Categorical aspects of cointegrals on quasi-Hopf algebras

We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra $H$, we give an explicit description of categorical cointegrals of the category ${}_H \mathcal{M}$ of left $H$-modules in terms of cointegrals on $H$. Provided that $H$ is unimodular, we also express the Frobenius structure of the `adjoint algebra' in the Yetter-Drinfeld category ${}^H_H \mathcal{YD}$ by using an integral in $H$ and a cointegral on $H$. Finally, we give a description of the twisted modified trace for projective $H$-modules in terms of cointegrals on $H$.


Introduction
Results on Hopf algebras have been reexamined from the viewpoint of the theory of tensor categories since a result on Hopf algebras generalized to tensor categories is expected to be useful in applications to, e.g., low-dimensional topology and conformal field theories. Integrals and cointegrals for Hopf algebras are introduced by Sweedler [Swe69] and play an important role in the study of Hopf algebras. Let H be a finite-dimensional Hopf algebra. Recently, we have observed that cointegrals on H appear in several results on finite tensor categories. For example, it is known that, if C is a unimodular finite tensor category, then there is a Frobenius algebra A in the Drinfeld center Z(C) of C arising from an adjunction between Z(C) and C [IM14,Shi17b]. The Frobenius structure of A is written in terms of integrals and cointegrals of H if C = H M fd is the category of finite-dimensional left H-modules. As another example, we mention that the space of modified traces on a pivotal finite tensor category C can be identified with the space of 'µ-symmetrized' cointegrals on H if C = H M fd and µ is the modular function on H [BBG17,FG18].
The above-mentioned results may be explained by the integral theory for finite tensor categories established by the second-named author [Shi17b,Shi18]. He introduced the notions of categorical integrals and categorical cointegrals for a finite tensor category C and showed that they can be identified with ordinary (co)integrals if C = H M fd . As demonstrated in [Shi18], several results in the Hopf algebra theory are extended to finite tensor categories by using categorical (co)integrals. However, a relation between modified traces and categorical cointegrals is still open. We also remark that an explicit description of a categorical cointegral is not known except the case where C = H M fd or if C is a fusion category.
In this paper, we discuss these problems in the case where C is the category H M of left modules over a finite-dimensional quasi-Hopf algebra H over a field k. In this case, the Drinfeld center Z(C) is identified with the category H H YD of Yetter-Drinfeld modules over H. By using the fundamental theorem for quasi-Hopf bimodules, we give a convenient expression of a right adjoint R : H M → H H YD of the forgetful functor from H H YD to H M (Theorem 5.3). As an application, we express the space of categorical cointegrals in terms of ordinary cointegrals on H (Theorem 5.11). Under the assumption that H is unimodular, we give an explicit description of the Frobenius structure of the algebra A = R(k) in terms of (co)integrals of H (Theorem 5.12). Finally, we show that µ-twisted modified traces on H M fd are expressed by 'µ-symmetrized' cointegrals on H, where µ is the modular function on H (Theorem 6.4).
Recently, several interesting examples of finite-dimensional quasi-Hopf algebras are introduced and studied from the viewpoint of logarithmic conformal field theories [FGR17,CGR17,Neg18]. We hope our abstract results are useful in future study of the representation theory of these algebras and their applications to conformal field theories.
This paper is organized as follows: In Section 2, we recall basic notions in the theory of monoidal categories and their module categories. In Section 3, we recall the definition of quasi-Hopf algebras from [Dri89,Kas95] and elementary results on the representation theory of quasi-Hopf algebras. We also collect useful identities in a quasi-Hopf algebra.
In Section 4, we review the integral theory for quasi-Hopf algebras. Let H be a quasi-Hopf algebra and, for simplicity, assume that H is finite-dimensional in this Introduction.  3). By using the above commutative diagram, we also show that the adjunction F ⊣ R is a co-Hopf adjunction (Theorem 5.5), the dual notion of a Hopf adjunction [BLV11]. We also express the monoidal structure of R and the structure of the algebra A = R(k) ∈ H H YD explicitly (Theorem 5.6). It turns out the algebra A is identical to the algebra H 0 given in [BCP05,BCP06]. We deduce some results on the algebra A from the general theory of monoidal categories (Corollaries 5.7, 5.8 and 5.9).
The above commutative diagram also shows that the functor L : H M → H H YD given by tensoring H ∨ ∈ H H M H is left adjoint to F (Theorem 5.10). Let µ : H → k be the modular function on H and regard it as a one-dimensional left H-module. The fundamental theorem for quasi-Hopf bimodules gives a relation between H and H ∨ , and this relation gives rise to a relation between the functors L and R. Specifically, there is an isomorphism of functors written by using a non-zero cointegral on H (Theorem 5.10). A categorical cointegral of H M , introduced in [Shi18], is an element of H H YD(R(µ), k) in our notation. By using the above isomorphism, we give a description of a categorical cointegral of H M in terms of a cointegral on H (Theorem 5.11). The above isomorphism implies that L ∼ = R when H is unimodular, i.e., µ is identical to the counit of H. It is known that if H is unimodular, then the algebra A is a Frobenius algebra in H H YD [Shi17b]. We write the Frobenius structure of A explicitly in terms of an integral in H and a cointegral on H.
In Section 6, we suppose that H has a pivotal element and study 'µ-twisted modified traces' on the pivotal monoidal category H M fd of finite-dimensional left Hmodules. Let H P fd be the full subcategory of H M fd consisting of projective objects. A µ-twisted modified traces on H M fd (cf. [GKP18,BBG17,FG18]) is a family of linear maps t = {t P : Hom H (P, µ ⊗ P ) → k} P ∈H P fd satisfying the µ-twisted cyclicity and being compatible with the partial pivotal trace in H M fd . According to [FG18], such a trace is constructed from a 'µ-symmetrized' cointegral on H if H is an ordinary Hopf algebra. The main result of this section is a generalization of this result to the case where H is a quasi-Hopf algebra (Theorem 6.4). The proof goes along almost the same way as [FG18] but also uses a certain technical result on cointegrals on a quasi-Hopf algebra discussed in Section 4.

Acknowledgment.
We are grateful to Peter Schauenburg for helpful discussion on Yetter-Drinfeld modules of the second kind. The second author (K.S.) is supported by JSPS KAKENHI Grant Number JP16K17568.
The natural isomorphisms Φ, l and r are called the associator, the left unit isomorphism and the right unit isomorphism, respectively. We always assume that the left and the right unit isomorphisms are identities. Although the associator is often assumed to be the identity in the study of monoidal categories, we do not so in this paper as our main example is the category of modules over a quasi-Hopf algebra.
Let C and D be monoidal categories. A monoidal functor from C to D is a triple (F, F (0) , F (2) ) consisting of a functor F : C → D, a morphism F (0) : ½ → F (½) in D and a natural transformation such that the equations hold for all objects X, Y, Z ∈ C. A monoidal functor F : C → D is said to be strong if F (0) and F (2) are invertible. It is said to be strict if F (0) and F (2) are identities.
Let F, G : C → D be monoidal functors. A monoidal natural transformation from F to G is a natural transformation ξ : Suppose that a strong monoidal functor F : C → D admits a right adjoint R : D → C. Let η : id C → RF and ε : F R → id D be the unit and the counit of the adjunction, respectively. The functor R has a monoidal structure given by The adjunction (F, R, ε, η) is in fact a monoidal adjunction in the sense that F and R are monoidal functors and η and ε are monoidal natural transformations.
2.2. Duality in a monoidal category. Let C be a monoidal category, and let X be an object of C. A left dual object of X is a triple (Y, e, c) consisting of an object Y ∈ C and morphisms e : Y ⊗ X → ½ and c : The morphisms e and c are referred to as the evaluation and the coevaluation, respectively. A left dual object of X is, if it exists, unique up to unique isomorphism in the following sense: If (Y, e, c) and (Y ′ , e ′ , c ′ ) are both left dual objects of X, then there exists a unique isomorphism f : Now we suppose that X has a left dual object (X ∨ , ev X , coev X ). Given an object A ∈ C, we denote by L A : C → C the functor defined by L A (V ) = A ⊗ V . The definition of a left dual object implies that the functor L X ∨ is left adjoint to L X . More precisely, there is a natural isomorphism If we use the graphical calculus (see, e.g., [Kas95]), then the natural isomorphism É and its inverse are expressed by string diagrams as follows: When we express a morphism by such a string diagram, we adopt the convention that a morphism goes from the top to the bottom of the diagram. The evaluation and the coevaluation are represented by a cup (∪) and a cap (∩), respectively. Although the graphical calculus is a very useful tool in the theory of monoidal categories, it hides the associator that should not be ignored in the study of quasi-Hopf algebras. For this reason, we use string diagrams only to give readers graphical intuition.
There is also a natural isomorphism for V, W ∈ C. The inverse of È is given by The isomorphism È and its inverse are expressed as follows: A right dual object of X is a triple (Y, e, c) consisting of an object Y ∈ C and morphisms e : X ⊗ Y → ½ and c : ½ → Y ⊗ X such that, in a word, the triple (X, e, c) is a left dual object of Y . Let C rev be the monoidal category obtained from C by reversing the order of the tensor product. A right dual object of X is nothing but a left dual object of X ∈ C rev . Thus, if ∨ X is a right dual object of X, then there are natural isomorphisms 2.3. Modules over an algebra. Let C be a monoidal category. A left C-module category is a category M endowed with a functor : C × M → M (called the action) and natural isomorphisms Ω X,Y,M : (X ⊗ Y ) M → X (Y M ) and obeying certain axioms similar to those for monoidal categories. A right C-module category and a C-bimodule category are defined analogously. We omit the definitions of C-module functors and their morphisms; see [EGNO15].
An algebra in C is a triple (A, m, u) consisting of an object A ∈ C and morphisms m : A ⊗ A → A and u : ½ → A in C such that the following equations hold: Dually, a coalgebra in C is a triple (C, ∆, ε) consisting of an object C ∈ C and morphisms ∆ : C → C ⊗ C and ε : C → ½ in C such that, in a word, (C, ∆, ε) is an algebra in C op . Now let M be a left C-module category. Given an object A ∈ C, we define the functor L Similarly, if C is a coalgebra in C, then the functor L C has a natural structure of a comonad on M.
Definition 2.1. Given an algebra A in C, we define the category A M of left Amodules in M to be the category of L A -modules (= the Eilenberg-Moore category of L A [ML98]). Given a coalgebra C in C, we define the category C M of left C-comodules in M to be the category of L C -comodules. The category of right (co)modules in a right C-module category is defined and denoted in an analogous way.
We note that C is a C-bimodule category by the tensor product. Thus, given an algebra A in C, the notions of a left A-module in C and a right C-module in C are defined. Let X be a left A-module in C with action ρ : A ⊗ X → X. If X has a left dual object (X ∨ , ev X , coev X ), then we define ρ ♯ : X ∨ ⊗ A → X ∨ to be the morphism corresponding to ρ via It is known that X ∨ is a right A-module in C by the action ρ ♯ . Graphically, the morphism ρ ♯ can be expressed as follows: There is a similar construction for comodules: Let C be a coalgebra in C, and let X be a right C-comodule in C with coaction δ : X → X ⊗ C. If X has a left dual object X ∨ , then X ∨ is a left C-comodule by the coaction defined to be the morphism corresponding to δ via 3. Quasi-Hopf algebras 3.1. Notation. Throughout this paper, we work over a fixed field k. Unless otherwise noted, the symbol ⊗ means the tensor product over k. For a vector space V over k and a positive integer n, we denote by V ⊗n the n-th tensor power of V over k. An element x ∈ V ⊗n is written symbolically as although an element of V ⊗n is a sum of the form i x i,1 ⊗· · ·⊗x i,n in general. Given a permutation σ on the set {1, . . . , n}, we write x σ(1) ··· σ(n) = x σ(1) ⊗ · · · ⊗ x σ(n) .
By a k-algebra, we always mean an associative unital algebra over the field k. If A is a k-algebra and x is an invertible element of the k-algebra A ⊗n , then we write its inverse as Given a vector space M , we denote its dual space by M * = Hom k (M, k). For ξ ∈ M * and m ∈ M , we often write ξ(m) as ξ, m . If M is a left A-module, then M * is a right A-module by the action ↼ given by ξ ↼ a, m = ξ, am for ξ ∈ M * , a ∈ A and m ∈ M . Similarly, if M is a right A-module, then M * is a left A-module by a ⇀ ξ, m = ξ, ma for a ∈ A, ξ ∈ M * and m ∈ M .

21
(3.24) by defining formulas (3.9)-(3.14). Thus The antipode of a Hopf algebra is known to be an anti-coalgebra map. To extend this result to quasi-Hopf algebras, it is convenient to introduce the twisting operation for quasi-Hopf algebras [Dri89,Kas95]. A gauge transformation of H is an invertible element F ∈ H ⊗2 such that ǫ(F 1 )F 2 = 1 = F 1 ǫ(F 2 ). Given a gauge transformation F of H, we define Then H F := (H, ∆ F , ǫ, φ F , S, α F , β F ) is a quasi-Hopf algebra, called the twist of H by F . Drinfeld [Dri89] introduced a special gauge transformation given as follows: We define , ∈ H ⊗2 by (3.26) Then and its inverse are given by the following formulas: The antipode of H is shown to be an isomorphism S : H op,cop → H of quasi-Hopf algebras. Namely, we have for all h ∈ H. The following equations are also useful: (3.31) For later use, we prove the following three lemmas: Thus we compute: This shows the first equation of (3.32). To prove the second one, we note: Thus we have Lemma 3.2. The following equations hold: Proof. The first equation is verified as follows: The second one is obtained by applying (3.33) to H op .
Lemma 3.3. The following equations hold: Proof. By (3.28) and (3.31), we have for all h ∈ H. We now verify (3.35) as follows: w ∈ W ). The vector space ½ := k is a left H-module through the counit ǫ : H → k.
The category H M of left H-modules is a monoidal category with the associator Φ, the left unit isomorphism l and the right unit isomorphism r given by For a finite-dimensional left H-module X, we define the left H-module X ∨ to be the vector space X ∨ = X * with the left H-module structure given by h · ξ = ξ ↼ S(h) for h ∈ H and ξ ∈ X ∨ . We fix a basis {x i } of X and let {x i } be the dual basis of {x i }. We define ev X and coev X by respectively, for ξ ∈ X ∨ and x ∈ X, where we have used the Einstein convention to suppress the sum over i. Equations (3.5) and (3.6) imply that (X ∨ , ev X , coev X ) is a left dual object of X.
Let X be a finite-dimensional left H-module with basis {x i }, and let {x i } be the dual basis of X * . As we have recalled in Subsection 2.2, there is a natural isomorphism for V, W ∈ H M . We express this isomorphism explicitly. Given a morphism f : Lemma 3.5. The isomorphism É and its inverse are given by Proof. For f ∈ Hom H (V, X ⊗ W ), ξ ∈ X ∨ and v ∈ V , we compute: To verify the expression for É −1 , we note that the equation The following lemma is proved in a similar manner: Lemma 3.6. The natural isomorphism and its inverse are given by A left H-module (co)algebra is a synonym for a (co)algebra in the monoidal category H M . As an application of Lemmas 3.5 and 3.6, we give a description of the dual (co)module in H M . For this purpose, we introduce the following notation: By Lemmas 3.5 and 3.6, we have for ξ ∈ X ∨ , a ∈ A and x ∈ X. Hence Part (i) is proved. To prove Part (ii), we let δ : X → X ⊗ C be the coaction of C on X and set δ ♮ = É X,C,X (δ). Then the coaction of C on X ∨ is computed as follows: For a finite-dimensional left H-module X, we also define the left H-module ∨ X to be the vector space ∨ X = X * with the left H-module structure given by h · ξ = ξ ↼ S(h) for h ∈ H and ξ ∈ ∨ X. We fix a basis {x i } of X and define ev ′ X and coev ′ X by ev ′ If we denote by Ù L cop and Ú R cop the elements Ù L and Ú R for H cop , respectively, then we have (3.50) by equations (3.24) and (3.32). By applying Lemma 3.7 to H cop , we obtain the following lemma:

4.
Integral theory for quasi-Hopf algebras 4.1. Integrals. The notions of integrals and cointegrals for Hopf algebras play an important role in the Hopf algebra theory and its applications. The integral theory for quasi-Hopf algebras is established by Hausser and Nill in [HN99]. The definition of integrals in a quasi-Hopf algebra is completely same as the case of Hopf algebras: The definition of cointegrals on a quasi-Hopf algebra is more complicated than the Hopf case. The original definition of Hausser and Nill is based on the fundamental theorem for quasi-Hopf bimodules. There are several formulas defining cointegrals, however, the original definition is convenient from the theoretical point of view. In this section, following [HN99, BC03, BC12], we review basic results on quasi-Hopf bimodules and (co)integrals for quasi-Hopf algebras.
for h, h ′ ∈ H, m ∈ M and n ∈ N . The unit object ½ is the base field k regarded as an H-bimodule by the counit of H. The associator of ( H M H ,⊗, ½) is given by (4.1) Thus we have a functor The category H H M H of left quasi-Hopf bimodules over H is defined to be the category of left H-comodules in H M H . This category can be identified with the category of right quasi-Hopf modules over K = H cop . Thus, by applying the fundamental theorem to K, we see that the functor is an equivalence of categories. A quasi-inverse of (4.6), which we denote by is given as follows: The functors (4.6) and (4.7) actually form an adjoint equivalence with the unit and the counit given respectively by respectively, for h, h ′ ∈ H, m ∈ M and ξ ∈ M ∨ , where the Einstein notation is used to suppress the sum over i. Analogously, the right dual object ∨ M of M is defined by ∨ M = M * as a vector space. The H-bimodule structure, the evaluation and the coevaluation are given by See    Proof. We have isomorphisms of right quasi-Hopf bimodules. By the fundamental theorem, we have an isomor-

Properties of cointegrals.
Let H be a finite-dimensional quasi-Hopf algebra. We note that the antipode of H is bijective if this is the case [BC03,Sch04]. By the fundamental theorem for quasi-Hopf bimodules, the map is an isomorphism in the category H M H H . We fix a non-zero left cointegral λ on H. Since the map Ξ is bijective, and since the antipode S of H is bijective, the map H → H * given by h → h ⇀ λ (h ∈ H) is also bijective. This means that the algebra H is a Frobenius algebra with Frobenius form λ.
We briefly recall basic results on Frobenius algebras. Let A be a Frobenius algebra with Frobenius form λ : A → k. By definition, the map Θ : A → A * defined by Θ(a) = a ⇀ λ (a ∈ A) is bijective. Given an algebra map χ : A → k, we define I χ = {t ∈ A | at = χ(a)t for all a ∈ A} and J χ = {f ∈ A * | a ⇀ f = χ(a)f for all a ∈ A}. Since Θ is an isomorphism of left A-modules, it induces an isomorphism between I χ and J χ . It is easy to see that J χ is spanned by χ. Hence I χ is spanned by t χ := Θ −1 (χ). If t ∈ I χ is a non-zero element, then it is a non-zero scalar multiple of t χ . Since we have λ(t) = 0 for all non-zero elements t ∈ I χ . The Nakayama automorphism of A (with respect to the Frobenius form λ) is the algebra automorphism ν : Let ν be the inverse of ν. Then we have for all a, b ∈ A. Thus we have ta = χ(ν(a))t for all t ∈ I χ and a ∈ A. Now we apply these results to the Frobenius algebra H.
Theorem 4.6. Let λ be a non-zero left cointegral on H. Then we have: (1) H is a Frobenius algebra with Frobenius form λ. The Nakayama automorphism of H with respect to λ is given by where µ is the modular function on H and Proof. We have proved that H is a Frobenius algebra with Frobenius form λ. We verify the given expression of the Nakayama automorphism. Since the map Ξ given by (4.18) is left H-linear, we have for all h, h ′ ∈ H. If we replace h and h ′ with S(h) and S(h ′ ), respectively, then we obtain λ(hh ′ ) = λ(h ′ S(S(h) ↼ µ)) as desired. Thus Part (1) is proved. To prove Part (2), we note that the space of left integrals in H is I χ with χ = ǫ in the above notation. By the above-mentioned results on Frobenius algebras, I ǫ is one-dimensional. Furthermore, if Λ is a non-zero left integral in H, then we have λ(Λ) = 0 and Λh = ǫ(ν(h))Λ = µ(h)Λ for all h ∈ H. Thus the proof of Part (2) is done. Part (3) is proved by applying the same argument to H op , which is also a Frobenius algebra with the same Frobenius form λ.
Let λ be a non-zero left cointegral on H. In summary, the bijectivity of the map Ξ given by (4.18) implies that H is a Frobenius algebra with Frobenius form λ. As we have seen in the proof of the above theorem, an expression of the Nakayama automorphism is obtained from the left H-linearity of Ξ. Other results in this subsection follow from general theory of Frobenius algebras.
The H-colinearity of Ξ has not been used yet. This property gives the following equation: Lemma 4.7. Let λ be a left cointegral on H. Then the equation We recall that δ is given by (4.1) with V = ∫ L , and the H-bimodule structure of ∫ L is given by xλy = µ(x)ǫ(y)λ for x, y ∈ H. Thus we have We fix a basis {e i } of H and let {e i } be the dual basis of {e i }. Then we have for all h ∈ H by (4.20). The first appearance of (4.21) seems to be the proof of [ (4.23) The first one is proved as follows: Setˇ = S( 2 ) ⊗ S( 1 ). Then, The second one is proved in a similar way. Now we suppose that λ ∈ H * satisfies (4.21). Then, for all h ∈ H, we have Lemma 4.9. Suppose that λ ∈ H * satisfies (4.22). Then we have for all left integrals Λ in H.

Proof. If Λ is a left integral in H, then we have
Let λ be a non-zero left cointegral on H. We have proved in Theorem 4.6 that H is a Frobenius algebra with Frobenius form λ. In other words, the map H → H * given by h → h ⇀ λ (h ∈ H) is bijective. The inverse of this map is given explicitly as follows: Theorem 4.10. We fix a non-zero left cointegral λ on H. Then the linear maps are bijective. Let Λ be a left integral in H such that λ, Λ = µ(β) −1 , and set (4.25) Then the inverses of Θ L and Θ R are given respectively by Since H is a Frobenius algebra with Frobenius form λ, the maps Θ L and Θ R are bijective. Let ν be the Nakayama automorphism of H with respect to λ. The defining formula (4.19) of ν implies the equation Θ R = Θ L • ν. Hence, To prove the formula for Θ −1 R , we note: Indeed, for all h ∈ H, we have Set Θ(ξ) = S( Λ 1 ) ξ, Λ 2 . By Lemmas 4.8 and 4.9, equation (4.24) holds. Hence, This implies Θ −1 R = Θ. The expression for Θ −1 L is obtained by (4.26) and the explicit description of the Nakayama automorphism ν given in Theorem 4.6. Now we give the following characterizations of cointegrals: Theorem 4.11. For λ ∈ H * , the following are equivalent: (1) λ is a left cointegral on H.
(2) ⇔ (4). To prove this, we require the following relations: (4.31) The first equation is proved as follows: The second one easily follows from the first one. Now we suppose that (2) holds. Then we have . for all h ∈ H. Thus (4) holds. If, conversely, (4) holds, then we prove that (2) holds as follows: We note that a right cointegral on H is just a left cointegral on the quasi-Hopf algebra H cop . By rephrasing the above theorem for H cop by using (3.10), (3.24), (3.32) and (3.50), we obtain the following theorem: Theorem 4.12. For λ ∈ H * , the following are equivalent: (1) λ is a right cointegral on H.
(2) For all h ∈ H, the following equation holds: (3) For all h ∈ H, the following equation holds: (4) For all h ∈ H, the following equation holds: (5) For all left integrals Λ in H, the following equation holds: such that the following equations hold for all h ∈ H and v ∈ V .
where φ ′ is a copy of φ. The map δ 1st V is called the coaction of the first kind. We denote by H H YD 1 the category of Yetter-Drinfeld modules of the first kind and k-linear maps that preserves the action and the coaction of H.
The trivial H-module ½ = k is a Yetter-Drinfeld module of the first kind by the coaction determined by δ 1st ½ (1) = 1 ⊗ 1. If V and W are Yetter-Drinfeld modules of the first kind, then their tensor product H-module V ⊗ W is a Yetter-Drinfeld module of the first kind by the coaction given by for v ∈ V and w ∈ W (see [Maj98, Proposition 2.2]). The category H H YD 1 is a monoidal category with this tensor product.
Given V ∈ H H YD 1 and X ∈ H M , we define σ V,X : where χ is the element of H ⊗4 given by The associator Ω is then given by hold for all h ∈ H and v ∈ V .
Schauenburg [Sch02] showed that there is an isomorphism Ψ : H H YD 1 → H H YD 2 of categories. The isomorphism Ψ is the identity on morphisms and keeps the underlying H-module unchanged. Given V ∈ H H YD 1 , the isomorphism Ψ replaces the coaction of V of the first kind with The inverse of the isomorphism Ψ replaces the second kind coaction of V ∈ H H YD 2 with δ 1st Specifically, the action of H and the second kind coaction of H on R(V ) are given by for a, h ∈ H and v ∈ V . By (5.11), the first kind coaction is given by As we have recalled in Subsection 2.1, the functor R is a monoidal functor as a right adjoint of the strict monoidal functor F . The structure morphisms are given as follows: Lemma 5.4. The morphism R (0) is determined by (5.17) The natural transformation R (2) is given by Proof. We note that the unit η is given by the coaction of the second kind. Since the first kind coaction of the trivial Yetter-Drinfeld module ½ = k is determined by δ 1st ½ (1) = 1 ⊗ 1, we have Thus (5.17) is proved. We verify (5.18). For simplicity, we write m = m H ⊗ m V ∈ H ⊗ V for an element m ∈ R(V ). The following equations hold: for h ∈ H and m ∈ R(V ). Indeed, the first one is proved as follows: The second one is proved as follows: The third one is proved as follows: For v ∈ R(X) and w ∈ R(Y ), we have where Ø ∈ H ⊗3 is given by We verify (5.18) as follows: respectively, for X ∈ D and M ∈ C, where i : id C → T U is the unit of U ⊣ T . We say that the monoidal adjunction U ⊣ T is co-Hopf if H (ℓ) and H (r) are invertible. We note that U ⊣ T is co-Hopf if and only if the comonoidal adjunction U op ⊣ T op is a Hopf adjunction in the sense of [BLV11]. Thus any results on a Hopf adjunction can be translated into a result on co-Hopf adjunction. A monoidal adjunction enjoys several favorable properties when it is co-Hopf. Thus it is important to know whether a given monoidal adjunction is co-Hopf. We shall consider the monoidal adjunction F ⊣ R : H M → H H YD given in Theorem 5.3.
. Hence the following linear map is well-defined: X,M . To see this, we remark that the equation holds for all m ∈ M . By using this equation instead of (3.1), one can verify that the equations hold for all m ∈ M in a similar way as (3.17) and (3.18). Now we prove that G (ℓ) is the inverse of H (ℓ) as follows: For h ∈ H, x ∈ X and m ∈ M , In a similar manner, one can verify that the linear map is well-defined and is the inverse of H Theorem 5.6. The action ⊲, the coaction δ 1st A of the first kind, the coaction δ 2nd A of the second kind, the multiplication ⋆, and the unit 1 A of the algebra A ∈ H H YD are given respectively by a ⋆ b = φ 1 aS(φ 1 φ 2 )αφ 2 φ 3(1) bS(φ 3 φ 3(2) ), (5.28) Thus our algebra A ∈ H H YD is identical to the algebra H 0 of [BCP05].
Proof. Equation (5.25) is obvious from the definition of the functor R. It also follows from the definition of R that the second kind coaction of A is given by for a ∈ A. Thus (5.27) is proved. We verify (5.26) as follows: For a ∈ A, (1) ) = (the right-hand side of (5.26)). Equation (5.28) is proved as follows: For a, b ∈ A, (2) ) (3.4),(3.5) = (the right-hand side of (5.28)).
Theorem 5.6 gives a category-theoretical origin of the algebra H 0 of [BCP05]. Since (5.25) is usually called the adjoint action of H, we call A the adjoint algebra. We now demonstrate that some properties of the adjoint algebra are derived from the general theory of monoidal categories. Let C be a monoidal category such that the forgetful functor Z(C) → C admits a right adjoint, say I : C → Z(C). It is known that the object I(½) is a commutative algebra in Z(C) (a proof for fusion categories is found in [DMNO13, Lemma 3.5], but the same proof can be applied for the general case). Applying this result to the monoidal category C = H M , we obtain: Corollary 5.7. The algebra A is commutative.
The algebra A acts on an object of the form R(V ), V ∈ H M , from the right by R (2) V,½ : R(V ) ⊗ A → R(V ). We denote by R(V ) A the right A-module obtained in this way. This construction gives rise to a functor Corollary 5.8. The functor K is an equivalence.
Proof. We have proved that F ⊣ R is a co-Hopf adjunction. Hence R op ⊣ F op is Hopf. The claim is proved just by applying the fundamental theorem for Hopf modules [BLV11] to the Hopf monad associated to this Hopf adjunction.
The vector space CF(H) := Hom H (A, ½) is called the space of class functions as it coincides with the space of class functions in the usual sense when H is a group algebra [Shi17a]. We introduce the binary operation ⋆ on CF(H) by for ξ, ζ ∈ CF(H) and a ∈ A.
Corollary 5.9. CF(H) is an associative unital algebra with respect to ⋆, and the map Proof. Let η be the unit of the adjunction F ⊣ R. By Theorems 5.5 and 5.6, we have ξ ⋆ ζ = ξ • F R(ζ) • F (η A ) for ξ, ζ ∈ CF(H). Namely, the binary operation ⋆ is in fact the composition of morphisms in the co-Kleisli category of the adjunction F ⊣ R. The adjunction isomorphism of F ⊣ R is given by where δ : H ∨ → H⊗ H ∨ is the left coaction of H given by Lemma 4.2 (i) with M = H. Specifically, the action and the second kind coaction on L(V ) are given by where χ ∈ H ⊗4 is given by (5.6). The elementω is actually the inverse of the element ω ∈ H ⊗ H ⊗ H ⊗ H op ⊗ H op . Hence the inverse of the natural isomorphism Ω is given by where Ø ′ = S(ω 1 )αω 2 ⊗ω 3 ⊗ω 4 S(β)S(ω 5 ). We prove (5.32) by showing Ø ′ = Ø as follows: (ii) By the fundamental theorem for quasi-Hopf bimodules, the map is an isomorphism of left quasi-Hopf bimodules. Thus the map is an isomorphism of Yetter-Drinfeld H-modules that is natural in the variable V ∈ H M . The map ξ V actually coincides with the map (5.33). Indeed, for a ∈ H, λ ∈ ∫ R and v ∈ V , we have Here, the last equality follows from the following computation: . We now establish analogous results for quasi-Hopf algebras as follows: Theorem 5.11. Let λ be a non-zero right cointegral on H, and define λ cat : A µ → ½, λ cat (a) = λ, αS(a) (a ∈ A µ ).

(5.35)
Then ∫ cat is the one-dimensional vector space spanned by λ cat .
Proof. Let λ be a non-zero right cointegral on H. Since ∫ R is the one-dimensional left H-module spanned by λ and is isomorphic to µ as a left H-module, there is an for all h, h ′ ∈ H by Theorem 4.6. We choose a basis {a i } of A and let {a i } be the dual basis. Using the Einstein convention, we compute for h, h ′ ∈ H. Hence, we finally obtain H). We note that λ is a left cointegral on H cop . Thus, by applying Theorem 4.10 to H cop , we see that the inverse of Θ L is given by for all h ∈ H. Since Θ L , S and are invertible, Θ is invertible. Explicitly, the inverse of Θ is given by for ξ ∈ A ∨ . Thus A is a self-dual object with evaluation e and the coevaluation 6. Modified trace 6.1. Modified trace on module categories. Let C be a rigid monoidal category. Then the assignment X → X * * canonically extends to a monoidal autoequivalence on C. A pivotal structure of C is an isomorphism X → X * * (X ∈ C) of monoidal functors. A pivotal monoidal category is a rigid monoidal category equipped with a pivotal structure. Now let C be a pivotal monoidal category with pivotal structure j : id C → (−) * * . For objects V, X, Y ∈ C, the (right) partial pivotal trace over V is the map . Graphically, the partial pivotal trace is expressed as follows: The partial pivotal trace is widely used to construct invariants of knots and closed 3-manifolds so-called quantum invariants [Tur94]. Although such a construction of quantum invariants works in a quite general setting, most of interesting quantum invariants originate from a semisimple k-linear pivotal monoidal category. In fact, the partial pivotal trace often vanishes in the non-semisimple case. To construct a meaningful invariant from a non-semsimple category, some authors considered a modification of the partial pivotal trace [BBG17,CGPT18,GKP18,FG18].
We suppose that C is a k-linear pivotal monoidal category with finite-dimensional Hom-spaces. In this paper, we mention a recent result of Fontalvo Orozco and Gainutdinov [FG18]. They introduced the notion of a module trace over a right C-module category M equipped with a module endofunctor Σ. For simplicity, we consider the case where M is a tensor ideal of C, that is, a full subcategory of C such that P ⊗ V and V ⊗ P belong to M whenever P ∈ M and V ∈ C. We moreover restrict ourselves to the case where Σ = D ⊗ (−) for some object D ∈ C. Then the notion of a module trace [FG18], which we call a D-twisted modified trace, is defined as follows: Definition 6.1. (a) Let M be a k-linear category, and let Σ : M → M be a k-linear endofunctor on M. A Σ-twisted trace [BKW16, Definition 2.1] on M is a family t = {t P : Hom C (P, Σ(P )) → k} P ∈M of k-linear maps such that the equation holds for all morphisms g : P → Q and f : Q → Σ(P ) in M. We denote by HH 0 (M, Σ) the class of Σ-twisted traces on M. We say that t ∈ HH 0 (M, Σ) is non-degenerate if the bilinear form is non-degenerate for all objects P, Q ∈ M.
(b) Let I be a tensor ideal of C, and let D be an object of C. A D-twisted module trace on I is a Σ-twisted trace on I, where Σ = D ⊗ (−), such that the equation holds for all objects P ∈ I, V ∈ C and morphisms f : One of the main results of [FG18] classifies D-twisted module traces on I in the case where C = H M fd for some finite-dimensional pivotal Hopf algebra H, I is the class of projective objects of C, and D is the one-dimensional left H-module associated to the modular function µ on H. According to [FG18], the space of such a trace is identified with the space of "µ-symmetrized" cointegrals on H. The aim of this section is to give the same description of such a trace in the case where H is a finite-dimensional quasi-Hopf algebra.
6.2. Pivotal quasi-Hopf algebras. We recall that a pivotal Hopf algebra is a Hopf algebra H equipped with a grouplike element ∈ H such that h −1 = S 2 (h) for all h ∈ H. If H is a pivotal Hopf algebra, then H M fd is a pivotal monoidal category by the pivotal structure given by . The definition of a pivotal quasi-Hopf algebra is a little more complicated than the ordinary case because of the fact that the canonical isomorphism (V ⊗ W ) ∨∨ ∼ = V ∨∨ ⊗ W ∨∨ is non-trivial in the quasi-Hopf case. hold for all element h ∈ H. A pivotal quasi-Hopf algebra is a quasi-Hopf algebra equipped with a pivotal element.
The category Vec := k M fd of finite-dimensional vector spaces over k is a pivotal monoidal category with the canonical pivotal structure j V : V → V * * given by The partial pivotal trace over X (with respect to the canonical pivotal structure j) is given by for v ∈ V , where {x i } is a basis of X, {x i } is the dual basis of {x i } and the Einstein summation convention is used to suppress the sum. Let H be a quasi-Hopf algebra. Given a pivotal element of H, we define the natural isomorphism g V : Lemma 6.3. The partial trace of f : V ⊗ X → W ⊗ X in C := H M fd is given by Proof. We recall that there is a canonical isomorphism Let f ♯ : V → (W ⊗X)⊗X ∨ be the morphism in H M fd corresponding to f : V ⊗X → W ⊗ X via this isomorphism. Although an explicit formula of f ♯ has been given in Lemma 3.6, we also express We fix a basis {x i } of X and let {x i } be the dual basis to {x i }. By the definition of the partial pivotal trace, we compute as follows: For v ∈ V , 6.3. Twisted modified trace on H M fd . Let A be a finite-dimensional algebra. We denote by A P fd the full subcategory of projective objects of A M fd . Given an algebra automorphism χ : A → A and a left A-module V , we define χ * (V ) to be the vector space V equipped with the left action · χ of A given by a · χ v = χ(a)v for a ∈ H and v ∈ V . It is easy to see that the assignment V → χ * (V ) extends to a k-linear autoequivalence on A M fd preserving the full subcategory A P fd . We say that a linear form t : According to [FG18], we denote by HH 0 (A, χ) the space of χ-symmetric linear forms on A. Given a χ-symmetric linear form t on A and P ∈ A P fd , we define the linear map t P : Hom A (P, χ * (P )) → k as follows: By the dual basis lemma, there are a natural number n and A-linear maps a i : A → P and b i : P → A (i = 1, . . . , n) such that n i=1 a i b i = id P . Choose such a system {a i , b i } n i=1 and define for f ∈ Hom A (P, χ * (P )). The family t = {t P } of k-linear maps is actually a χ * -twisted trace on A P fd and this construction gives an isomorphism HH 0 (A, χ) → HH 0 ( A P fd , χ * ) of vector spaces. Moreover, a χ * -twisted trace is non-degenerate if and only if the corresponding element of HH 0 (A, χ) is a non-degenerate linear form on A. Now we consider the case where A = H is a finite-dimensional pivotal quasi-Hopf algebra and the automorphism χ is given by χ(h) = h ↼ µ for h ∈ H. By abuse of notation, we write HH 0 (H, µ) := HH 0 (H, χ) and HH 0 ( H P fd , µ) := HH 0 ( H P fd , χ * ).
We note that P := H P fd is a tensor ideal of H M fd . Again by abuse of notation, we denote by µ the one-dimensional left H-module associated to µ. Let HH mod 0 (P, µ) be the class of µ-twisted module traces on P. Since the autoequivalence χ * on P is identified with µ ⊗ (−), the class HH mod 0 (P, µ) is the subspace of HH 0 (P, µ) consisting of elements satisfying the module trace property (6.2). By the reduction lemma [FG18, Lemma 2.9], to check the module trace property, it is enough to verify that the equation is a subspace of HH 0 (H, µ). We call ∫ µ-sym the space of µ-symmetrized cointegrals on H. We recall that there is an isomorphism between HH 0 (H, µ) and HH 0 ( H P fd , µ). Now the main result of this section is stated as follows: Theorem 6.4. The isomorphism HH 0 (H, µ) ∼ = HH 0 ( H P fd , µ) restricts to an isomorphism ∫ µ-sym ∼ = HH mod 0 ( H P fd , µ). We fix a µ-symmetric linear form t ∈ HH 0 (H, µ) and then define t ∈ HH 0 ( H P fd , µ) from t as in the previous subsection. To prove the above theorem, we first derive a necessary and sufficient condition for t satisfying (6.8) in terms of the linear form t. The following lemma is useful: Lemma 6.5. For a left H-module X, we define the linear map θ X by where X 0 is the vector space X regarded as a left H-module by ǫ. Then the family θ = {θ X } is a natural isomorphism. The inverse of θ X is given by holds for all ξ ∈ H * and x, y ∈ H. We now compute the left and the right hand sides of (6.11).
Claim 6.6. The left-hand side of (6.11) is equal to t(x)ξ(y).
Proof. Let {h i } n i=1 be a basis of H, and let {h i } be the dual basis of {h i }. For each i, we define t(x h i , y ξ, h i ) = t(x)ξ(y).
Claim 6.7. The right-hand side of (6.11) is equal to µ(φ 1 ) ξ, Õ R 2 φ 3 x (2) Ô R 2 y t, (Õ R 1 ↼ µ)φ 2 x (1) Ô R 1 . (6.12) Proof. For simplicity, we set f := Φ −1 µ,H,H • Γ ′ ξ,x,y and r := ptr C H,µ⊗H|H (f ). By the definition of the trace t, the right-hand side of (6.11) is equal to t(r(1)). To compute r(1), we note: Hence, for all h ∈ H, we have Now we define f by (6.6) with V = H and W = µ ⊗ H. Then, We fix a basis {h i } of H and let {h i } be the dual basis of {h i }. We identify µ ⊗ H with H as a vector space. By Lemma 6.3, we compute r(1) as follows: r(1) = ptr Vec H,H|H ( f )(1) (by identifying µ ⊗ H with H) Hence the right-hand side of (6.11) coincides with (6.12) The above two claims show that the trace t satisfies (6.11) if and only if the equation holds for all h ∈ H. Now we prove Theorem 6.4.