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Two characterizations of finite quasi-Hopf algebras

Let H be a finite-dimensional quasibialgebra. We show that H is a quasi-Hopf algebra if and only if the category of its finite-dimensional left modules is rigid if and only if a structure theorem for Hopf modules over H holds. We also show that a dual structure theorem for Hopf modules over a coquasibialgebra H holds if and only if the category of finite-dimensional right H-comodules is rigid; this is not equivalent to H being a coquasi-Hopf algebra.


Introduction
Let H be a bialgebra over the field k. It was shown by Ulbrich [12] that H is a Hopf algebra if and only if the monoidal category M H f of finite-dimensional right Hcomodules is rigid, that is, if every finite-dimensional H-comodule has a dual object within the category M H f . In particular, if H is a finite-dimensional bialgebra, then H is a Hopf algebra if and only if the category H M f of finite-dimensional left H-modules is rigid.
It is a natural question whether the same holds for quasibialgebras: Drinfeld's definition of a quasibialgebra H ensures that the category H M is, just like in the bialgebra case, a monoidal category. And Drinfeld's definition of a quasiantipode is motivated by the fact that the category of finite-dimensional modules over a quasi-Hopf algebra is a rigid monoidal category. However, if we try to prove the converse, then we run into difficulties. The key problem is that the underlying functor H M → M k to the category of k-vector spaces is monoidal if H is a bialgebra, and monoidal functors automatically preserve dual objects. If H is only a quasibialgebra, then the underlying functor H M → M k is still compatible with tensor products, but not coherent in the sense of the definition of a monoidal functor. Thus it is not clear that the functor preserves duals. That the problem is really serious was shown in [9] by an example based on a construction of Yongchang Zhu [13]: There is a coquasibialgebra H such that the category M H f is rigid, although H is not a coquasi-Hopf algebra. The existence of a coquasiantipode is ruled out quite drastically by the fact that a finite-dimensional H-comodule and its dual object may have different dimensions. One result of this paper will be that all is well in the finite-dimensional case: A finite-dimensional quasibialgebra H is a quasi-Hopf algebra if and only if H M f is rigid.
Another well-known criterion says that a bialgebra H is a Hopf algebra if and only if the structure theorem for Hopf If the structure theorem for Hopf modules holds, it is very easy to check that the canonical map is bijective. For a quasi-Hopf algebra H, Drinfeld's paper [3] contains an analog of the canonical map H ⊗ H → H ⊗ H. However, the proper anolog is given by a more complicated formula; in particular, one already needs a quasiantipode to even write down the map (or its inverse), and it seems to have no analog for quasibialgebras. We shall show that the problem is just as serious as that with the first criterion mentioned above: Given a coquasibialgebra H we will prove in Section 2 that the structure theorem for Hopf modules holds -that is, a certain functor M H → H H M H is a a category equivalence -, if and only if the category M H f is rigid. For a finite-dimensional quasi-Hopf algebra this provides a new and rather conceptual proof for the structure theorem of Hausser and Nill -in fact our proof needs hardly any unpleasant calculations with the quasibialgebra structure and its axioms, and none at all with the quasiantipode. Of course the result also provides examples of coquasibialgebras that satisfy the structure theorem for Hopf modules, while they do not have a coquasiantipode.
On the other hand, if H is a finite-dimensional quasibialgebra, Theorem 3.1 shows that all is well, that is, the structure theorem for Hopf modules is equivalent to the existence of a quasiantipode.
In Section 4 we will show that the quasiantipode of a finite-dimensional quasi-Hopf algebra is a bijection. This was first proved by Bulacu and Caenepeel [2]. We will give a rather different proof.
Throughout the paper, we work over a base field k.

Duality and the structure of Hopf modules
Throughout this section, we let H be a coquasibialgebra. That is, (H, ∆, ε) is a coassociative coalgebra, endowed with a (nonassociative) multiplication ∇ : H ⊗ H → H which is a coalgebra map, a grouplike unit element 1 ∈ H, and a convolution invertible form φ : H ⊗ H ⊗ H → k, the associator, satisfying the identities φ(g ⊗ 1 ⊗ h) = ε(g)ε(h), for d, f, g, h ∈ H. We have used Sweedler notation in the form ∆(h) = h (1) ⊗ h (2) . We will also use Sweedler The axioms ensure that the category M H of right H-comodules is a monoidal category in the following way: For V, W ∈ M H , the tensor product V ⊗ W over k is an H-comodule with the codiagonal comodule structure induced by multiplication in H; the associativity isomorphism Φ : (1) ). Since the opposite of a coquasibialgebra and the tensor product of two coquasibialgebras are coquasibialgebras as well, the category H M H of H-H-bicomodules is also a monoidal category. This time the associativity isomorphism Φ : (1) ). It is a key observation that H (which is not associative as a k-algebra) is an associative algebra within the monoidal category H M H , that is, we have Thus we can use the general theory of algebras and modules in monoidal categories, see Pareigis [6,7], to do (or rather avoid) calculations with the multiplicative structure of H.
In particular, there is a well-defined notion of (say, left) H-module within the monoidal category H M H . We denote the category of such modules by H H M H , and call its objects Hopf modules. We note that for any where µ denotes the module structure of M . We will abbreviate this module structure by a dot attached to the tensorand M , i.e. write . .M . ⊗ . P . for it, with the upper dots indicating on which tensor factors we have a codiagonal coaction, and the upper dot indicating where the action takes place; note, though, that the actual formula for the action involves both tensorands through the action of the associators.
Taking M = H as a special case we obtain the left adjoint P → . .H . ⊗ . P . to the underlying functor H H M H → H M H . As a particular case, we can consider a right H-comodule V as a bicomodule with the trivial comodule structure on the left, and apply the above construction to obtain a functor The formally dual version of this functor (for a quasibialgebra) was studied by Hausser and Nill [4], who also proved that it is an equivalence in case H is a quasi-Hopf algbebra. Moreover, H H M H is a monoidal category, and L is a monoidal functor. Hausser and Nill show this in the dual case using the assumption that H is quasi-Hopf, the quasibialgebra case is treated in [10]. We shall say for short that the structure theorem for Hopf modules holds if L is a category equivalence. In this section we shall give a different proof of the structure theorem for Hopf modules than the ones in [4,10], under the weaker assumption that M H f is a rigid monoidal category.
We start by stating several facts on the functor L and the category H H M H that are formally dual to facts proved and used in [4] and [10]. We will not give the formally dual proofs, but will indicate how Proposition 2.4 follows from more abstract reasons without any work. The dual statement of the following is observed between Corollary 3.9 and Lemma 3.10 of [4]. See also [10,Lem.and In This follows from the following more general statement: Proof. It is obvious that the isomorphism is left and right H-colinear. H-linearity is a small calculation: Denoting the respective actions by h((m ⊗ n) ⊗ v) and The two associativity isomorphisms both map m We shall repeat briefly the abstract argument:  [4] or [10] to show that L is monoidal, the reader might not want to get into C-category theory. In this case one may dualize the proof of [10, Prop.3.6] to obtain: H M H is a monoidal functor. Let C be a monoidal category. Recall that a dual object of V ∈ C is a triple (V ∨ , ev, db) in which V ∨ ∈ C, and ev : V ∨ ⊗ V → I and db : I → V ⊗ V ∨ are morphisms such that the two compositions are identities. If (F , ξ) : C → D is a monoidal functor, and (V ∨ , ev, db) is a dual object of V in C, then F (V ∨ ) is a dual object of F (V ) in D, with evaluation and coevaluation Let V ∈ M H be finite-dimensional. We can endow the dual vector space V * with a canonically corresponding left H-comodule structure defined by is an H-bimodule map.
It is obviously left and right H-colinear, and it is H-linear by the calculation for g, h ∈ H. To check the identities for a dual object, we have to bear in mind the canonical identifications and can calculate x i ⊗ y i for some x i ∈ V and y i ∈ V ∨ . The latter generate a finite-dimensional subcomodule U ⊂ V ∨ , and it is straightforward to check that the map db ′ : k → V ⊗U induced by db, and the restriction ev ′ : U ⊗V → k of the evaluation ev : V ∨ ⊗ V → k make U a dual object of V , whence U = V ∨ . Now assume (2). We need to show that L is essentially surjective.  (2) in Theorem 2.6 is fulfilled if H is a coquasi-Hopf algebra. We will not recall the axioms of a coquasiantipode in any detail, but shall merely say that it involves an anti-coalgebra endomorphism S of H that allows us to endow V ∨ := V * , the k-linear dual of V ∈ M H f , with an H-comodule structure, and extra structure elements that make V ∨ into a left dual of V in M H f . If W ∼ = V * as at the end of the proof of Theorem 2.6, then we see that V ∨ ∼ = W S , the right H-comodule obtained from the left H-comodule W along S. Thus we have:

Finite Quasi-Hopf algebras
Recall that a quasibialgebra H is an associative algebra with an algebra map ∆ : H → H ⊗ H called comultiplication, an algebra map ε : H → k that is a counit for ∆, and an invertible element φ ∈ H ⊗ H ⊗ H such that hold for all h ∈ H. We will write ∆(h) =: A finite-dimensional quasibialgebra is the same as the dual of a finite-dimensional coquasibialgebra (historically, vice versa would be more to the point).
A quasiantipode (S, α, β) for a quasibialgebra H consists of an anti-algebra endomorphism S of H, and elements α, β ∈ H, such that hold in H, for h ∈ H. A quasi-Hopf algebra is a quasibialgebra with a quasiantipode. Note that we disagree in this definition with Drinfeld who requires S to be a bijection. We will return to this in Section 4 where we give another proof for a recent result of Bulacu and Caenepeel, which says that the antipode of a finite-dimensional quasi-Hopf algebra is automatically bijective.
The main result of this section characterizes finite quasi-Hopf algebras via rigidity of their module category, or the structure theorem for Hopf modules. The functor R in the theorem is the formal dual to (and older than) the functor L in the preceding section; it is due to Hausser and Nill [4]. We shall recall some details in the proof.  (2) and (3) follows by duality from Theorem 2.6. The implication (1) =⇒ (2) is the motivation for the definition of a quasiantipode in Drinfeld's paper [3]; the question whether one should require bijectivity of the antipode is actually irrelevant here. To recall some details: If V is a finite-dimensional H-module, then a dual object for V in H M is the dual vector space V * with the H-module structure given by the transpose of the action via S, and the evaluation and coevaluation maps ev : We shall now prove (3)  In particular, for all g, h, ℓ ∈ H. In particular γ(h) = βS(h) for β := γ(1). We define ϑ : We may summarize the three variants of H-linearity in the formula ϑ((g ⊗ h)ξ(j (1) ⊗ j (2) )) = (g ⊗ h (2) )ϑ(ξ)(S(h (1) ) ⊗ j) for g, h, j ∈ H and ξ ∈ H ⊗ H, in which all multiplications are now in the algebra H ⊗ H.
Our short proof took advantage of the structure theorem for Hopf modules as well as the isomorphism ϑ. It may be worthwhile to note that one does not really need the full generality of the structure theorem, but can use more directly the information contained in the map ϑ: For V ∈ M H , we can define