with the coaction modified by $S^{2i}$.

2.2 The center of the opposite bimodule category.

We recall from [3] that a left-right anti-Yetter-Drinfeld module $M$ over a Hopf algebra $H$ is a left $H$-module and a right $H$-comodule satisfying

[TABLE]

It is stable if $m_{1}m_{0}=m$ for all $m\in M$.

Recall, for example from [7], the notions of the opposite bimodule category and of the center of a bimodule category.

Proposition 2.7**.**

The category of $aYD$-modules for $H$ is equivalent to $\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})$.

Proof.

Let $M$ be an $aYD$-module and define the central structure

[TABLE]

[TABLE]

Note that $\tau$ is invertible with $\tau^{-1}f(w)=S^{-1}(w_{1})f(w_{0})$. Define a map $``\tau\otimes Id":\mathop{\rm Hom}\nolimits(W,M\otimes H)\to\mathop{\rm Hom}\nolimits(W,M\otimes H)$ by

[TABLE]

so that if $f(w)=w~{}^{{}^{\left(1\right)}}\otimes w~{}^{{}^{\left(2\right)}}$ then

[TABLE]

A direct computation (using the $aYD$ condition (2.6)) demonstrates that the following diagram commutes:

[TABLE]

and since $``\tau\otimes Id"$ restricted to $\mathop{\rm Hom}\nolimits(W,M)\otimes H$ is actually $\tau\otimes Id$, so in fact $\tau:\mathop{\rm Hom}\nolimits^{l}(W,M)\to\mathop{\rm Hom}\nolimits^{r}(W,M)$ and it is an isomorphism in $\mathcal{M}^{H}$.

Observe that $\tau$ is natural in $W$, i.e., if $\phi:W^{\prime}\to W$ is a morphism in $\mathcal{M}^{H}$ then $\phi(w_{0})\otimes w_{1}=\phi(w)_{0}\otimes\phi(w)_{1}$ so that $w_{1}f(\phi(w_{0}))=\phi(w)_{1}f(\phi(w)_{0})$ and $\tau(f\circ\phi)=\tau f\circ\phi$.

If $\theta:M\to M^{\prime}$ is a map of $aYD$-modules, then

[TABLE]

so that a map of $aYD$-modules induces a map of central elements.

To check the commutativity of

[TABLE]

is to check that going along the bottom and obtaining $w\otimes v\mapsto(w_{1}v_{1})f(w_{0}\otimes v_{0})$ is the same as the long way around which gives $w\otimes v\mapsto w_{1}(v_{1}f_{w_{0}}(v_{0}))=w_{1}(v_{1}f(w_{0}\otimes v_{0}))$; and they are the same by the usual $H$-action axiom. Similarly, the unitality of the action implies that $k\triangleright M\to M\triangleleft k$ is the identity since $\tau:m\mapsto 1m$.

What has been shown so far is that if $M$ is an $aYD$-module, then $(M,\tau)\in\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})$ and any $\theta:M\to M^{\prime}$ a morphism of $aYD$-modules induces a morphism between the corresponding central elements.

Conversely, let $M\in\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})$ so that we have natural isomorphisms $\tau:\mathop{\rm Hom}\nolimits^{l}(W,M)\to\mathop{\rm Hom}\nolimits^{r}(W,M)$. Note that

[TABLE]

where $W_{\alpha}\subset W$ is a finite dimensional sub-comodule since any $w\in W$ is contained in such an $W_{\alpha}$ and so

[TABLE]

So we have a $\tau:\mathop{\rm Hom}\nolimits(W,M)\to\mathop{\rm Hom}\nolimits(W,M)$ that satisfies a version of all the properties that make the original $\tau$ so useful. Denote by $r$ the composition

[TABLE]

so that $m\mapsto\tau(h\mapsto\epsilon(h)m)$. Define

[TABLE]

Note that we needed to use $\mathop{\rm Hom}\nolimits(H,M)$ instead of $\mathop{\rm Hom}\nolimits^{l}(H,M)$ since $h\mapsto\epsilon(h)m$ is not in $\mathop{\rm Hom}\nolimits^{l}(H,M)$.

By the unitality of $\tau$ we have $ev_{1}\circ\tau=ev_{1}$ so that

[TABLE]

Furthermore by the “associativity” of $\tau$, i.e., the diagram (2.8) and its naturality, we have

[TABLE]

Let $\theta:M\to M^{\prime}$ be a map in the center, then we have

[TABLE]

where the left square commutes trivially and the right one commutes by definition, so that

[TABLE]

Before proving that the $H$-action defined above satisfies the $aYD$-module condition (2.6) we will show that the definition of the action from $\tau$ and vice versa are mutually inverse. Let an $H$-action be given, then we set $\tau f(w)=w_{1}f(w_{0})$ so that the action becomes

[TABLE]

i.e., the original action. On the other hand if $\tau:\mathop{\rm Hom}\nolimits(W,M)\to\mathop{\rm Hom}\nolimits(W,M)$ is given and we defined the action by $hm=r_{m}(h)=\tau(x\mapsto\epsilon(x)m)(h)$, then we obtain the following. Let $f\in\mathop{\rm Hom}\nolimits(W,M)$, consider $f\otimes\epsilon\in\mathop{\rm Hom}\nolimits(\underline{W}\otimes H,M)$ where $\underline{W}$ is a trivial $H$-comodule. Note that the co-action map $\rho_{W}$ is a morphism in $\mathcal{M}^{H}$ from $W$ to $\underline{W}\otimes H$. So $\tau(f\otimes\epsilon\circ\rho_{W})=\tau(f\otimes\epsilon)\circ\rho_{W}$ and the former is $\tau f$ while the latter is

[TABLE]

So that no matter if we start with a $\tau$ or an $H$-action, we always have

[TABLE]

Now recall the diagram (2.7), and note that it now commutes essentially by definition. Let $W=H$ and keep in mind the formula (2.10). We now get that for any $f\in\mathop{\rm Hom}\nolimits(H,M)$ we have

[TABLE]

and let us apply it to $f(h)=\epsilon(h)m$ to obtain

[TABLE]

so that

[TABLE]

and $M$ satisfies the $aYD$-module condition (2.6).

∎

Recall that we denote by $\mathcal{Z}^{\prime}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})$ the full subcategory that consists of objects such that the identity map $Id\in\mathop{\rm Hom}\nolimits(M,M)^{H}$ is mapped to itself via

[TABLE]

We have a straightforward corollary:

Corollary 2.8**.**

The category of $saYD$-modules for $H$ is equivalent to $\mathcal{Z}^{\prime}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})$.

Proof.

Recall that an $aYD$-module $M$ is stable if $m_{1}m_{0}=m$ for all $m\in M$. On the other hand considering $\tau:\mathop{\rm Hom}\nolimits(M,M)\to\mathop{\rm Hom}\nolimits(M,M)$ we see that according to (2.10) we have $\tau Id(m)=m_{1}m_{0}$ and so $\tau Id=Id$ if and only if $M$ is stable. ∎

Thus we have established the following:

Corollary 2.9**.**

The category of $saYD$-modules for $H$ is equivalent to the category of representable symmetric $2$-contratraces on $\mathcal{M}^{H}$ via

[TABLE]

Contrast that with the ${{}_{H}\mathcal{M}}$ case considered in [5] where the category of representable symmetric $2$-contratraces is equivalent to the more unusual $saYD$-contramodules.

2.3 A functor from $(s)aYD$-modules to $(s)aYD$-contramodules

This section is motivated by the adjunction on cyclic cohomology of [7] that we explain below. Given an $saYD$-module $M$, i.e., a representable symmetric $2$-contratrace $\mathop{\rm Hom}\nolimits(-,M)^{H}$, as a special case of the theory developed in [7], we obtain an $H$-module $\widehat{M}$ such that $\mathop{\rm Hom}\nolimits_{H}(-,\widehat{M})$ is a representable symmetric $2$-contratrace.

We will need to recall from [2] that a right $C$-contramodule $N$, where $C$ is a counital coassociative coalgebra, is equipped with the contraaction

[TABLE]

satisfying

[TABLE]

for any $f\in\mathop{\rm Hom}\nolimits(C\otimes C,N)$ and

[TABLE]

for any $n\in N$. Furthermore, a left-right $aYD$-contramodule $N$ is a left $H$-module and a right $H$-contramodule such that for all $h\in H$ and any linear map $f\in\mathop{\rm Hom}\nolimits(H,N)$ we have

[TABLE]

It is called stable, i.e., an $saYD$-contramodule, if for all $n\in N$ we have $\alpha(r_{n})=n$ where $r_{n}(h)=hn$.

We will also recall the definitions from [7]: if $M$ is an $aYD$-module then

[TABLE]

has a left $H$-action via

[TABLE]

and furthermore $\widehat{M}$ has a contraaction $\alpha:\mathop{\rm Hom}\nolimits(H,\widehat{M})\to\widehat{M}$ defined as follows. Let $\theta\in\mathop{\rm Hom}\nolimits(H,\widehat{M})$ be viewed as $h\mapsto\theta_{h}(-)$ then

[TABLE]

It is not hard to check all these statements directly (note that the $aYD$-module condition for $M$ is only used to ensure that the action (2.15) preserves the $H$-comodule morphisms inside $\mathop{\rm Hom}\nolimits(H,M)$), and most importantly we can also check that $\alpha$ is compatible with the action in the $aYD$-contramodule sense, i.e., the identity (2.14) holds.

The constructions above describe a functor

[TABLE]

from $(s)aYD$-modules to $(s)aYD$-contramodules. Furthermore, the functor (2.17) is a special case of the pullback of contratraces [7] and so we have the following:

Proposition 2.10**.**

Given an $H$-module algebra $A$ and a $saYD$-module $M$, we have an isomorphism of cyclic cohomologies:

[TABLE]

where the theories considered are of the derived type.

We denote by $\widehat{HC}^{n}_{H}(A,\widehat{M})$ the cyclic cohomology obtained from an algebra $A$ and a $saYD$-contramodule $\widehat{M}$ via the associated representable symmetric $2$-contratrace $\mathop{\rm Hom}\nolimits_{H}(-,\widehat{M})$ on ${{}_{H}\mathcal{M}}$, while $HC^{n,H}(A\rtimes H,M)$ denotes the Hopf-cyclic cohomology of an $H$-comodule algebra $A\rtimes H$ with coefficients in a $saYD$-module $M$ obtained from the representable symmetric $2$-contratrace $\mathop{\rm Hom}\nolimits(-,M)^{H}$ on $\mathcal{M}^{H}$.

Remark 2.11**.**

In light of the Corollary 2.9 that shows the equivalence between $saYD$-modules and representable symmetric $2$-contratraces on $\mathcal{M}^{H}$ and [5] where a similar result is demonstrated for $saYD$-contramodules and ${{}_{H}\mathcal{M}}$, the Proposition 2.10 is a concrete realization of the pullback of representable contratraces of [7].

3 An adjoint pair of functors.

We will now analyze the functor $M\mapsto\widehat{M}$ with a view towards establishing some sufficient conditions for it being an equivalence. Consider the category $\mathcal{M}^{H}$ of right $H$-comodules and we are interested in comparing it to the category $\widehat{\mathcal{M}^{H}}$ of right $H$-contramodules. It turns out that the functor $M\mapsto\widehat{M}$, that appeared in [7] motivated by the pullback of contratraces has already appeared in the literature on comodule-contramodule correspondences [6], but considered without the extra $H$-module structure that we need. We will abuse notation somewhat and not usually distinguish between $\widehat{(-)}:\mathcal{M}^{H}\to\widehat{\mathcal{M}^{H}}$ of [6] and the upgraded version of [7] mentioned above (2.17). When we do want to emphasize the difference, the latter will be denoted by $\widehat{(-)}_{H}$.

Furthermore, $\widehat{(-)}$ has a left adjoint [6]

[TABLE]

where $N^{\prime}=H\odot_{H}N$ is the cokernel of the difference between the maps $Id\otimes\alpha$ and $h\otimes f\mapsto h^{2}\otimes f(h^{1})$ between $H\otimes_{k}\mathop{\rm Hom}\nolimits(H,N)$ and $H\otimes_{k}N$:

[TABLE]

The comodule structure on $N^{\prime}$ is given by

[TABLE]

When $H$ is finite dimensional then $N^{\prime}=H\otimes_{H^{*}}N$ so that the notation $\odot_{H}$ is a bit misleading.

The adjunctions are

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Remark 3.1**.**

Just as the functor $M\mapsto\widehat{M}$ was upgraded from the functor between comodules and contramodules to a functor between $aYD$-modules and $aYD$-contramodules by converting an $H$-action on $M$ to an $H$-action on $\widehat{M}$, we can do the same to its left adjoint directly. Namely, define an $H$-action on $H\otimes_{k}N$ via

[TABLE]

then one can check that if $N$ is an $aYD$-contramodule, then the action is well defined on the cokernel $H\odot_{H}N$ and gives $N^{\prime}$ the $aYD$-module structure.

We will now conceptually investigate if the adjoint pair of the functors above is compatible with the extra structure that we require. More precisely, $\mathcal{M}^{H}$ is a tensor category in the usual way with

[TABLE]

for $m\otimes n\in M\otimes N$ with $M,N\in\mathcal{M}^{H}$. Thus $\mathcal{M}^{H}$ is a bimodule category over itself.

On the other hand if $N\in\widehat{\mathcal{M}^{H}}$ and $T\in\mathcal{M}^{H}_{fd}$, i.e., $T$ is a finite dimensional $H$-comodule, then we can define a natural contramodule structure on both $N\otimes T$ and $T\otimes N$. Namely, due to the finite dimensionality of $T$, we represent elements of $\mathop{\rm Hom}\nolimits(H,N\otimes T)$ by $f\otimes t$ with $f\in\mathop{\rm Hom}\nolimits(H,N)$ and $t\in T$, then

[TABLE]

and similarly

[TABLE]

which makes $\widehat{\mathcal{M}^{H}}$ into a bimodule category over $\mathcal{M}^{H}_{fd}$.

The following is the key technical result of this section. It describes the exact nature of the compatibility of $M\mapsto\widehat{M}$ with the $\mathcal{M}^{H}_{fd}$-bimodule structure on both sides.

Proposition 3.2**.**

Let $W\in\mathcal{M}^{H}$ and $T,L\in\mathcal{M}^{H}_{fd}$ then we have:

[TABLE]

[TABLE]

a natural isomorphism in $\widehat{\mathcal{M}^{H}}$.

Proof.

Recall that $\mathop{\rm Hom}\nolimits^{L}(H,W)$ has a left $H^{*}$-action and a right $H$-contraaction and they commute. Namely,

[TABLE]

and

[TABLE]

One quickly checks that the map

[TABLE]

[TABLE]

is an isomorphism of both $H^{*}$-modules and $H$-contramodules, where $\overline{T}$ has the usual $H^{*}$ structure, but is considered trivial for the purposes of defining the $H$-contraaction on the right hand side.

On the other hand

[TABLE]

[TABLE]

is also an isomorphism of both structures where $\underline{T}$ has trivial $H^{*}$ structure but non-trivially modifies the contraaction on the right hand side.

So as $H$-contramodules we have:

[TABLE]

where $\mathop{\rm Hom}\nolimits_{H^{*}}(k,T\otimes V)\simeq\mathop{\rm Hom}\nolimits_{H^{*}}(k,V\otimes T^{S^{2}})$ is due to the rigidity of $\mathcal{M}^{H}_{fd}$ and the isomorphism $T^{**}\simeq T^{S^{2}}$.

Analogously we have:

[TABLE]

In the latter we have used the analogues of (3.6) and (3.7); namely the isomorphisms:

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

The result now follows after tracing through the isomorphisms. ∎

Denote by $\mathcal{M}^{H\#}$ the $\mathcal{M}^{H}_{fd}$ bimodule category with

[TABLE]

and by ${{}^{\#}\widehat{\mathcal{M}^{H}}}$ the $\mathcal{M}^{H}_{fd}$ bimodule category with

[TABLE]

then we immediately obtain the following as a Corollary of Proposition 3.2:

Corollary 3.3**.**

The functors

[TABLE]

and

[TABLE]

are bimodule functors over $\mathcal{M}^{H}_{fd}$ and so induce functors between the corresponding centers of bimodule categories.

Proof.

The claim about $\widehat{(-)}$ is immediate from Proposition 3.2. Since $\mathcal{M}^{H}_{fd}$ is rigid, the statement about $(-)^{\prime}$ follows from the one about $\widehat{(-)}$ through adjunction juggling, since they are adjoint functors.

∎

Remark 3.4**.**

The adjunction manipulations mentioned in the proof of Corollary 3.3 can be traced through to obtain an explicit analogue of Proposition 3.2 for the functor $N\mapsto N^{\prime}$. Namely, for $N\in\widehat{\mathcal{M}^{H}}$ and $T,L\in\mathcal{M}^{H}_{fd}$ we have a natural isomorphism in $\mathcal{M}^{H}$:

[TABLE]

[TABLE]

As in [4], we have central interpretations of $aYD$ objects.

Lemma 3.5**.**

The center of $\mathcal{M}^{H\#}$ is equivalent to the category of anti-Yetter-Drinfeld modules, namely

[TABLE]

Proof.

The proof proceeds very much like that of Proposition 2.7 and so we provide only a sketch. Let $M\in{{}_{H}\mathcal{M}^{H}}$, i.e., it is both a left module and a right comodule, and let $T\in\mathcal{M}^{H}_{fd}$. Consider the map

[TABLE]

[TABLE]

It is an isomorphism with inverse $m\otimes t\mapsto t_{0}\otimes S(t_{1})m$. It is a map in $\mathcal{M}^{H}$ if and only if $M\in aYD\text{-mod}$. It is immediate that $(M,\tau)\in\mathcal{Z}_{\mathcal{M}^{H}_{fd}}(\mathcal{M}^{H\#})$.

Conversely, let $M\in\mathcal{M}^{H}$ such that we have natural isomorphisms $\tau_{T}:T\otimes M\to M\otimes T^{S^{2}}$ for all $T\in\mathcal{M}^{H}_{fd}$. Now proceed in a by now familiar fashion. We need an action $H\otimes M\to M$ which we obtain via a limit over finite dimensional subcoalgebras $C\subset H$, i.e.,

[TABLE]

and the latter contains $(Id\otimes\epsilon_{C})\circ\tau_{C}$. ∎

Remark 3.6**.**

Note that what these limit arguments demonstrate is that in contrast to the $H$-module case considered in [5], the $H$-comodule case is much easier as it reduces to the rigid category $\mathcal{M}^{H}_{fd}$. More precisely, Proposition 2.7 shows that $\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})\simeq aYD\text{-mod}$ by essentially showing that $\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{Hop})\simeq\mathcal{Z}_{\mathcal{M}^{H}_{fd}}(\mathcal{M}^{Hop})$, but the latter is clearly $\mathcal{Z}_{\mathcal{M}^{H}_{fd}}(\mathcal{M}^{H\#})$, which as we saw above is equivalent to $\mathcal{Z}_{\mathcal{M}^{H}}(\mathcal{M}^{H\#})$.

Lemma 3.7**.**

The center of ${{}^{\#}\widehat{\mathcal{M}^{H}}}$ is equivalent to the category of anti-Yetter-Drinfeld contramodules, namely

[TABLE]

Proof.

Repeat the proof of Lemma 3.5 verbatim with the exception that

[TABLE]

[TABLE]

is a map in $\widehat{\mathcal{M}^{H}}$ if and only if $N\in aYD\text{-ctrmd}$. ∎

We summarize this section with the following Theorem.

Theorem 3.8**.**

The following diagram commutes:

[TABLE]

Recall that for $M\in aYD\text{-mod}$ we equip $\widehat{M}$ with (2.16) and (2.15), whereas for $N\in aYD\text{-ctrmd}$ we equip $N^{\prime}$ with (3.2) and (3.3).

Proof.

For the $\widehat{(-)}$ case we have the map $T\otimes M\to M\otimes T^{S^{2}}$ with $t\otimes m\mapsto t_{1}m\otimes t_{0}$ mapping to $\mathop{\rm Hom}\nolimits(H,T\otimes M)^{H}\to\mathop{\rm Hom}\nolimits(H,M\otimes T^{S^{2}})^{H}$ with $t\otimes f\mapsto t_{1}f\otimes t_{0}$ which maps under the identification of Proposition 3.2 to $T^{S^{2}}\otimes\mathop{\rm Hom}\nolimits(H,M)^{H}\to\mathop{\rm Hom}\nolimits(H,M)^{H}\otimes T$ with $t_{0}\otimes f(S^{2}(t_{1})-)\mapsto t_{2}f(-t_{1})\otimes t_{0}$ and the latter coincides with $t\otimes g\mapsto t_{1}\cdot g\otimes t_{0}$.

For the adjoint $(-)^{\prime}$ we have $T^{S^{2}}\otimes N\to N\otimes T$ with $t\otimes n\mapsto t_{1}n\otimes t_{0}$ mapping to $H\odot_{H}(T^{S^{2}}\otimes N)\to H\odot_{H}(N\otimes T)$ with $h\otimes t\otimes n\mapsto h\otimes t_{1}n\otimes t_{0}$ which identifies with $T\otimes(H\odot_{H}N)\to(H\odot_{H}N)\otimes T^{S^{2}}$ with $t_{0}\otimes S(t_{1})h\otimes n\mapsto hS(t_{1})\otimes t_{2}n\otimes t_{0}$ under the isomorphisms of Remark 3.4 and the latter coincides with $t\otimes(x\otimes m)\mapsto t_{1}\cdot(x\otimes m)\otimes t_{0}$. ∎

In the end we see that $((-)_{H}^{\prime},\widehat{(-)}_{H})$ is an adjoint pair between $aYD\text{-mod}$ and $aYD\text{-ctrmd}$ extending the result of [6].

3.0.1 Stability.

Recall that in order to obtain cyclic cohomology we need to consider the coefficients in stable anti-Yetter-Drinfeld modules or contramodules. We now address the preservation of the stability conditions under the adjoint pair of functors of the previous section.

Recall the map

[TABLE]

[TABLE]

with the inverse $m\mapsto S^{-1}(m_{1})m_{0}$; it defines an element $\sigma\in\mathop{\rm Aut}\nolimits(Id_{aYD\text{-mod}})$. Similarly, there is a

[TABLE]

[TABLE]

with the inverse $n\mapsto\alpha(h\mapsto S^{-1}(h)n)$; it defines an element $\sigma\in\mathop{\rm Aut}\nolimits(Id_{aYD\text{-ctrmd}})$.

It is an easy calculation to see that $\widehat{\sigma_{M}}:\widehat{M}\to\widehat{M}$ coincides with $\sigma_{\widehat{M}}:\widehat{M}\to\widehat{M}$ and also $(\sigma_{N})^{\prime}=\sigma_{N^{\prime}}$. For example to prove the latter equality observe that the left hand side is $h\otimes n\mapsto h\otimes\alpha(r_{n})=h^{2}\otimes r_{n}(h^{1})=h^{2}\otimes h^{1}n=(h\otimes n)_{1}(h\otimes n)_{0}$ which is the right hand side.

Recall that $saYD\text{-mod}$ is the full subcategory of $aYD\text{-mod}$ that consists of $M$ such that $\sigma_{M}=Id_{M}$. The definition of $saYD\text{-ctrmd}$ is identical. We have proved the following Corollary to Theorem 3.8:

Corollary 3.9**.**

The functors $((-)_{H}^{\prime},\widehat{(-)}_{H})$ is an adjoint pair between $saYD\text{-mod}$ and $saYD\text{-ctrmd}$.

4 A comodule-contramodule correspondence.

Here we will address the question of $\widehat{(-)}$ (equivalently $(-)^{\prime}$) being an equivalence. Note that in light of the preceding discussion if $\widehat{(-)}:\mathcal{M}^{H}\to\widehat{\mathcal{M}^{H}}$ is an equivalence, then so is $\widehat{(-)}_{H}:aYD\text{-mod}\to aYD\text{-ctrmd}$ and also $\widehat{(-)}_{H}:saYD\text{-mod}\to saYD\text{-ctrmd}$.

As usual, let us consider $k$ as the trivial $H$-comodule, and let $J=\widehat{k}$ be its contramodule image under $\widehat{(-)}$. Note that this is nothing but the two-sided ideal in $H^{*}$ consisting of right integrals [8]. Namely, $\chi\in J$ if and only if we have $\chi(h^{1})h^{2}=\chi(h)1$ for all $h\in H$. Strictly speaking it is left integrals that are considered in [8] but if $\chi$ is a left integral then $\chi(S(-))$ is right and vice versa. It is known [1] that $\text{dim}J\leq 1$ and if $J\neq 0$ then $S$ is invertible, which we have been assuming anyhow.

Remark 4.1**.**

Dually, we may consider $k$ as the trivial contramodule, i.e., $\alpha:H^{*}\to k$ is evaluation at $1\in H$. Let $K=k^{\prime}$ and note that $K=H/I$ where $I$ is generated by $\mu(h^{1})h^{2}-\mu(1)h$ for $\mu\in H^{*}$ and $h\in H$. Thus $K^{*}=I^{\perp}=\{\chi\in H^{*}|\mu(1)\chi(h)=\mu(h^{1})\chi(h^{2})\forall h\}$ and the latter is the ideal of left integrals.

We are ready for the first negative result:

Lemma 4.2**.**

If $J=0$ then $\widehat{(-)}$ is not an equivalence.

Proof.

Obviously we have that $\widehat{k}=0$, but furthermore, by Proposition 3.2 we have that for $M\in\mathcal{M}^{H}_{fd}$, $\widehat{M}\simeq M^{S^{2}}\otimes J=0$. ∎

On the other hand let us assume that $J\neq 0$. Let $\widehat{\mathcal{M}^{H}}_{rfd}$ denote the full subcategory of $\widehat{\mathcal{M}^{H}}$ consisting of finite dimensional, rational contramodules. By analogy with the $H^{*}$-module case, we say that a finite dimensional contramodule $M$ is rational if the structure map $\alpha$ factors through $\mathop{\rm Hom}\nolimits(C,M)$ for some $C$ a finite dimensional subcoalgebra of $H$.

Lemma 4.3**.**

Let $J\neq 0$ then $\widehat{(-)}:\mathcal{M}_{fd}^{H}\simeq\widehat{\mathcal{M}^{H}}_{rfd}$.

Proof.

Again, for $M\in\mathcal{M}_{fd}^{H}$ we have that $\widehat{M}\simeq M^{S^{2}}\otimes J$. Note that by [8] the contramodule $J$ is rational and thus so is $\widehat{M}$. On the other hand any rational finite dimensional contramodule is essentially a comodule (see Lemma 5.6) and so $(-\otimes{{}^{*}J})^{S^{-2}}$ is the inverse of $\widehat{(-)}$. ∎

The above Lemma should be considered as in general a negative result. Namely, if exotic, i.e., non-rational contramodules are possible, then the equivalence fails. More precisely, let us consider the possibility of exotic contramodule structures on $k$. Let $\chi\in J$ and observe that

[TABLE]

Since by [8], as $x$ ranges over $H$, the functional $\chi(xS(-))$ ranges over $H^{*rat}$ so if

[TABLE]

then $\exists\mu\in H^{*rat}$ such that $\mu\cdot 1=c\neq 0$. So that for any $\eta\in H^{*}$ we have $\eta\cdot 1=\eta\mu\frac{1}{c}=\eta(\mu_{1})\mu_{0}\frac{1}{c}$ and so the action of $H^{*}$ on $k$ factors through $C^{*}$ and the structure on $k$ is necessarily rational. On the other hand if $\chi(1)=0$ then it is possible that the whole of $H^{*rat}$ acts trivially without $H^{*}$ doing the same, resulting in an exotic structure.

This suggests two possibilities for $\widehat{(-)}$ being an equivalence:

• •

$\exists\chi\in J$ with $\chi(1)\neq 0$.

• •

$H$ is finite dimensional.

Note that the second case may appear trivial at first, but it isn’t. It is true that there is no difference between $H$-comodules, $H^{*}$-modules and $H$-contramodules in the case when $H$ is finite dimensional. However, we are not interested in the naive identification of the categories, rather the $\widehat{(-)}$ one. The latter functor is the one that translates the equivalence between comodules and contramodules to the equivalence between the $saYD$ versions that we need. Of course given all the work already done on this matter, the conclusion is easy to obtain, so we start with this case.

Proposition 4.4**.**

Let $H$ be finite dimensional, then $\widehat{(-)}$ is an equivalence, and so is $\widehat{(-)}_{H}$.

Proof.

From [8] we know that $J\neq 0$. Furthermore, for $M\in\mathcal{M}^{H}$ we have $M=\varinjlim M_{i}$ with $M_{i}\in\mathcal{M}^{H}_{fd}$ so that $\widehat{M}=\mathop{\rm Hom}\nolimits(H,M)^{H}=\mathop{\rm Hom}\nolimits(H,\varinjlim M_{i})^{H}$ which by the finite dimensionality of $H$ is $\varinjlim\mathop{\rm Hom}\nolimits(H,M_{i})^{H}\simeq\varinjlim(M_{i}^{S^{2}}\otimes J)=M^{S^{2}}\otimes J$. Since there are no exotic contramodules here this proves the equivalence. ∎

Moving on to the first case we get by [8] that the $\chi(1)\neq 0$ condition is actually very strict. Namely, we have that $H$ is such that as a coalgebra $H=\bigoplus_{i}C_{i}$ where $C_{i}$ are finite dimensional simple subcoalgebras. Let $\epsilon_{i}$ denote the counit of $C_{i}$ with $\epsilon=\sum\epsilon_{i}$. For $x\in H$ let $x=\sum_{i}x_{i}$ denote its decomposition with respect to that of $H$.

Theorem 4.5**.**

The category of $H$-comodules and $H$-contramodules are equivalent. The former consists of $\bigoplus_{i}M_{i}$ and the latter of $\prod_{i}M_{i}$ where $M_{i}$ are right $C_{i}$-comodules, i.e., $M_{i}\in\mathcal{M}^{C_{i}}$.

Proof.

The assertion about the comodules is immediate. Now let $M$ be an $H$-contramodule, define $\alpha_{i}:M\to M$ via $\alpha_{i}(m)=\alpha(\epsilon_{i}(-)m)$. Note that

[TABLE]

Let $M_{i}=\alpha_{i}(M)$ and consider $\beta:M\to\prod M_{i}$ such that

[TABLE]

and $\iota:\prod M_{i}\to\mathop{\rm Hom}\nolimits(H,M)$ via

[TABLE]

We have that

[TABLE]

On the other hand we have that $\beta\alpha\iota((m_{i})_{i})=(\alpha_{i}(\alpha(x\mapsto\sum\epsilon_{j}(x)m_{j})))_{i}$ and so we need to show that

[TABLE]

but the latter is

[TABLE]

Thus $\beta:M\simeq\prod_{i}M_{i}$ and using this identification we see that $\alpha:\mathop{\rm Hom}\nolimits(H,M)\to M$ becomes

[TABLE]

[TABLE]

so that if we denote by $\alpha^{i}:\mathop{\rm Hom}\nolimits(C_{i},M_{i})\to M_{i}$ the map $\alpha^{i}(f)=\alpha(h\mapsto f(h_{i}))$ then we see that the original $\alpha$ identifies with $\prod_{i}\alpha^{i}:\prod_{i}\mathop{\rm Hom}\nolimits(C_{i},M_{i})\to\prod_{i}M_{i}.$ It is immediate that $\alpha^{i}$ is a $C_{i}$-contramodule structure on $M_{i}$ and since $C_{i}$ is finite dimensional is the same as a $C_{i}$-comodule structure.

Conversely, given the data of $\rho_{i}:M_{i}\to M_{i}\otimes C_{i}$ we can define

[TABLE]

[TABLE]

and assemble the $\alpha^{i}$ into an $\alpha:\mathop{\rm Hom}\nolimits(H,\prod_{i}M_{i})\to\prod_{i}M_{i}$ that satisfies the contramodule axioms.

Now let $\phi:M\to N$ be a map of contramodules and let $m\in M$ with $m=(m_{i})_{i}$ under the $\beta$ identification, then

[TABLE]

so that $\phi=\prod_{i}\phi_{i}$ with $\phi_{i}:M_{i}\to N_{i}$. It is immediate that $\phi_{i}\in\mathop{\rm Hom}\nolimits(M_{i},N_{i})^{C_{i}}$ and that conversely, any such $(\phi_{i})_{i}$ data can be reassembled into a $\phi:M\to N$ a map of contramodules.

∎

Remark 4.6**.**

The proof of Theorem 4.5 demonstrates a difference between $H$-contramodules and $H^{*}$-modules. While there is a forgetful functor from the former to the latter, the contramodule condition is better behaved than the $H^{*}$-module one in the case of the infinite dimensional $H$. Considering finite dimensional contramodules, that at first glance appear to be given an action indistinguishable from that of an $H^{*}$-module, it is the associativity that is strictly strengthened in the contramodule case. More precisely, there exist exotic $1$-dimensional $kG^{*}$-modules (for example given by non-principal ultrafilters on $G$), yet any $1$-dimensional $kG$-contramodule is supported at some $g\in G$, just as is the case for $kG$-comodules. The difference is due to the fact that in the contramodule case we have the freedom to work with the full $(H\otimes H)^{*}$ as opposed to only $H^{*}\otimes H^{*}$. Of course in the case when $H$ is finite dimensional all three categories: $H$-contramodules, $H$-comodules and $H^{*}$-modules are equal.

We need to connect the above to our adjoint pair of functors.

Proposition 4.7**.**

The correspondence

[TABLE]

of Theorem 4.5 is given, up to equivalence, by the adjoint functor pair $(\widehat{(-)},(-)^{\prime})$. Thus $\widehat{(-)}$ is an equivalence and so is $\widehat{(-)}_{H}$.

Proof.

Observe that

[TABLE]

where $A_{i}=C^{*}_{i}$ is a unital simple finite dimensional algebra. Let

[TABLE]

be given by $\mu_{i}(a)(b)=\text{tr}_{A_{i}}(l_{ab})$, i.e., it is the trace of left multiplication by $ab\in A_{i}$. Note that $\mu_{i}$ is an $A_{i}$-bimodule map and $\mu_{i}(1)(1)=\text{tr}_{A_{i}}(1)=\text{dim}A_{i}\neq 0$ since $\text{char}k=0$, so that $\mu_{i}$ is an isomorphism by the simplicity of $A_{i}$. So $\widehat{M}\simeq\prod_{i}\mathop{\rm Hom}\nolimits_{A_{i}}(A_{i},M_{i})\simeq\prod_{i}M_{i}$.

Similarly

[TABLE]

∎

4.1 The case of $H=kG$.

Let $G$ be an infinite discrete group. We ask that $G$ be infinite as otherwise all of our considerations here become more or less trivial. Let $M$ be a $kG$-contramodule, i.e., we view $kG$ as a counital coalgebra with $\Delta(g)=g\otimes g$ and $\epsilon(g)=1$. We have the following corollary of Theorem 4.5:

Corollary 4.8**.**

The category of $kG$-contramodules $\widehat{\mathcal{M}^{kG}}$ is equivalent to the category of $G$-graded vector spaces $\text{Vec}_{G}$. The equivalence is given by

[TABLE]

Compare this with the well known equivalence

[TABLE]

where $\Gamma_{c}$ are global sections with compact support.

Proof.

Note that $kG=\bigoplus_{g\in G}kg$ with $kg=k$ as coalgebras. ∎

It is well known that the category of anti Yetter-Drinfeld modules for $kG$ (since $S^{2}=Id$ it coincides with the category of Yetter-Drinfeld modules, and thus with the center of the monoidal category of $kG$-modules) is equivalent to the category $\text{Vec}_{G/G}$ of $G$-equivariant $G$-graded vector spaces. More precisely, the $kG$-comodule part of the structure gives the $G$-grading, and the $kG$-module part gives the $G$-action, while the Yetter-Drinfeld compatibility ensures that the action obeys

[TABLE]

We have an immediate Corollary to Proposition 4.7:

Corollary 4.9**.**

The category of $aYD$-contramodules for $kG$ is equivalent to the category of $G$-graded $G$-equivariant vector spaces via

[TABLE]

We now would like to address the question of stability. A stable $aYD$-module for $kG$ is known to be $G$-graded $G$-equivariant vector space with the stability condition translating into

[TABLE]

for all $x\in G$ and all $m_{x}\in M_{x}$. Denote by $\text{Vec}^{\prime}_{G/G}$ the full subcategory of $\text{Vec}_{G/G}$ consisting of objects for which (4.1) holds. We have another immediate Corollary to Proposition 4.7:

Corollary 4.10**.**

The functor

[TABLE]

is an equivalence.

We can now restate the Proposition 2.10 more elegantly in the case of $H=kG$.

Proposition 4.11**.**

Let $A$ be a $G$-equivariant algebra, and $\mathcal{M}\in\text{Vec}^{\prime}_{G/G}$. Then

[TABLE]

Remark 4.12**.**

While the right hand side of the above Proposition is definition invariant, the left hand side uses the definition of [7] and not the more classical one used in [4].

4.2 A new “coefficient”.

Since the introduction of coefficients in symmetric $2$-contratraces in [4], there remained an obvious question: do these simply generalize the already well known coefficients in $saYD$-modules or contramodules to other settings, or do these traces furnish examples of coefficients that had not yet been considered even in the classical theories? In [7] we gave a derived version of the definition of cyclic cohomology with coefficients that restricted the possible symmetric $2$-contratraces to the left exact ones. The results obtained in [5] immediately tell us that in the case of $H$-module algebras we need to look beyond the representable symmetric $2$-contratraces if we are to obtain anything but the usual $saYD$-contramodule coefficients. In the present paper, Corollary 2.9 implies the same about $H$-comodule algebras, i.e., we need a non-representable contratrace to get away from the usual $saYD$-module coefficients. We will construct one below.

Let $G$ have infinitely many conjugacy classes (such as when $G=\mathbb{Z}$ for example). Let $\mathcal{M}_{\left<g\right>}\in\text{Vec}^{\prime}_{G/G}$ be supported on the conjugacy class $\left<g\right>$, for example we can let

[TABLE]

with the trivial $G$-action. Then each $\mathcal{M}_{\left<g\right>}$ yields a representable left exact symmetric $2$-contratrace

[TABLE]

yet

[TABLE]

is an example of a non-representable, left exact symmetric $2$-contratrace on $\mathcal{M}^{H}$. Note that taking $\mathcal{M}$ to be the superposition of all $\mathcal{M}_{\left<g\right>}$’s would result in $V\mapsto\prod_{\left<g\right>}\mathcal{F}_{\left<g\right>}(V)$.

5 Periodicities.

In this section we revisit the $YD_{i}$-modules from [4] and see that under the conditions that we have been looking at in this paper, there is nothing new that arises and we still only have the Yetter-Drinfeld and the anti-Yetter-Drinfeld modules and contramodules; this is the first observed periodicity. In addition, we examine a natural symmetry on these objects and observe that it too is periodic; this we refer to as the second periodicity.

We recall the definition of $YD_{i}$-modules:

Definition 5.1**.**

Let $M$ be a left module and a right comodule over $H$, and let $i\in\mathbb{Z}$. We say that $M$ is a $YD_{i}$-module if

[TABLE]

for $h\in H$ and $m\in M$.

Remark 5.2**.**

Equivalently, we can define $YD_{i}$-modules by requiring that the comodule structure map $M\to M\otimes H$ is $H$-equivariant with respect to the $H$-structure on the right hand side given by $x\cdot(m\otimes h)=x^{2}m\otimes x^{3}hS^{-1-2i}(x^{1})$.

Note that $YD_{-1}$-modules are anti-Yetter-Drinfeld modules, while $YD_{0}$-modules are Yetter-Drinfeld modules.

We can rephrase the above a little more conceptually. Let $\mathbb{Z}$ act on $\mathcal{M}^{H}$ with $1\cdot M=M^{S^{2}}$ so that we may consider the monoidal category $\mathcal{M}^{H}\rtimes\mathbb{Z}$. We get an immediate generalization of Lemma 3.5:

Lemma 5.3**.**

We have an equivalence of monoidal categories:

[TABLE]

There are a few consequences of the above. First, if $M\in YD_{i}\text{-mod}$ and $N\in YD_{j}\text{-mod}$ then $M\otimes N\in YD_{i+j}\text{-mod}$ with the usual comodule structure, but ${{}_{S^{-2i}}N}\otimes M$ as an $H$-module, i.e.,

[TABLE]

but

[TABLE]

Second, if $M\in YD_{i}\text{-mod}$ then so is $1\cdot M={{}_{S^{-2}}M}^{S^{2}}\in YD_{i}\text{-mod}$. Third, $\mathcal{M}^{H}$ has internal Homs, and so does $\mathcal{M}^{H}\rtimes\mathbb{Z}$, i.e.,

[TABLE]

and the same for right Homs. Consequently, $\mathcal{Z}_{\mathcal{M}^{H}_{fd}}(\mathcal{M}^{H}\rtimes\mathbb{Z})$ has internal Homs as well. In particular $\mathcal{M}^{H}_{fd}$ is rigid, so is $\mathcal{M}_{fd}^{H}\rtimes\mathbb{Z}$ with $(V,i)^{*}=((-i)V^{*},-i)$ and ${{}^{*}(V,i)}=((-i){{}^{*}V},-i)$ and so is $\mathcal{Z}_{\mathcal{M}^{H}_{fd}}(\mathcal{M}_{fd}^{H}\rtimes\mathbb{Z})$. Thus if $M\in YD^{fd}_{i}\text{-mod}$ then we have elements $M^{\star}$ and ${{}^{\star}M}$ in $YD_{-i}^{fd}\text{-mod}$ that are its right and left duals.

Just as we have generalized $aYD\text{-mod}$ to $YD_{i}\text{-mod}$, we can do the same to $aYD\text{-ctrmd}$.

Definition 5.4**.**

Let $M$ be a left $H$-module and a right $H$-contramodule, we say that $M$ is a $YD_{i}$-contramodule if the contramodule structure $\alpha:\mathop{\rm Hom}\nolimits(H,M)\to M$ is $H$-equivariant with respect to the $H$-action on the left given by

[TABLE]

where $h\in H$ and $f\in\mathop{\rm Hom}\nolimits(H,M)$.

Note that $YD_{1}$-contramodules are $aYD$-contramodules.

5.1 The first periodicity.

We can easily generalize the content of Section 3 as follows. We have the Proposition/Definition below.

Proposition 5.5**.**

Let $M$ be a $YD_{i-1}$-module, define $\widehat{M}=\mathop{\rm Hom}\nolimits(H,M)^{H}$ and equip the latter with a left $H$-action via

[TABLE]

and an $H$-contraaction as before (2.16). Let $N$ be a $YD_{i+1}$-contramodule, define $N^{\prime}=H\odot_{H}N$ and equip the latter with a left $H$-action via

[TABLE]

and an $H$-coaction as before (3.2).

This defines an adjoint pair of functors $((-)_{H}^{\prime},\widehat{(-)}_{H})$:

[TABLE]

Let us again (see Remark 4.1 and the preceding discussion) consider the trivial $YD_{0}$-module $k$ from which we obtain by the Proposition 5.5 the object $J=\widehat{k}$ which is now seen to be in $YD^{fd}_{2}\text{-ctrmd}$. Conversely, again considering the trivial $YD_{0}$-contramodule $k$, we obtain $K=k^{\prime}$ which is now seen to be in $YD^{fd}_{-2}\text{-mod}$. If we denote by $YD_{i}^{rfd}\text{-ctrmd}$ the full subcategory of $YD_{i}\text{-ctrmd}$ that consists of objects that as contramodules are in $\widehat{\mathcal{M}^{H}}_{rfd}$ then $J\in YD_{2}^{rfd}\text{-ctrmd}$. Observe that we have an easy Lemma:

Lemma 5.6**.**

We have an equivalence (equality actually) of categories:

[TABLE]

that does not change the underlying vector space $M$, nor the $H$-action, and sends the coaction to the contraaction:

[TABLE]

[TABLE]

As a consequence, we have $\iota^{-1}J\in YD_{2}^{fd}\text{-mod}$ which is the dual of $K\in YD_{-2}^{fd}\text{-mod}$.

Remark 5.7**.**

If $H$ is a Hopf algebra with $J\neq 0$ then both $YD_{i}\text{-mod}$ and $YD_{i}\text{-ctrmd}$ are $2$-periodic, i.e.,

[TABLE]

and the same for contramodules.

For a finite dimensional $H$, the functor $\widehat{(-)}_{H}$ is essentially the periodicity above. Not so for the infinite dimensional case.

5.2 The second periodicity.

Recall our discussion of stability in Section 3.0.1. We observe that the

[TABLE]

that was used to define stability for $aYD$-modules (and its contramodule variant) can be generalized, with an interesting difference, to an arbitrary $i$ for both modules and contramodules. More precisely,

[TABLE]

i.e., for $M\in YD_{i-1}\text{-mod}$ we have $\sigma_{M}:M\to{{}_{S^{2i}}M^{S^{-2i}}}$, with $m\mapsto S^{2i}(m_{1})m_{0}$ and the inverse $m\mapsto S^{-1}(m_{1})m_{0}$, is an identification in $YD_{i-1}\text{-mod}$.

Remark 5.8**.**

The above implies that if $M$ is a Yetter-Drinfeld module, then it is canonically isomorphic to ${{}_{S^{-2}}M^{S^{2}}}$ as a Yetter-Drinfeld module. More generally, the action of $\mathbb{Z}$ on $YD_{i-1}\text{-mod}$ factors through $\mathbb{Z}/i\mathbb{Z}$. We will see below that the same holds for $YD_{i+1}\text{-ctrmd}$.

Note that the $M\mapsto{{}_{S^{-2}}M^{S^{2}}}$ symmetry of $YD_{i}$-modules also exists for $YD_{i}$-contramodules, i.e., ${{}_{S^{-2}}N^{S^{2}}}$ has its $H$-action modified by $S^{-2}$ and $\alpha^{S^{2}}(f)=\alpha(f(S^{2}(-)))$. Then we have

[TABLE]

i.e., for $N\in YD_{i+1}\text{-ctrmd}$ we have $\sigma_{N}:N\to{{}_{S^{2i}}N^{S^{-2i}}}$, with $n\mapsto\alpha(r_{n})$ and the inverse $n\mapsto\alpha(h\mapsto S^{2i-1}(h)n)$, is an identification in $YD_{i+1}\text{-ctrmd}$.

Furthermore, the generalized functor $\widehat{(-)}_{H}$ of Proposition 5.5 is compatible with these symmetries, namely the diagram of isomorphisms

[TABLE]

commutes in $YD_{i+1}\text{-ctrmd}$, where $M\in YD_{i-1}\text{-mod}$.