Monadic cointegrals and applications to quasi-Hopf algebras

Abstract

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1,\dots ,A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1}\to A_i\left(D\right)$, where D is the distinguished invertible object of $\mathcal{C}$.

We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L}={\int }^X{X}^{\vee }\otimes X$ in $\mathcal{C}$ studied by Lyubashenko (1995).

Our main motivation stems from the application to finite dimensional quasi-Hopf algebras H. For the category of finite-dimensional H-modules, we relate the four monadic cointegrals (two of which require H to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called γ-symmetrised cointegrals in the pivotal case, for γ the modulus of H.

For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$. In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of S and T which uses the monadic cointegral.

Introduction

Let H be a finite-dimensional Hopf algebra over an algebraically closed field k. There are two notions of cointegrals for H, namely left cointegrals and right cointegrals. These are elements $\mathit{\lambda }\in H^{⁎}$ which satisfy, for all $h\in H$,

$$2)(\mathit{\lambda }\otimes \mathrm{id})\circ \mathrm{\Delta }(h)=\mathit{\lambda }(h)1\text{(right cointegral)},3)(\mathrm{id}\otimes \mathit{\lambda })\circ \mathrm{\Delta }(h)=\mathit{\lambda }(h)1\text{(left cointegral)}.$$

The unusual numbering will be explained below.

If H is a pivotal Hopf algebra, that is, if it is equipped with a group-like element g expressing the square of the antipode via conjugation, one can introduce two more notions of cointegrals, so-called γ-symmetrised left and right cointegrals [2], [20]. Here $\mathit{\gamma }\in H^{⁎}$ denotes the modulus of H, which encodes the difference between left and right integrals for H. The defining equation for $\mathit{\lambda }\in H^{⁎}$ to be a γ-symmetrised left/right cointegral is

$$1)(\mathit{\lambda }\otimes \mathrm{id})\circ \mathrm{\Delta }(h)=\mathit{\lambda }(h){\mathit{g}}^{-1}\text{(right }\mathit{\gamma }\text{-symmetrised cointegral)},4)(\mathrm{id}\otimes \mathit{\lambda })\circ \mathrm{\Delta }(h)=\mathit{\lambda }(h)\mathit{g}\text{(left }\mathit{\gamma }\text{-symmetrised cointegral)}.$$

Such γ-symmetrised cointegrals, and in particular the special case where γ is given by the counit, have applications to modified traces and to quantum invariants of links and three-manifolds [2], [3], [13]. Furthermore, γ-symmetrised cointegrals are an example of g-cointegrals for a group-like g as introduced in [34].

The category ${}_H\mathcal{M}$ of finite-dimensional left H-modules is a finite tensor category in the sense of [15], in particular it has left and right duals. If H is in addition pivotal, then ${}_H\mathcal{M}$ becomes a pivotal tensor category [1]. One can now ask if it is possible to describe the above two (or four, in the pivotal case) notions of cointegrals in terms of the category ${}_H\mathcal{M}$ in a such a way that it generalises to arbitrary (pivotal) finite tensor categories $\mathcal{C}$. This is indeed possible, and has been done for left cointegrals in [38] using Hopf comonads on $\mathcal{C}$. In the present paper we define analogous notions for all four variants of cointegrals, working instead with Hopf monads.

Hopf monads on a rigid monoidal category $\mathcal{C}$ were introduced in [9]. These are monads M on $\mathcal{C}$, equipped with extra structure: First, the functor $M:\mathcal{C}\to \mathcal{C}$ is equipped with a lax comonoidal structure, i.e. there are morphisms

$$\text{(comultiplication)}M(X\otimes Y)\to M(X)\otimes M(Y),\text{natural in }X,Y,\text{(counit)}M(\mathbf{1})\to \mathbf{1},$$

satisfying certain conditions. The multiplication and the unit of the monad M have to be comonoidal as well. Finally, M has (unique) left and right antipodes, again given by certain natural transformations, see Section 2.1 for details.

A module over a monad M is an object $V\in \mathcal{C}$ together with an action $M(V)\to V$ of the monad. For a Hopf monad, the category ${\mathcal{C}}_M$ of M-modules is again a rigid monoidal category [9].

Let now $\mathcal{C}$ be a finite tensor category. The four monads we are interested in are all given in terms of coends. Namely, for $V\in \mathcal{C}$ we set

$$A_1\left(V\right)=\stackrel{X\in \mathcal{C}}{\int }{}^{\vee }X\otimes (V\otimes X),A_2\left(V\right)=\stackrel{X\in \mathcal{C}}{\int }{X}^{\vee }\otimes (V\otimes X),A_3\left(V\right)=\stackrel{X\in \mathcal{C}}{\int }(X\otimes V)\otimes {}^{\vee }X,A_4\left(V\right)=\stackrel{X\in \mathcal{C}}{\int }(X\otimes V)\otimes X^{\vee }.$$

Here ${}^{\vee }X$ denotes the right dual and $X^{\vee }$ the left dual of an object $X\in \mathcal{C}$ (see Section 1.5 below for our conventions). Finiteness of $\mathcal{C}$ ensures existence of these coends. The index $1,\dots ,4$ indicates the “position” of the duality symbol ∨.

$A_2$ and $A_3$ are Hopf monads [10, Sec. 5.4], called central monads, and are isomorphic as Hopf monads. If $\mathcal{C}$ is pivotal, then $A_1$ and $A_4$ are also Hopf monads, and all four $A_i$ are isomorphic as Hopf monads, see Proposition 2.4. In the following, if we discuss properties of $A_1$ and $A_4$, we will implicitly assume that $\mathcal{C}$ is in addition pivotal.

We define four types of monadic cointegrals as follows. Denote by $D\in \mathcal{C}$ the distinguished invertible object of $\mathcal{C}$.¹ The tensor unit $\mathbf{1}\in \mathcal{C}$ is an $A_i$-module via the counit. $A_i(D)$ is an $A_i$-module by the monad multiplication. A monadic cointegral for $A_i$ is an intertwiner of $A_i$-modules ${\mathit{\lambda }}_i:\mathbf{1}\to A_i(D)$. This means that ${\mathit{\lambda }}_i$ is a morphism in $\mathcal{C}$ which satisfies

where ${ϵ}_i$ and ${\mu }_i$ denote the counit and multiplication of $A_i$, respectively.

In the case $D=\mathbf{1}$, this definition of monadic cointegral agrees with the definition of a cointegral for a Hopf monad that first appeared in [9, Sec. 6.3]. For $i=2$ (and general D), the above definition is related by an isomorphism to the notion of “categorical cointegral” defined in [38] (Corollary 2.13).

Theorem 1.1

For a finite tensor category $\mathcal{C}$, non-zero monadic cointegrals for $A_2,A_3$ (and for $A_1,A_4$ if $\mathcal{C}$ is pivotal) exist and are unique up to scalar multiples.

This follows from a corresponding result in [38], see Proposition 2.10.

We now come to a key observation, which explains the reason for introducing four slightly different Hopf monads $A_i$, even though they are all isomorphic. Namely, for H a finite-dimensional Hopf algebra and $\mathcal{C}={}_H\mathcal{M}$, each monad $A_i$ has a particularly natural realisation (Example 2.5). With respect to this realisation one finds, firstly, that as a vector space $A_i(D)=H^{⁎}$, so that a λ as in (1.2) is an element of $H^{⁎}$, and, secondly, the equivalences

$$\mathit{\lambda }\text{ is a mon.}\text{coint.}\text{for }\{\begin{array}{c}{A}_1\hfill \\ A_2\hfill \\ A_3\hfill \\ A_4\hfill \end{array}\iff \mathit{\lambda }\text{ is a }\{\begin{array}{c}\text{right}\mathit{\text{γ}}\text{-sym.}\hfill \\ \text{right}\hfill \\ \text{left}\hfill \\ \text{left}\mathit{\text{γ}}\text{-sym.}\hfill \end{array}\text{coint. for }H\text{ ,}$$

see Example 2.9. As before, in cases 1 and 4 we require H to be pivotal.

The universal properties of the coends $A_i$ give isomorphisms between the various spaces of monadic cointegrals, and hence also between the various spaces of cointegrals for H. This will be useful in our application to quasi-Hopf algebras.

For Hopf algebras, the relation between integrals and cointegrals is very simple: passing to the dual Hopf algebra exchanges the two notions. For quasi-Hopf algebras, this is no longer the case as the definition of a quasi-Hopf algebra is not symmetric under duality. While integrals for a quasi-Hopf algebra H are defined in the same way as for Hopf algebras, cointegrals $\mathit{\lambda }\in H^{⁎}$ are markedly more complicated.

The definition of left/right cointegrals for a quasi-Hopf algebra H is given in [25], and that of γ-symmetrised left/right cointegrals in [39], [6] and in Section 4.2 below.

Fix a finite-dimensional quasi-Hopf algebra H over an algebraically closed field. As for Hopf algebras, ${}_H\mathcal{M}$ is also a finite tensor category, and each of the monads $A_i$ on ${}_H\mathcal{M}$ has a natural realisation such that the underlying vector space of $A_i(D)$ is $H^{⁎}$ in all four cases, cf. Section 3.3. With these realisations, we describe the monadic cointegrals for H via equations involving quasi-Hopf data. For example, an element $\mathit{\lambda }\in H^{⁎}$ which is an H-module intertwiner $\mathbf{1}\to A_2(D)$ is a monadic cointegral for $A_2$ if and only if the equation (3.26) holds.

Our main result is a generalisation of the relations in (1.3) for quasi-Hopf algebras, namely the precise relation between the monadic cointegrals and the quasi-Hopf cointegrals from [25], [39], [6]:

Theorem 1.2

We have the bijections from the various types of cointegrals $\mathit{\lambda }\in H^{⁎}$ for the finite-dimensional quasi-Hopf algebra H to the corresponding types of monadic cointegrals in ${}_H\mathcal{M}$ as shown in Table 1.

This is shown in Theorem 4.1, Theorem 4.4.

Remark 1.3

Analogous to [38] one can introduce monadic integrals by noting that $A_i(\mathbf{1})$ is a coalgebra and defining left/right monadic integrals for $A_i$ to be left/right comodule morphisms in $\mathcal{C}(A_i(D),\mathbf{1})$. In the quasi-Hopf case, monadic integrals are elements of $H^{⁎⁎}\cong H$, and one finds that the left/right monadic integrals for $A_i$ are the usual left/right integrals for the quasi-Hopf algebra H (for $i=1,2$), and the right/left integrals (for $i=3,4$). In the Hopf case, a similar result was also noted in [38].

A slightly different notion of integrals for Hopf monads has already appeared earlier in the literature [9]. If $\mathcal{C}$ is braided, then by [9, Ex. 5.4] this notion agrees with the definition of cointegrals for Hopf algebras in braided categories as in [26]. By arguments analogous to those given in Section 6, it in turn also agrees with the monadic integrals from the beginning of this remark. Here we will not go into the details of either of these points and focus instead on the study of monadic cointegrals.

Let $\mathcal{C}$ now be a braided finite tensor category. In [28], [29] the coend $\mathcal{L}={\int }^{X\in \mathcal{C}}{X}^{\vee }\otimes X$ was studied and shown to be a Hopf algebra in $\mathcal{C}$. One can use the braiding of $\mathcal{C}$ to construct a natural family of isomorphisms

$${\xi }_V:A_2(V)\to \mathcal{L}\otimes V.$$

This provides an isomorphism $A_2\cong \mathcal{L}\otimes ?$ of Hopf monads.

For Hopf algebras H in braided categories, one can define integrals and cointegrals just as for Hopf algebras over a field, up to one additional subtlety. Namely, integrals are certain morphisms from a so-called object of integrals $\mathrm{Int}H\in \mathcal{C}$ to H. Conversely, cointegrals are certain morphisms $H\to \mathrm{Int}H$, see [26] and Section 6. For example, a left integral for H is a morphism $\mathrm{\Lambda }:\mathrm{Int}H\to H$ such that

commutes. Here ε and m denote the counit and multiplication of H.

We find that monadic cointegrals for $A_2$ are related to left integrals for $\mathcal{L}$, and that the object of integrals for $\mathcal{L}$ is $\mathrm{Int}\mathcal{L}={}^{\vee }D$, the right dual of the distinguished invertible object of $\mathcal{C}$ (Proposition 6.1):

Proposition 1.4

Let $\mathcal{C}$ be a braided finite tensor category. Then $\mathit{\Lambda }:{}^{\vee }D\to \mathcal{L}$ is a left integral for $\mathcal{L}$ if and only if

is a monadic cointegral for $A_2$.

Proposition 6.1 also contains a similar statement for the relation of right integrals of $\mathcal{L}$ with monadic cointegrals for $A_2$.

An important application of $\mathcal{L}$ and its integrals arises in the case that $\mathcal{C}$ is modular, that is, a (not necessarily semisimple) finite ribbon category whose braiding satisfies a non-degeneracy condition called factorisability (see Section 7).

In this case, one can define a projective representation of the genus-g surface mapping class group on the Hom-space $\mathcal{C}({\mathcal{L}}^{\otimes g},1)$ [30], as well as a non-semisimple variant of the Reshetikhin-Turaev topological field theory [13], [12]. The integral for $\mathcal{L}$ (which is two-sided for $\mathcal{C}$ modular) enters in both constructions.

Let us specialise to the case that $\mathcal{C}={}_H\mathcal{M}$ for H a finite-dimensional quasi-triangular quasi-Hopf algebra which is in addition ribbon (and so in particular pivotal). In Section 6.3 we explicitly relate left integrals for $\mathcal{L}$, right monadic cointegrals in ${}_H\mathcal{M}$, and right cointegrals for H. In Section 7.2 we assume furthermore that H is factorisable (as defined in [7]), which is equivalent to ${}_H\mathcal{M}$ being factorisable [18]. In this case, $D=\mathbf{1}$ and both integrals and cointegrals for $\mathcal{L}$ are two-sided. We may take $A_2(\mathbf{1})=\mathcal{L}$ and by the above proposition, integrals for $\mathcal{L}$ are precisely the same as monadic cointegrals for $A_2$.

We present the projective representation of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$ as an example for the mapping class group actions mentioned above by giving the action of the S and T generators, simplifying the corresponding expressions in [18]. Denote by $Z(H)$ the centre of H and write $\mathit{\alpha }Z=\{\mathit{\alpha }z|z\in Z(H)\}$, where α is the evaluation element of H. Recall that $\mathcal{L}=H^{⁎}$ as a vector space. One finds that via the natural isomorphism $H\cong H^{⁎⁎}$ one has $\mathit{\alpha }Z\cong \mathcal{C}(\mathcal{L},\mathbf{1})$. In Proposition 7.1 we compute the action of S and T on $\mathit{\alpha }Z$ to be, for $z\in Z(H)$,

$$\mathbf{S}.(\mathit{\alpha }z)=〈\mathit{\lambda }|{\hat{\omega }}_{1}z〉{\hat{\omega }}_2,\mathbf{T}.(\mathit{\alpha }z)={\mathit{v}}^{-1}\mathit{\alpha }z.$$

Here, λ is a monadic cointegral for $A_2$, ${\hat{\omega }}_{1,2}$ are the components of the Hopf-pairing $\omega :\mathcal{L}\otimes \mathcal{L}\to \mathbf{1}$, and v is the ribbon element of H, see Section 7 for details.

Finally, let us note that the construction in [12] of a three-dimensional topological field theory from a modular category $\mathcal{C}$ uses the integral for $\mathcal{L}$ and the modified trace on (the projective ideal in) $\mathcal{C}$. For $\mathcal{C}={}_H\mathcal{M}$ with H a factorisable ribbon quasi-Hopf algebra, monadic cointegrals for $A_2,A_3$ provide the integral for $\mathcal{L}$, and monadic cointegrals for $A_1,A_4$ provide the modified trace via the construction using symmetrised cointegrals in [39], [6]. An important class of factorizable quasi-Hopf algebras as inputs for such topological field theories comes from the fundamental examples of logarithmic conformal field theories [23], [19], [11], [22], [33].

The fact that monadic cointegrals provide integrals for $\mathcal{L}$ and modified traces in a uniform setting was one of the motivations to carry out the present investigation. Another motivation was that because of their direct categorical interpretation, in certain situations monadic cointegrals for quasi-Hopf algebras may be easier to use than those in the left column of Table 1.

For quasi-Hopf algebras, a relation between right cointegrals for H and categorical cointegrals in the sense of [38] was derived in [39]. The main theorem in the present paper (Theorem 1.2) is an analogous result for the four types of monadic cointegrals. Comparing to [39], we work in a dual setting that uses central Hopf monads instead of comonads. This last choice of monads over comonads is merely conventional. However, our approach to the application of monadic cointegrals to quasi-Hopf cointegrals is quite different from the one in [39]: the latter uses a detour via the category of Yetter-Drinfeld modules, while we follow a more direct route. In our approach, we found the monadic setting better suited to make the connection to [25] than the comonadic picture.

A more conceptual relation between these two pictures is described in Remark 2.14.

Outline of the paper. In Section 2 we start by reviewing the definition of a Hopf monad. Then, the central monads and their Hopf structures are described. We define four versions of monadic cointegrals, and show that they are related by isomorphisms to the categorical cointegral considered in [38].

Section 3 contains our conventions for (pivotal) quasi-Hopf algebras. We specialise the definition of the various monadic cointegrals to the category of modules over a quasi-Hopf algebra, and we review the definition and some properties of left and right cointegrals for quasi-Hopf algebras from [25], [4].

Our main theorem showing that quasi-Hopf cointegrals are equivalent to monadic cointegrals is formulated in Section 4. The main ideas of the proof are outlined, while technical details are deferred to Appendix A.

Examples of monadic cointegrals for quasi-Hopf algebras are given in Section 5.

In Section 6 we consider integrals for Hopf algebras in braided finite tensor categories, and we relate left and right integrals for $\mathcal{L}$ to monadic cointegrals for $A_2$. As an example, we treat finite-dimensional quasi-triangular quasi-Hopf algebras.

With $\mathcal{C}={}_H\mathcal{M}$ the category of finite-dimensional modules over a finite-dimensional factorisable ribbon quasi-Hopf algebra H, in Section 7 we express the $SL(2,\mathbb{Z})$-action on the centre of H using the monadic and the quasi-Hopf cointegral.

Throughout this paper we fix an algebraically closed field k. Following [15], by a finite tensor category we mean a k-linear abelian category that

• has finite-dimensional Hom-spaces, and every object is of finite length,

• possesses a finite set of isomorphism classes of simple objects,

• has enough projectives,

• is rigid monoidal, such that the tensor product functor is k-bilinear and the monoidal unit 1 is simple.

We denote the left and the right dual of an object X by $X^{\vee }$ and ${}^{\vee }X$, respectively. The corresponding evaluations and coevaluations are

satisfying the familiar zig-zag equalities. We do not assume that $\mathcal{C}$ is strict monoidal, and (compositions of) coherence isomorphisms will be indicated.

Our conventions for string diagrams are as follows. We read them from bottom to top, and coherence isomorphisms will usually not be drawn.

Left and right coevaluation and evaluation for the object $X\in \mathcal{C}$ are drawn as

respectively, so that in our conventions for duals and string diagrams, arrows on the duality maps for left (right) duals point to the left (right).

A functor $F:\mathcal{C}\to \mathcal{D}$ between monoidal categories is lax comonoidal if there is a natural transformation $F_2$ and a morphism $F_0$,²

$$F_2(X,Y):F(X\otimes Y)\to FX\otimes FY,F_0:F\mathbf{1}\to \mathbf{1},$$

satisfying certain coherence conditions so that coalgebras in $\mathcal{C}$ are mapped to coalgebras in $\mathcal{D}$. For that reason we will commonly refer to $F_2$ and $F_0$ as the comultiplication and the counit of the lax comonoidal functor F. If $F_2$ and $F_0$ are isomorphisms then F is called a strong comonoidal functor.

Similarly, a functor $F:\mathcal{C}\to \mathcal{D}$ between monoidal categories is lax/strong monoidal if $F^{\text{op}}:{\mathcal{C}}^{\text{op}}\to {\mathcal{D}}^{\text{op}}$ is lax/strong comonoidal. The corresponding natural transformation $F_2$ and the morphism $F_0$ we call the multiplication and the unit, respectively.

A natural transformation $\phi :F\Rightarrow G$ between two comonoidal functors is called comonoidal if it commutes with the comonoidal structures. That is, if

$$G_2(X,Y)\circ {\phi }_{X\otimes Y}=({\phi }_X\otimes {\phi }_Y)\circ F_2(X,Y)\text{and}{F}_0=G_0\circ {\phi }_{\mathbf{1}}$$

is true for all objects $X,Y$.

Monoidal natural transformations between monoidal functors are defined similarly.

A rigid category $\mathcal{C}$ is called pivotal if there is a monoidal natural isomorphism $\delta :{\mathrm{id}}_{\mathcal{C}}\Rightarrow {(?)}^{\vee \vee }$, i.e. from the identity functor on $\mathcal{C}$ to the double dual functor. The monoidal structure of the double dual is given in terms of the natural isomorphism

as the composition

$$V^{\vee \vee }\otimes W^{\vee \vee }\stackrel{{\gamma }_{V^{\vee },W^{\vee }}}{\to }{(W^{\vee }\otimes V^{\vee })}^{\vee }\stackrel{{\left({\gamma }_{W,V}^{-1}\right)}^{\vee }}{\to }{(V\otimes W)}^{\vee \vee }.$$

Note that the existence of the pivotal structure δ is equivalent to requiring that the left and the right dual functor be isomorphic as monoidal functors. Indeed, given δ we can form the isomorphism

Conversely, given a natural monoidal isomorphism ${}^{\vee }X\cong X^{\vee }$, we have

$$X^{\vee \vee }\cong {({}^{\vee }X)}^{\vee }\cong X$$

where the second isomorphism is

We will suppress some of the tensor product symbols to shorten expressions, e.g. in the above expression we only left those tensor symbols between objects necessary to make the assignment of duals unambiguous. As a string diagram (1.12) simply reads