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Integrals for braided Hopf algebras

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. The fully braided version of Radford's formula for the fourth power of the antipode is obtained. Connections of integration with cross-product and transmutation are studied. The results apply to topological Hopf algebras, e.g. a torus with a hole, which do not have additive structure.


Introduction
The notion of integrals for finite-dimensional Hopf algebras was first introduced by Larson and Sweedler in an attempt to generalize the notion of Haar measures on groups. In their seminal paper [17] they prove that integrals always exist for finite dimensional Hopf algebras and give a Our main goal in this article is to extend the above outlined classical results for Hopf algebras to the framework of categories. Specifically, we will show that for any braided Hopf algebra, H , in a braided, rigid, monoidal category C with split idempotents, the analogues of integrals and moduli are defined and, similarly, satisfy existence and uniqueness assertions. Equipped with these tools we shall then derive the respective generalization of Radford's formula for these types of categories, which in Theorem 3.6 will take on the form Here the morphism u 0 −2 is defined in Figure 1 and, the Ω is constructed from the square of the braiding of H with Int H -the object of the integrals, see Lemma 2.4. We shall also see that all three multiplicands in the right hand side are Hopf algebra automorphisms commute with each other.
Hopf algebras in tensor categories and their properties have been investigated by a number of other people. Let us briefly review some of the previous contributions to the subject, which we build up on and generalize, and explain how our results are threaded into this development.
The idea of an algebra in a tensor category, see [16], is straightforwardly extended to that of a Hopf algebra, if that category is also symmetric. Most proofs of analogous assertions for categorical Hopf algebras in the symmetric case are formally identical to those in the classical situation of a linear Hopf algebra over k. The first proofs of existence and uniqueness of integrals in this setting are due to Drinfel'd. It is immediate from definitions that in a symmetric category the elements u 0 −2 and Ω (Int H) H are trivial (i.e. units) so that also the categorical version of Radford's formula will differ from the original one only in the interpretations of where the automorphisms live.
A nice way of illustrating the calculations, leading to the uniqueness of integrals and the Radford formula, has been given by Kuperberg [15]. He uses diagrammatic techniques that are essential for the construction of his 3-manifold invariants. In this language the generalizations to the symmetric categories follow immediately without much further explanation.
The notion of Hopf modules for ordinary Hopf algebras, which combines an action with a compatible coaction on the same space, can be similarly generalized without much difficulty to a Hopf algebra living in an abelian, symmetric, monoidal category. The classical results on Hopf modules can be rederived again by "imitation" in the categorical framework, where, similarly, the modules appear as abstract objects. See [28].
In the theory of Hopf algebras in braided categories a variety of complications arise due to the fact that the isomorphism between the tensor product of two objects and the transposed product is no longer canonical, i.e., also no longer coherent. However, a transposition isomorphism is needed, e.g., to formulate the compatibility axiom between multiplication and comultiplication for bialgebras.
If for the latter the braid isomorphism is employed one is left with the definition of a braided Hopf algebra, as it was given and first studied by Majid [21].
In the case where the braided tensor category is also rigid and abelian, integrals for such Hopf algebras were investigated by Lyubashenko [19], leading to general proofs for the existence of integrals as well as for the invertibility of the object of integrals for a braided Hopf algebra. The results in [19] follow from the study of Hopf modules in abelian braided monoidal categories. This strategy generalizes the approach to integrals via Hopf modules as proposed by Sweedler in [30].
As for symmetric categories we shall prove here analogously the existence and uniqueness of integrals for braided Hopf algebras in braided monoidal categories, but in addition we shall drastically weaken the condition of abelianness. More precisely, we shall no longer assume that the category has kernels or direct sums. As has already been pointed out in remarks in [2,5] it will suffice that our category has split idempotents. This means roughly that the category contains for a given projection, P , in an End-set the image of P as an object, and that P factors through this object.
The property of split idempotent is central in Karoubi's definition of pseudo-abelian categories [11], which will be further discussed later on. It is easy to satisfy this condition taking the Karoubian envelope of a category [11].
Besides existence and uniqueness of integrals we will also prove under our very general assumptions the invertibility of the object of integrals Int H , which, in the case of abelian, symmetric categories, has already been pointed out by Drinfel'd.
Furthermore, we will generalize the results on Hopf modules for abelian, symmetric, monoidal category as in [28] to any braided monoidal category with split idempotents.
Many of the algorithmic parts of the proofs will be done in a similar diagrammatic language as in [15]. Only now the diagrams are no longer plat graphs, but projections of graphs that distinguish between over and under crossings. As a result we will often encounter additional special elements resulting from non trivial full twists, such as u 0 −2 and Ω (Int H) H , which enter the Radford formula.
It is worth mentioning that our results apply to ordinary Hopf algebras and Hopf modules over a commutative ring R , if they are projective R-modules of finite rank (cf. [9]).
Our interest in braided Hopf algebras in non-abelian, braided categories has primarily been triggered by realizing their crucial rôle in the most recent discoveries in three dimensional topology related to quantum physics. In particular, integrals turned out to be the algebraic objects that are stringently associated to elementary surgery data in the construction of invariants of surgically presented 3-manifolds. Although the point of view of integrals was not used in the more computational approach in [31], it is inevitable in the construction of the non-semisimple analogues of invariants of 3-manifolds as in [20] and, more generally, 3-dimensional topological quantum field theories as in [14].
The fact that braided Hopf algebras and their integrals are inseparable from 3-dimensional topology in this approach, naturally leads one to identifying the torus with one hole as such a Hopf algebra in the braided category of 3-dimensional cobordisms between one holed surfaces. As a non trivial example of our generalized theory, we will precisely identify the integrals and all ingredients to the Radford formula for this braided Hopf algebra and category, via their explicit presentations in a tangle category as in [12].

Summary of Contents:
In Section 2 we review the necessary preliminaries and standard notations pertaining to rigid braided categories and braided Hopf algebras. We shall further define categories with split idempotents and discuss basic properties of invertible objects. In Section 3 results about Hopf modules are applied to prove the existence of ( Int H)-valued(-based) integrals on (in) H . There the object Int H of Int H is characterized by a universal property implying uniqueness, and will turn out to be invertible. The fully braided version of Radford's formula for the fourth power of the antipode is derived. Section 4 starts with an exposition of several results about Hopf modules, that are useful for our purposes. We then continue to prove the main results in this section, notably the invertibility of the object of integrals, the relationships of integrals with modular group-like elements and the antipode, and, eventually, the generalized Radford formula. In Section 5 we shall develop the example of a braided Hopf algebra in a category of tangles, and explain its applications to topological field theory. This algebra can be functorially mapped to representatives of another important class of braided Hopf algebras, namely coends in abelian rigid braided categories. Section 6 is devoted to connections of integration with cross-products and transmutations. In Section 7 we shall attempt to define external Hopf algebras, and discuss duality properties for Hopf bimodules. We present an explicit equivalence between the categories of Hopf H-bimodules and Hopf H ∨ -bimodules given by the tensor product with Int H. From these duality considerations we are able to obtain two more proofs of the generalized Radford formula.
ev : They are subject to the condition that the following compositions between evaluations and coevaluations are all equal the identity morphism in End(X) : This data allows us to introduce the transposes , f t : Y ∨ → X ∨ and t f : ∨ Y → ∨ X , of a given morphism, f : X → Y in C .
For a morphism, f , in a rigid braided monoidal category its trace tr 8 f is defined using Figure 1, the subscript being motivated by the resemblance with the digit 8. The dimension associated to this trace is dim 8 (X) := tr 8 (id X ) ∈ End(1I) . We will also need the natural automorphism u 0 −2 : X → X defined via the diagram in Figure 1. The notation is borrowed from [18].

Categories with split idempotents
M. Karoubi introduced in [11] a class of Banach categories, which he calls pseudo-abelian. If we drop, irrelevant for us, Banach and additive structures his definition reduces to the following: An idempotent, e = e 2 : X → X , in a category, D , is said to be split, if there exists an object, X e , and morphisms, i e : X e → X and p e : X → X e , such that e = i e • p e and id Xe = p e • i e . If every idempotent in D is split then we say that D admits split idempotents.
For a given category, C , there exists a universal embedding, C i − → C , such that the category C admits split idempotents. Moreover, C can be chosen universal in sense that for any category, D , with split idempotents every functor, F : C → D , factors in the form F = C i − → C G − → D , such that the functor G is unique up to an isomorphism of functors. The category C is called the Karoubi enveloping category of C. According to Karoubi [11] it is realized as the category with objects, X e = (X, e) , where X is an object in C and e : X → X is an idempotent in C . The morphisms in C are defined by It is noted in [19] that if C is a (braided) monoidal category then the category C can be equipped with a (braided) monoidal structure: In this case i is a (braided) monoidal functor. Furthermore, if the category C is rigid, so is C , and the dual objects of (X, e) are (X ∨ , e t ) and ( ∨ X, t e) . From now on we assume that our braided rigid category C admits split idempotents.

Hopf algebras
Recall that a Hopf algebra, H ∈ C , [21] is an object, H ∈ Obj C , together with an associative multiplication, m : H ⊗ H → H , and an associative comultiplication, ∆ : H → H ⊗ H , obeying the bialgebra axiom Moreover, H shall have a unit, η : 1I → H , a counit, ε : H → 1I , an antipode, S : H → H , and an inverse antipode, S −1 : H → H , which shall satisfy axioms analogous to the classical case.
A left (resp. right) module over an algebra, H , is an object, M ∈ C , equipped with an associative action, µ ℓ : H ⊗ M → M (resp. µ r : M ⊗ H → M ). The category of left (resp. right) H-modules will be denoted by H C (resp. C H ). The morphisms in H C are those from C that are equivariant with respect to µ ℓ . A left (resp. right) comodule over a coalgebra, H , is an object, M ∈ C , equipped with an associative coaction, ∆ ℓ : M → H ⊗ M (resp. ∆ r : M → M ⊗ H). The category of left (resp. right) H-comodules will be denoted by H C (resp. C H ).
If (C, ⊗, 1I, Ψ) is a braided monoidal category we shall denote by C = (C, ⊗, 1I, Ψ) the same monoidal category with the mirror-reversed braiding Ψ X,Y := Ψ Y,X −1 . For a Hopf algebra, H , in C we further denote by H op (resp. H op ) the same coalgebra (resp. algebra) with opposite multiplication µ op (resp. opposite comultiplication ∆ op ) defined as follows:

Integrals and the generalized Radford formula
This section contains the detailed statement of the main result of our paper, namely the braided, categorical version of Radford's formula, as well as the precise definitions of left and right integrals in a category and of the respective (co)moduli entering this formula. We shall defer all proofs to Section 4. In the first paragraph on integrals we will introduce a set of canonical projections, ∨ Π • • ∈ End C (H) , which will guarantee the existence of integrals for a braided Hopf algebra, H , in a category with split idempotents. We shall also illustrate the generalizations in Radford's formula in the example of the category of Z/n-graded vector spaces equipped with a braiding given by Heisenberg-type relations. Here the moduli will turn out to be units, whereas the additional elements u 0 −2 and Ω (Int H) H are going to be non-trivial.

Integrals for Hopf algebras
The following definition directly generalizes the classical one by rewriting the defining formula into a relation of operations given by morphisms.
Right X-valued integrals on H are defined analogously.
A left X-based integral in H is a morphism, f : X → H , such that Right X-based integrals in H are defined similarly.
In other words, a left (resp. right) X-valued integral is a homomorphism from the left (resp. right) regular H-comodule to the trivial H-comodule X . A left (resp. right) X-based integral is a homomorphism from the trivial H-module X to the left (resp. right) regular H-module.
We shall consider only bialgebras with invertible antipode and call them Hopf algebras. The next proposition not only asserts that a Hopf algebra in a category with split idempotents always admits integrals, but also that the objects of the integrals are invertible, and can be chosen the same for all cases.
For proofs of this and the following results, see Section 4. It follows immediately from the universality of integrals that the object Int H is defined uniquely up to a unique isomorphism.
Notice that H is a coequalizer for a pair of left ∨ H-actions on H , given by the formulae: Concretely, this means that H is a universal morphism for which Also the right integral H on H can be identified as a coequalizer for a similar pair of actions. The integrals in H are equalizers in a dual fashion. In the case of abelian categories this property served as the definition of integrals, see [19]. In order to prove the existence of integrals in the case of a category with split idempotents (Proposition 3.1), we construct the following four idempotents (the next lemma will be derived as a special case of Corollary 4.4): The morphisms which split these idempotents are precisely the integrals we are looking for. Some of their properties are described in the following theorem.
for certain invertible c ℓ ℓ , c ℓ r , c r ℓ , c r r ∈ End(1I). The idempotents from Lemma 3.2 are H-valued and H-based integrals. They are split by Int H as follows: is a right H-valued integral on H and a left H-based integral in H, is a left H-valued integral on H and a right H-based integral in H, is a left H-valued integral on H and a left H-based integral in H, is a right H-valued integral on H and a right H-based integral in H.

For any of the idempotents
The following lemma is a useful tool for recognizing the object Int H of the integrals as the object of non-degenerate pairings of a special form.

The generalized Radford formula
In the following lemma group-like elements, a and α , are extracted, which characterize the, e.g., right integral as a homomorphism with respect to a left regular action. These special elements are essential ingredients in the generalized Radford formula. Lemma 3.5. There exists a unique group-like element, a : 1I → H , and a unique multiplicative functional, α : H → 1I , such that the following identities hold: I.e. integrals are (co)module morphisms between H with the regular structure and Int H with the structure determined by the group-like element a (multiplicative functional α).
We shall use the following notations for morphisms in End(H) : Given this the precise form of our main result is as follows: Theorem 3.6 (The generalized Radford formula).
where the monodromy action Ω K of an invertible object, K , is defined in Lemma 2.4, and the morphism u 0 −2 is defined on Figure 1. The first proof of Theorem 3.6 is given in Section 4. It follows the scheme of the original proof given by Radford [26] for usual Hopf algebras, and adapted by Kuperberg [15] to the case of symmetric rigid monoidal categories. We use several presentations of the antipode and the inverse antipode via integrals.
The second proof of Theorem 3.6, given in Section 7, is based on the properties of Hopf bimodules. The generalized Radford formula follows from the equivalence of two presentations of the Hopf H ∨ -bimodule structure on X ⊗ H H ∨ , where X is a Hopf H-bimodule.
In the case of usual Hopf algebras the object of the integrals is a one-dimensional vector space. In the case of Z/2-graded Hopf algebras the object of the integrals can be odd or even. In braided categories there are more possibilities as the following example shows.
Example 3.1 (Non-trivial object of integrals). Let C = Z/n-grad-Vect(k) be the category of Z/n-graded k-vector spaces with the braiding ψ : Here q ∈ k is a primitive n th root of unit (a root of the n th cyclotomic polynomial φ n ). Notice that the category C is also ribbon with respect to the ribbon twist ν : Denote by X = X 1 a selected one-dimensional vector space concentrated in degree 1. The tensor algebra T (X) , equipped with a comultiplication induced by ∆(x) = x ⊗ 1 + 1 ⊗ x for a basis vector, x ∈ X 1 , constitutes a braided Hopf algebra in the braided category C .
Applying the bialgebra axiom, which contains the braiding ψ , we find iteratively the comultiplication and the antipode as given next: For any 0 < k < n the q-binomial coefficient n k q is divisible by φ n (q) . Therefore, ∆x n = x n ⊗ 1 + 1 ⊗ x n , and the span of x n is a coideal. Hence, H = T (X)/(X ⊗n ) is a rigid Hopf algebra in C .
Using the definition of integrals one easily verifies that x ⊗n−1 → x n−1 , is a right and left X ⊗n−1 -based integral in H . Since X ⊗n−1 is one-dimensional, the above integrals are universal and Int H = X ⊗n−1 ≃ X ∨ . In particular, (Int H) ⊗n ≃ 1I .
One concludes that all idempotents ∨ Π • • coincide and are given by x k → δ k n−1 x n−1 . The element a is the unit, the functional α is the counit. The monodromy Ω , the map on the right-hand side of the generalized Radford formula turns out to be Ω

Proofs of the main results
The proof of Theorem 3.3 we will follow in this section relies heavily on the theory of Hopf modules, which will be discussed in the first paragraph. The projections ∨ Π • • are explicitly constructed, and their images identified as integrals. In the following paragraph we prove the invertibility of the universal object of the integrals, permitting the conclusion of the proof of Theorem 3.3. In subsequent parts of this section we discuss the ingredients of the generalized Radford formula, namely the special group-like elements, and the action of the antipode on the integrals. With this preparation the proof of the formula for the forth order of the antipode is finally only a matter of combining several identities in the right way.

Hopf modules
From [19] we know that the Structure Theorem of Hopf modules [29] also holds for Hopf algebras in braided, monoidal categories, if there exist (co-)equalizers. In [5,2] it is shown that in order to prove the Structure Theorem in a braided monoidal category, C , it suffices to assume that C admits split idempotents.
In a monoidal category, D , which admits split idempotents the subcategory of (co-)modules over a (co-)algebra admits split idempotents, too. Similar facts hold for the categories of crossed modules and categories of Hopf (bi-)modules over a bialgebra in a braided, monoidal category with split idempotents. Definition 4.1.

1.
A left Hopf module, X , (resp. right-left Hopf module) over a bialgebra, H , in C is a left (resp. right) H -module and a left H -comodule, such that the action is a comodule morphism. Here the (co-)actions of H on X ⊗ H and H ⊗ X are determined by the braided, diagonal tensor (co-)action, given that H is an H -(co-)module via the regular (co-)action. As illustrated in Figure 4a) the axiom implied by this condition can be thought of as a "polarized" version of the bialgebra, which has to hold besides the comodule and module axioms for X . We shall denote by H H C (resp. H C H ) the subcategory of Hopf modules, whose objects are the left (resp. right-left) Hopf modules, and whose morphisms are the left-(resp. right-)H-module-left-H-comodule homomorphisms. The definitions for C H H and H C H are analogous.

2.
A two-fold Hopf module, X = (X, µ ℓ , µ r , ∆ ℓ ) , is an object, which is a H-bimodule in the category of left H-comodules, or in the language of Hopf modules: X ∈ Obj( H H C) and X ∈ Obj( H C H ) . The two-fold Hopf modules together with the H-bimodule-left-Hcomodule morphisms form the category of two-fold Hopf modules H H C H . The remaining three types of two-fold Hopf modules are defined in similar ways.
3. An object, (X, µ r , µ ℓ , ∆ r , ∆ ℓ ) , is called an H-Hopf bimodule if (X, µ r , µ ℓ ) is a H-bimodule, and (X, ∆ r , ∆ ℓ ) is a H-bicomodule in the category of H-bimodules, where the regular (co-)action on H and the diagonal (co-)action on tensor products of modules are used. Hopf bimodules together with the H-bimodule-H-bicomodule morphisms form the category which will be denoted by H H C H H .
It follows directly from the definition that any bialgebra, H , is a Hopf bimodule over itself with the regular actions and the regular coactions. Next we give the construction of the morphism Π ℓ ℓ (X) : X → X . Idempotency follows from straightforward application of the Hopf algebra and Hopf module axioms as depicted in Figure 4. In the same way the object, through which Π ℓ ℓ (X) : X → X factors, is identified as a (co)invariance.
Lemma 4.1. Let H be a Hopf algebra in C and (X, µ l , ∆ l ) be a left H-Hopf module. Then the following hold: (see also Figure 4b)) is an idempotent in End C (X). Figure 4: Hopf modules and projections are equalizer and coequalizer respectively. Hence H X is at the same time an object of invariants and coinvariants of X.
The following lemma embodies the categorical generalization of the Fundamental Theorem of Hopf modules, which can be thought of as the main reason for the structural rigidity of Hopf algebras. In the classical case it essentially states that all Hopf modules are free. The version below can be derived by translating the classical proof into the diagrammatic language, and generalizing from there to the braided situation. • H ⋉ X is an object H ⊗ X with Hopf module structure given by ∆ ℓ := ∆ ⊗ id X and µ ℓ := µ ⊗ id X , called standard Hopf module, and H ⋉ f := id H ⊗ f as morphism in C ; • H X splits the idempotent Π ℓ ℓ (X) from the previous lemma, and H f : are mutually inverse Hopf module morphisms.
Our strategy for extending the result from the previous two lemmas for Π ℓ ℓ (X) to the other projections is to find appropriate functors that turn left actions into right actions and map Π ℓ ℓ (X) to the respective Π • • (X) . To this end suppose X is an object in C equipped with some or all of the H-module and H-comodule structures from above, namely µ ℓ , µ r , ∆ ℓ , and ∆ r . We shall introduce the following notations for the same object X with modified (co)actions: ) to be the object equipped with the H ∨ -(resp. ∨ H-) (co)actions as depicted below: Hence the so defined operations ( , and [∨] ( ), identify, e.g. for ( ) [op] , a module-object X ∈ Obj(C) for a Hopf algebra H to be also a comodule in X ∈ Obj(C) for H [op] , where C is the same monoidal category as before but with mirrored braiding. It is easy to see that, similarly, an H−module morphism in C is at the same time an H [op] −comodule morphism in C with the modified coaction as above. Thus in this way we obtain functors between the (co)module subcategories of C as specified next. As functors of only the monoidal category C they act like identity on objects and morphisms.
For any functor from the above list with the source category H H C H H we keep the same notation for other functors, which act by the same rule on categories of Hopf modules or two-fold Hopf modules. We now find the desired results for the morphisms ∨ Π • • (X) : For a Hopf bimodule, X , the endomorphisms Π • • (X) and ∨ Π • • (X) , as given in Figure 4, are idempotents.
The objects that split idempotents Π • • (X) (resp. ∨ Π • • (X)) are all isomorphic. A representative of the corresponding class of isomorphic objects shall be denoted by Inv X (resp. by Int X ).
Proof. All assertions about the endomorphisms can be reduced to the ones for the idempotent from Lemma 4.1 using the identities below: be morphisms, which split the idempotent Π ℓ ℓ (X) (resp. Π ℓ r (X) ). Then by Lemma 4.1 both morphisms i and i ′ are equalizers of the pair ∆ ℓ , η ⊗ id X : X → H ⊗ X . Thus by the universal property of equalizer we have Y ≃ Y ′ . Other functors can be obtained as compositions of the functors from Lemma 4.3. For a Hopf H-bimodule X we consider the underlying object equipped with (co)module structures over the dual Hopf algebra H ∨ or ∨ H as shown in Figure 5.
Corollary 4.5. The following functors, whose actions on objects are as in Figure 5, and which are identity on morphisms, define the following equivalences of categories: Proof. The functors ( ) • between two-fold Hopf modules are "glued" from the following composite functors between Hopf modules:

H∨
Other functors are reversed forms of ( ) • .
Note that X • is not a Hopf bimodule. The left-right Hopf module axiom is not satisfied. In Section 7 we will introduce a construction, which turns X • into a Hopf H ∨ -bimodule.  Figure 6. Similarly, we can construct Hopf module isomorphisms corresponding to three other types of integrals. The inverse morphisms are given by the formulae

Invertibility of the object of integrals
In the case of self-dual Hopf algebras and bosonic integrals, meaning Int H ≃ 1I , these isomorphisms turn into Fourier transforms [19]. The braided Fourier transform H → H ⊗ ∨ H defined in [7] (where the H-valued integral is used) factorizes over our H F. Given these isomorphisms we shall now prove Proposition 3.1 and Theorem 3.3 : Figure 6: Proof of Proposition 3.1. In order to prove invertibility of Int H , we consider the Hopf module isomorphisms F H : is an isomorphism of standard Hopf modules in Hop Hop C. From this we conclude that We also compute,

Group-like elements
Let us now give the derivation of the special group-like elements (or moduli) exploiting universality properties of integrals, and discuss their properties as needed in the Radford formula.
For the same reason there exists a unique morphism, f , which makes the following diagram commutative:  Moduli and Comoduli are dual to each other in the following sense: There is a natural way in which the group-like elements act on the spaces Hom C (H, X) and Hom C (X, H) . Specifically, for any morphisms, f : H → X and g : X → H , we shall use the following abbreviated notations for these actions: Specialized to the integrals we obtain the following relations: c ℓ r c r ℓ = ρc ℓ ℓ c r r , where ρ := α • a ∈ Aut(1I) . Proof. In the case of the left equation from (4.4), we can see from the diagrammatic calculation in Figure 8, where the third equality follows from (3.1), that the left hand side is also a right integrals on H . Hence a. H is "proportional" to H in the sense of Proposition 3.1 by an element in in Aut(IntH) . The fact that the latter is given by the indicated isomorphisms c ∈ Aut(1I) is seen by composing both sides with • H . The remaining three relations between integrals are found similarly. Composition of these equalities with • H and (3.4) yield (4.6).
The scheme of this proof was taken from [26,15].
For a fixed choice of Int H all integrals are defined uniquely up to multiplication by elements of Aut(1I) . The normalizations can be chosen, such that any three of the constants c ℓ ℓ , c ℓ r , c r ℓ , and c r r are unit.

Integrals and the antipode
Before we consider the action of the antipode on integrals and idempotents, we need a presentation of S as given in the following two technical but easy lemmas: Figure 8: Proof of the right integral property.  Proof. This follows from the equation derived in Figure 11, to which we apply Lemma 4.8.
Besides the elements c • • we need another invertible element, σ ∈ Aut(1I) , which determines the self braiding of Int H . It is defined by either of the following conditions, which are equivalent by Lemma 2.1 and Theorem 3.3: Using cyclicity, tr 8 (f g) = tr 8 (gf ) , one can rewrite the latter expression for σ in many equivalent forms. For example, we have σ −1 = tr 8 ∆ op • µ • (S 2 ⊗ id) . The action of S on the integrals is now as follows: Figure 10: Figure 11: Proof of Lemma 4.9 Proposition 4.10.
Proof. The composition of both sides of the first identity from Figure 10 with H ⊗ id (Int H) gives the first identity from (4.8). All other identities are found analogously.
Using Theorem 3.3 we then find immediately the following relations between the special idempotents and the antipode:

Proof of the generalized Radford formula
Proof of Theorem 3.6. The general scheme of our proof is taken from [26,15]. The basic steps in the proof in the braided case are illustrated in Figures 12-14. The identity in Figure 12a) is a corollary of the identities in Figure 10, rewritten for H op and for H op . The first identity in Figure 12b) follows from the first or from the second identity in Figure 10, rewritten for H op or H op respectively. The second identity is the first one rewritten for (H op ) op .
The first identity in Figure 12c) is a result of combining the identities in Figure 10. The second is obtained by application of the identity in Figure 12b).
In Figure 13 the letter f denotes a morphism in End(H ⊗3 ) , presented by the framed diagram in Figure 12c). The first two lines of the former figure are obtained, e.g., in a similar way as the derivation in Figure 8.
Denoting x := ad α • S 4 • u 0 −2 • ad a , we can rewrite the middle equation of Figure 13 in a symbolic form, using new morphisms, a , b , c , and d , which can be visualized via Figure 12c). The left hand side of the equation there can be transformed to the first line in Figure 14, where u 0 2 denotes the inverse to u 0 −2 . Hence, the first equation in the last line of Figure 14 holds. Using Figure 12b) we get the last equation of Figure 14.

Remark 4.2.
Under the assumption that the antipode in H is an isomorphism, Proposition 3.1 implies that the object Int H is invertible. Conversely, if we suppose that the object Int H , which splits the idempotent ∨ Π r ℓ (H) , is invertible, then S −1 exists and can be derived from the first identity in Figure 12b). Figure 12: Proof of the generalized Radford formula (part 1) Figure 13: Proof of the generalized Radford formula (part 2) Figure 14: Proof of the generalized Radford formula (part 3)

Examples of braided Hopf algebras
Let us furnish in this section the general theory around Radford's formula that we have developed so far with a couple of prominent examples of braided, monoidal categories that are not abelian. The first is a variant of the category of surfaces and 3-cobordisms, for which the integrals are found even without the property of split idempotents. The second is any rigid, braided tensor category with split idempotents, in which the coend X∈C X ⊗ X ∨ exists. We shall describe in both cases the Hopf algebras structure, integrals, and other elements explicitly. These two categories also figure prominently in topological quantum field theories, where the task is to functor one into the other.

A topological example of a Hopf algebra
Crane and Yetter have shown in [8] that a 3-dimensional topological quantum field theory assigns a braided Hopf algebra to a torus with one hole. Yetter has explained in [32] that a torus with one hole T is a Hopf algebra in a braided category of cobordisms between oriented connected surfaces with one hole. The 3-dimensional manifolds which are the structure morphisms of T are described in both articles [8,32]. Independently Kerler [13] recovered this structure in a very similar category, Cob 3 (1) , whose objects are specially selected standard surfaces, and the morphisms are homeomorphisms classes of triples (M, ψ, σ) , where M is a 3-cobordism, ψ the homeomorphisms between ∂M and the standard surfaces, and σ the signature of a four-manifold bounding M in a standard fashion. Using a somewhat different topological language, not only the usual braided tensor structure and the Hopf algebra structures associated to the one-holed torus are found, but in addition to the elements in [8,32] also the cobordisms assigned to canonical Hopf pairings and the integrals are identified precisely in [13]. In particular, the integrals are directly interpreted as the algebraic elements intrinsically associated to elementary surgery. It is also realized in [13] that Cob 3 (1) , restricted to connected surfaces, is generated by the morphisms of the braided Hopf algebra structure together with the additional elements. (The original generators from [8,32] only produce cobordisms embeddable into R 3 ). I.e., a purely algebraic category can be defined that is freely generated by a Hopf algebra object, and which surjects onto Cob 3 (1) . It is further conjectured in [13] that there is some such definition of the algebraic category, which is in fact isomorphic to Cob 3 (1) for connected surfaces.
In this section we shall first review the explicit presentations of the cobordisms that are used for integrals and other Hopf algebra structures using another category, T C , of a topological and combinatorial nature, which is equivalent to Cob 3 (1) , as proven by Kerler [12]. The category T C is a subquotient of the category RT of ribbon tangles in [0, 1] × R 2 . Unlike the ordinary category of surfaces and 3-cobordisms, its central extension Cob 3 (1) admits interesting representations in braided categories built with the help of coends, as we will see in the second paragraph of this section.
Recall [31] that the objects of RT are non-negative integers, and the morphisms are ribbon tangles. For our purposes we define ribbon (or framed) tangles to be ambient isotopy classes of smooth embeddings of rectangles Furthermore, the images of the rectangles shall be tangent to the strip [0, 1] × R × 0 at these intervals, and the induced isomorphism of the tangent plane to the rectangle and the tangent plane to [0, 1] × R × 0 shall preserve the orientations.
In the drawings of ribbon tangles below we will represent them only by threads, tacitly assuming the blackboard framing, and explicitly insert powers of 2π-twists ν only when necessary. The tensor multiplication of objects is the addition of the non-negative integers. The unit object is 0. The composition (resp. the tensor product) of morphisms is given by vertically stacking tangle diagrams (resp. horizontally juxtaposing tangle diagrams) [31]. With these structures it is easily seen that RT is a braided, rigid, monoidal category.
Definition 5.1. The category T C of tangle-cobordisms is a subquotient of RT , whose objects are even, non-negative integers and morphisms of the form 2n → 2m are equivalence classes of ribbon tangles obeying the following property †: Property † there are precisely n + m rectangles, each of them connects two consecutive intervals of the same distinguished line, which is either 0 × R × 0 or 1 × R × 0 .
A morphism in T C can be thought of as an equivalence class of morphisms in RT . The corresponding equivalence relation is generated by the following moves. All of them are stable under composition and tensor product and preserve Property † so that T C inherits these structures from RT .
The identity morphism is the tangle (5.1) A tangle 2n → 2m represents a cobordism between surfaces of genus n and m . Note also that (TD1) is the 2-handle-slide move of any ribbon segment over a 0-framed, closed, unknotted ribbon.
The braiding in T C is the braiding from RT composed on one or both sides with the identity morphism (5.1) to ensure the condition †. See for example Figure 15.
The evaluations and the coevaluations in T C are those in RT composed with the identity Figure 15: Braiding in T C morphism (5.1), as depicted below for 2 = 2 ∨ : ev : 2 + 2 → 0 coev : 0 → 2 + 2 We assert that 2 ∈ T C has the structure of a Hopf algebra. Indeed, a set of structure morphisms that fulfills all the axioms of a braided Hopf algebra is given if we choose for the multiplication and the comultiplication the following two diagrams, , m : 2 + 2 → 2 ∆ : 2 → 2 + 2 for the unit and the counit the ones depicted next, and for the antipode and its inverse the pictures below.
The bialgebra axiom is verified by employing the handle-slide move (TD1). All other Hopf algebra axioms are proven straightforwardly. The projections of Lemma 3.2 for this Hopf algebra are easily found to be the following tangles: By Theorem 3.3 the form of these diagrams entails that the object Int 2 = 0 of integrals is the unit object, and that the integrals are given by the pieces Figure 16: The handle-slide move with arbitrary framing The generalized Radford formula from Theorem 3.6 yields for the Hopf algebra H = 2 the identity In fact, one can prove directly the stronger result S 2 = ν : 2 → 2 .
Finally, let us mention that the surgery calculus from [12] implies the Fenn Rourke and also more general two-handle slides. Here the case where the closed ribbon, over which to slide, is still an unknot, but may have arbitrary framing: The handle-slide move with arbitrary framing shown on Figure 16 holds in T C .
This follows from results of Kerler [12] and, alternatively, can be deduced directly from relations (TD1)-(TS3) in the category T C .

Coends as Hopf algebras
Let (C, ⊗, 1I) be a braided rigid category with split idempotents. Assume that the coend F = X∈C X ⊗ X ∨ exists in C . See, e.g., Mac Lane's book [16] for the definition of a coend.
Whenever the coend F exists in C , it is a Hopf algebra [19,21] with the structure morphisms given by the following pictures or commutative diagrams. The multiplication is determined by the condition that the diagram on the right of the following figure commutes for any pair of objects, (L, M ) . I.e., m F is the lift of the dinatural transformation defined by the braid on the left hand side.
The unit is 1I = 1I ⊗ 1I ∨ i 1 I − → F . The comultiplication ∆ on F is uniquely determined by the condition that the next equation shall hold for any object X .
The counit ε is given by the equation One also finds that the antipode S : F → F exists and is invertible. It is defined via the diagram A pairing of Hopf algebras, ω : F ⊗ F → 1I , is given by the tangle below. See also [19].
We shall call the category C modular in case the pairing ω is side-invertible (i.e., nondegenerate). Finally, it is easily seen that C is also a ribbon category.

Proposition 5.2 ([14]
). Let C be a modular category. Then the object of the integrals of the Hopf algebra F is isomorphic to 1I , its integrals are two-sided, i.e., F = F : F → 1I and F = F : 1I → F , and they are related by the equation In the case of abelian categories this was shown by Lyubashenko [19]. By Theorem 3.3 the number is invertible. We assume that it has a square root in End 1I so that we can rescale F in a way, The following theorem is a special case of the results proven in [14], where abelianness was assumed although not needed for the construction of the TQFT-functor. We shall also outline the proof, in which a lot of complications are avoided, since we have restricted ourselves to the tangles associated to connected, one-holed surfaces. For more detailed arguments the reader is referred to the original exposition. Theorem 5.3 (see also [14]).
Let C be a modular category and assume that the number in (5.6) is 1. Then there is a unique monoidal functor Φ : T C → C , for which Φ(2) = F and Φ( 2 ) = F , compatible with the braiding and ribbon twists, and which carries the Hopf algebra structure of 2 to the Hopf algebra structure of F . Outline of Proof: (a) Denote by DoubleRT the subcategory of RT , which has even, nonnegative integers as objects and ribbon tangles satisfying Condition † of Definition 5.1 as morphisms. With no loss of generality we may assume that C is a strict monoidal category with ∨ X = X ∨ , X ∨∨ = X and u 2 0 = 1 (see [18,14]). To begin with, we construct a functor Φ : DoubleRT → C following [19,20].
SetΦ(2n) = F ⊗n . Consider an arbitrary tangle, T : 2n → 2m ∈ DoubleRT , draw its planar diagram T ′ with threads, crossings and ribbon twists, ν , and mark an absolute maximum and on each of these maxima a closed thread with ends attached to the target (bottom) line. For an arbitrary family, X 1 , . . . , X n , of objects of C one can then construct a morphism φ(T ′ ) : X 1 ⊗ X ∨ 1 ⊗ · · · ⊗ X n ⊗ X ∨ n → F ⊗m by assigning braidings to crossings, evaluations or counits to minima, coevaluations to ordinary maxima, and the integral-element F : 1I → F to the special absolute maxima. By the universal property of the coend the morphism φ(T ′ ) factorizes through φ(T ′ ) : F ⊗n ≃ X 1 ,...,Xn∈C In fact, this morphism depends only on T and not on the choice of maxima in T ′ , since, for example, In order to see that settingΦ(T ) =φ(T ′ ) gives the desired functorΦ , we notice that the counit of F is determined by the evaluation as in equation (5.3).
(b) We want to check that ribbon tangles, equivalent under moves (TD1)-(TS3), are sent to the same morphisms byΦ : By the definition of integrals or by Property (3.1) of F we have If we now compose this identity with the coaction δ of F with respect to an arbitrary object Y ∈ C , pictured below, for any X, Y ∈ Obj C. Therefore,Φ is stable under Move (TD1).
To prove the remaining relation (TS3) notice that by the handle-slide moves (TD1) and (TS2). Therefore, using (TD1) once again, we get The non-degeneracy of the form ω implies now Move (TS3). This completes the construction of the functor Φ : T C → C .
(c) The graphical formulae for the structure maps for 2 differ from those for F by expressions as in (5.1). Since we have proven (TS3), the structure morphisms of 2 are sent by Φ to structure morphisms of F . The meaning of Theorem 5.3 is that all calculations using the handle-slide move with arbitrary framing (see Proposition 5.1) are valid for F . As a corollary one gets modular relations as in [19], representations of mapping class groups of surfaces with one hole as in [20,25], and, eventually, topological quantum field theories [14].

Integrals and related structures
The purpose of this section is to explain how the structures of integrals and moduli are modified by various procedures that can be applied to create new braided Hopf algebras from known ones. Often new Hopf algebras are obtained by taking some type of product between two given ones. The first example of such a product is the Heisenberg double H(H) = H# ∨ H , where a Hopf algebra, H , is combined with its dual. This is of particular interest since Hopf modules of H Figure 17: will turn out to be in a one-to-one correspondence with ordinary modules of H(H) . We shall also discuss a very general type of cross product, A ⋉ B , between two braided Hopf algebras.
Finally, we will also describe the procedure of "transmuting" the coproduct structure of a Hopf algebra, an idea, which has its origin in the Tannaka Krein theory for braided categories.

Heisenberg double, Hopf modules and integration.
For a Hopf algebra, H , in C with a dual, ∨ H , one can define the algebra H(H) = H# ∨ H , called the Heisenberg double, in the following way: ∨ H becomes an algebra in the category C Hop with right H op -modules structure given by the composition The multiplication on the corresponding smash-product algebra H# ∨ H is shown on Figure 17a). Similarly, H becomes an algebra in ( ∨ H)op C . The corresponding smash-product algebra H# ∨ H is the same. Note that the Heisenberg double H(H) is isomorphic to the "matrix algebra" of H , which is The existence of such an isomorphism for an ordinary Hopf algebra is a special case of the duality theorem from [6]. See [3] for the braided setting. The category of Hopf H-modules is identified with the category of H(H)-modules as follows: Proposition 6.2. The categories H H C and (H# ∨ H) C are isomorphic. The isomorphism is the identity on the underlying objects and morphisms in C , and it turns a Hopf H-module, (X, µ ℓ , ∆ ℓ ) , into an (H# ∨ H)-module, for which the action given by the composition In [7] an approach to integration on braided Hopf algebras involving Heisenberg doubles is developed. There the following two idempotents ("vacuum projectors") in H(H) play a crucial role: Note that the elements E and E act on a left H(H)-module (≡ left Hopf H-module), X , in the following way: Integrals, cross products, and transmutation.
Cross products and transmutation are basic constructions that allow us to obtain new braided Hopf algebras from given ones [23,24]. Here we use generalizations of these constructions from [2], and show that integrals on cross product Hopf algebras and on transmutation Hopf algebras are obtained in a simple way from integrals on the initial Hopf algebras.

Proposition 6.3 ([2]
). Let A be a Hopf algebra in C , and B be a Hopf algebra in the category DY (C) A A of crossed modules. (A crossed module is an object with both module and comodule structures satisfying the compatibility condition presented on Figure 18a)).
Then A ⋉ B with underlying object A ⊗ B , multiplication, comultiplication, and antipode as shown on the Figure 18b) is a Hopf algebra in C .
Int B is an invertible object in C and, therefore, the A -(co)module structures on Int B are given by the formulae for a certain multiplicative functional, α B/A , and some group-like element, a B/A . The crossed module axiom for Int B then takes on the form The notion of braided quantum groups (quasitriangular braided Hopf algebras) was introduced in [23], and the basic theory was developed there. We use the slightly modified definitions from [2], which reflect the symmetry between the two coalgebra structures under consideration. A pair of Hopf algebras, (A, A) , in C ×C and a bialgebra copairing, R : 1I → A op ⊗ A , in C , which is invertible as an algebra element, define a braided quasitriangular Hopf algebra, (A, A, R) , if A and A only differ in their comultiplication and antipode, i.e., if A = (A, µ, η, ∆, ε, S) and A = (A, µ, η, ∆, ε, S) , and if the identity (Ψ A,A • ∆) · R = R · ∆ holds. See Figure 19a) for an illustration of the latter. Similar to the case of ordinary quantum groups it is shown in [2] that the antipode of any quasitriangular Hopf algebra of the form (A, A, R) is invertible. In particular, we have For a quasitriangular bialgebra, (A, A, R) , we define the category C O(A,A) as the full subcategory of the category C A of A-right modules, whose objects (X, µ r ) satisfy the identity indicated in Figure 19b). This category is identified with a full braided monoidal subcategory of DY (C) A A [2,23]. I.e., every module (X, µ r ) in C O(A,A) becomes a crossed module, (X, µ r , ∆ r ) , Figure 19: in DY (C) A A through the coaction ∆ r := (µ r ⊗ id A ) • (id X ⊗ R) . Braiding in the category is given by the diagram in Figure 19c). The transmutation H of a Hopf algebra H is the underlying algebra of H with a new comultiplication, ∆ H , and a new antipode S H , as defined in Figures 19e-f). There we set In addition to this observation from [2], we shall next determine also the integrals of H : determine explicitly the integrals on T ∧ C (X) and derive from general integration theory some properties of the objects X ∧n .
First let us outline the braided combinatorics, as in [22,4], in an additive braided monoidal category, C . The canonical epimorphism of the braid group B n into the permutation group S n admits a section S n − → B n , which maps the standard generators to the standard generators and is uniquely determined on other elements by the property where ℓ(σ) is the length (of the minimal decomposition) of σ . Since the category C is braided, there is a canonical mapping B n → End(X ⊗ n ) . The image of σ ∈ S n under the composition S n → B n → End(X ⊗ n ) is denoted by σ C (X) . Now let π = (j 1 , . . . , j r ) be any N-partition of j , i.e., j = j 1 + · · · + j r and j 1 , . . . j r ∈ N . We consider the shuffle permutations S j π ⊂ S j . For every l ∈ {1, . . . , r} they are mapping the j l elements in {1, . . . , j} to {j 1 + · · · + j l−1 + 1, . . . , j 1 + · · · + j l } without changing their order. The set of inverse permutations of S j π is denoted by S π j . For every partition π = (j 1 , . . . , j r ) of j , any object X ∈ Obj(C) and any λ ∈ Aut(1I) we define braided multinomials as endomorphisms of X ⊗ j in C by the following identities π j X; λ = j 1 . . . j r j X; λ := σ∈S π j λ ℓ(σ) σ C (X) , j π X; λ = j j 1 . . . j r X; λ := σ∈S j π λ ℓ(σ) σ C (X) .
For any 0 ≤ k ≤ j we shall use abbreviated notations as follows: j k X; λ := j (k,j−k) X; λ , k j X; λ := (k,j−k) j X; λ and j X; λ ! := j 1...1 X; λ = 1...1 j X; λ . In this language we then derive by successive application of eq. (7.3) the identities [j]! = 2 1 ⊗ id X ⊗ j−2 • 3 2 ⊗ id X ⊗ j−3 • · · · • j j − 1 (7.4) and [j + k]! = ([j]! ⊗ [k]!) • j + k j , (7.5) Definition 7.1. Let I be N or Z/n , considered as a discrete category. We consider the functor category C I as an I-graded category over C . The objects of C I are given by I-tuples, X = (X 0 , X 1 , . . . ) , of objects, X j ∈ Obj(C) , where j ∈ I . The morphisms of C I are of the formf = (f 0 , f 1 , . . . ) :X →Ŷ , where f j : X j → Y j is a morphism in C for all j ∈ I . The category C I naturally inherits a braided monoidal structure from C . The unit object in C I is given by1I = (1I, 0, 0, . . . ) , the tensor product is defined by (X ⊗Ŷ ) n = k+l=n X k ⊗ Y l , where n, k, l ∈ I , and the tensor product of I-graded morphisms is built analogously. Besides the natural braided structure, the category C I actually admits a family of braidings given by (Ψ (λ) X,Ŷ ) n = k+l=n λ k l Ψ X k ,Y l , for any λ ∈ Aut C (1I) , if I = N, and for any λ ∈ Aut C (1I) with λ n = 1, if I = Z/n. 1. For a given object, X , in an additive braided category, C , the tensor algebra {X ⊗n } in C N admits two Hopf algebra structures, T C (X) = {X ⊗n }, µ, η, ∆, ε and • ε , which are given as follows: µ n,m ∼ = id X ⊗ n+m : X ⊗ n ⊗ X ⊗ m → X ⊗ n+m , ∆ n,m ∼ = n + m n X; λ : X ⊗ n+m → X ⊗ n ⊗ X ⊗ m , • µ n,m ∼ = n n + m X; λ : X ⊗ n ⊗ X ⊗ m → X ⊗ n+m , • ∆ n,m ∼ = id X ⊗ n+m : X ⊗ n+m → X ⊗ n ⊗ X ⊗ m , η n = Now let C be a rigid braided monoidal abelian category with a biadditive tensor product. In this case the unit object 1I is semisimple, see [9], so that we can suppose without restriction of generality that End C (1I) is a field. Rigidity implies that the tensor product is exact. In this situation we have that for any Hopf algebra morphism, f , the intermediate object of an epi-mono decomposition, f = im f • coim f , can be equipped with a Hopf algebra structure, such that both, im f and coim f , are Hopf algebra morphisms. This allows us to make the following definition: Definition 7.2. We will denote by T ∧ C (X) = {X ∧n } the Hopf algebra determined by the epimono decompositionÂ Let us suppose that λ n = 1 , [n|X]! = 0 , and [n − 1|X]! = 0 . In this case also X ∧k = 0 for any k ≥ n , and T ∧ C (X) can be considered as a Hopf algebra in the rigid braided monoidal abelian category C Z/n .
Proof. We have that ∨ Π • • T ∧ C (X) n−1 = id X ∧(n−1) , because m T ∧ C (X) k,ℓ = 0 for k + ℓ ≥ n . The condition that End C Int(T ∧ C (X)) has no nontrivial idempotents implies that Int(T ∧ C (X)) has only one nonzero component.
The Hopf algebra from Example 3.1 was obtained as T ∧ (k) for λ = q a primitive n th root of unit. Our results show similarities between the Hopf algebras T ∧ (k) and T ∧ (X) in the most general case. This follows from the fact that the matrix element of [n + m + 1|X ⊕ Y ]! , which is a morphism from X ⊗k ⊗ Y ⊗ℓ to X ⊗k ⊗ Y ⊗ℓ , equals to [k|X]! ⊗ [ℓ|Y ]! and other matrix elements are either of such type (up to isomorphism) or zero.
Remark 7.2. Suppose X lives in the category DY (C) H H of crossed modules over a Hopf algebra, H , and d : H → H ⋉ X defines a first order differential calculus. Then H ⋉ T ∧ C(X) is a Hopf algebra in C Z/n , and d has a unique extension, d ∧ , on H ⋉ T ∧ C(X) , which turns H ⋉ T ∧ C(X) into a differential Hopf algebra (braided De Rham complex) [4]. Integrals on H ⋉ T ∧ C(X) are calculated by the formulae (6.2) for integrals on cross products. They are nonzero only on one component H ⋉ X ∧(n−1) and play the role of the integration over a volume form.
• Let X, µ X a homomorphism of right H ∨ -comodules implies (and is equivalent to) the generalized Radford formula. This gives the third proof of Theorem 3.6.
Remark 7.4. The left dual Hopf H-bimodule to X ∈ Obj H H C H H is obtained by application of the functor ∨ ( ) in C to the Hopf H ∨ -bimodule X • ⊗ H H ∨ . As a result we obtain an object, Int H ⊗ ∨ X , equipped with tensor product H-(co)module structures, where Int H is considered with trivial (co)actions, and the structures of • ( ∨ X) are modified by means of a H , α H , Ω Int H H .