Transparency condition in the categories of Yetter-Drinfel d modules over Hopf algebras in braided categories

Transcription

1 São Paulo Journal of athematical Sciences 8, 1 (2014), Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories. Femić IERL, Facultad de Ingeniería, Universidad de la República, Julio errera y Reissig 565, ontevideo, Uruguay. address: bfemic@fing.edu.uy Abstract. We study versions of the categories of Yetter-Drinfel d modules over a opf algebra in a braided monoidal category C. Contrarywise to espalov s approach, all our structures live in C. This forces to be transparent or equivalently to lie in üger s center Z 2(C) of C. We prove that versions of the categories of Yetter-Drinfel d modules in C are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel d double D() C for finite. We obtain that these categories polarize into two disjoint groups of mutually isomorphic braided monoidal categories. We conclude that if Z 2(C), then D() C embeds as a subcategory into the braided center category Z 1( C) of the category C of left -modules in C. For C braided, rigid and cocomplete and a quasitriangular opf algebra such that Z 2(C) we prove that the whole center category of C is monoidally isomorphic to the category of left modules over Aut( C) - the bosonization of the braided opf algebra Aut( C) which is the coend in C. A family of examples of a transparent opf algebras is discussed. 1. Introduction Yetter introduced in [27] crossed bimodules generalizing to opf algebras the notion of crossed modules over finite groups, which appeared in topology. These new objects are modules and comodules over a opf algebra over a commutative ring with a certain compatibility condition. In [11] they were used to generate solutions to the Yang-axter equation and accordingly were called Yang-axter modules. Yetter s construction and 2000 athematics Subject Classification. 16T05, 18D10. Key words: braided monoidal category, braided opf algebra, Yetter-Drinfel d modules, center category. 33

2 34. Femić its variations were studied in [23] where they were termed Yetter-Drinfel d structures. The initial Yetter s category is denoted by YD. For a finite-dimensional opf algebra ajid proved that the category D () of modules over the Drinfel d double D () op is isomorphic to YD. In [22, Proposition 2.4] the analogous result to the former is proved for the left-right version of the Yetter-Drinfel d category: D() YD, where D() ( op ). Another, categorical interpretation of the Yetter-Drinfel d categories is that they can be seen as the center (or the inner double) of the category of modules over the opf algebra. The center construction (which to any monoidal category assigns a braided monoidal category) is a special case of Pontryagin dual monoidal category, [13]. As observed by Drinfel d [8] and proved in [15, Example 1.3] and [10, Theorem XIII.5.1] the left (resp. right) center of the category of left modules over is isomorphic to YD (resp. YD ). For the details on the center construction we refer to [10]. In Radford biproduct opf algebra [20], ajid observed that is a opf algebra in the category YD. If is quasitriangular, a left -module can be equipped with a left -comodule structure in such a way that one gets a Yetter-Drinfel d module. In this particular case, the opf algebra is named bosonization in [17]. The reversed process - recovering a braided opf algebra out of an ordinary one - was studied in [17, Section 2] and is called mutation. Yetter-Drinfel d modules through their equivalence with opf bimodules, [24], emerge in Woronowicz s approach to bicovariant differential calculi on quantum groups, [26]. The first order differential calculi over a opf algebra over a field consist of a derivation d Ω 1 (), where Ω 1 () is the bicovariant bimodule and has a structure of a opf bimodule. Another and exotic appearance of left-right Yetter-Drinfel d modules we find in 3Dtopological quantum field theories, [6, Theorem 3.4]. Some of the above-mentioned constructions have been generalized to any braided monoidal category. For a opf algebra in a braided monoidal category C which admits split idempotents the equivalence of the categories of opf bimodules and of Yetter-Drinfel d modules YD(C) was proved in [2]. In the same paper the authors prove that the category of bialgebras in YD(C) is isomorphic to the category of admissible pairs in C. The proof relies on the previously generalized Radford-ajid theorems to the braided case, [1, Theorems and 4.1.3]. The former result provides a natural and easy description for the Radford-ajid criterion for when a opf algebra is a cross product. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

3 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 35 In this paper we study categories of Yetter-Drinfel d modules over a opf algebra in a braided monoidal category C with a different approach than in [1]. oreover, we address the question of their isomorphism with the categories of left and right modules over the Drinfel d double in C. When studying the monoidal structures of the respective categories, one is tempted to impose the symmetricitity of the base category C as a necessary condition. To avoid this obstacle, espalov works in [1] both with C and with its opposite and co-opposite categories, C op and C cop respectively, and with a category C. The opposite category of C has the same objects as C, but the arrows go in the reversed order. The braiding in C op is given by X Y Φ Y,X Y X, where Φ is the braiding of C. The category C cop has reversed tensor product and the braiding X cop Y Y X Φ Y,X X Y Y cop X. The category C has the same tensor product and its braiding is Φ 1. Contrarywise, in the present paper we work only with the base category C and investigate which conditions we have to impose in order that the construction works. We find that it is sufficient to require that the braiding Φ of C fulfills Φ,X Φ 1 X, for every X C. This condition we have encountered also in [7]. It had already appeared in the literature in [4] and [18, Definition 2.9]. In the terminology of the former reference we have that is transparent, while due to the latter belongs to üger s center Z 2 (C) {X C Φ Y,X Φ X,Y id X Y for all Y C} of the braided monoidal category C. The notation Z 1 (C) üger reserved for the center of the monoidal category C that we mentioned above. If Φ X,Y Φ 1 Y,X for some X, Y C, we say that Φ X,Y is symmetric. As a particular case of the bicrossproduct construction (with trivial coactions) in braided monoidal categories, [29], we study the Drinfel d double D() of in C. We obtain that D() ( op ) in C is a bicrossproduct opf algebra for finite, if Φ, is symmetric. Equivalent conditions for when D() is (co)commutative are given. We prove that the category of modules over D() in C is isomorphic to that of Yetter-Drinfel d modules over in C if is transparent. In particular, we get that the two diagrams D()C YD(C) C D() YD(C) 1 YD(C) op YD(C) cop YD(C) op cop and 2 copyd(c) op YD(C) op copyd(c) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

4 36. Femić commute as arrows of mutually isomorphic braided monoidal categories. Our goal in this paper is not to prove that all the above Yetter-Drinfel d categories are braided monoidally isomorphic, as it was proved in [1, Corollary 3.5.5] under the previously mentioned suppositions. Rather, we set up a different approach and investigate how far we can get in the study of the above categories. espalov proved in [1, Proposition 3.6.1] that the category of left-left (resp. right-right) Yetter-Drinfel d modules in C is braided monoidally isomorphic to a subcategory of the center of the category of left -modules (resp. right -comodules). We differentiate the left and the right center category and observe that the mentioned category isomorphism can be extended to the categories in the rectangular diagrams 1 and 2 above yielding two polarized groups of mutually isomorphic braided monoidal categories: Zl C( C) Zr C ( C) Zr C (C ) Zl C(C ) Zr C ( C) Zl C( C) and Zr C (C ) Zl C(C ). As for the relation between the centers Z 1 and Z 2 in the notation of üger, we obtain in particular that if Z 2 (C), then D() C Z 1,l ( C) (and similarly C D() Z 1,r (C )). For the whole center category of a braided, rigid and cocomplete category C ajid proved Z 1,l (C) C Aut(C) in [14], where Aut(C) is the coend opf algebra in C. For a quasitriangular opf algebra C [16, Definition 1.3] such that Z 2 (C) we obtain Z 1,l (C ) C Aut(C ) as monoidal categories, where Aut(C ) is the bosonization of the braided opf algebra Aut(C ) in C. When C V ec and is a finite-dimensional quasitriangular opf algebra, this recovers the known isomorphism Z l ( ) D (). We point out that a similar result to ours was proved in [5] where the authors work with opf monads and construct a Drinfel d double in a fully non-braided setting. At the end we present a family of transparent opf algebras in braided monoidal categories which support our constructions. The paper is organized as follows. In Section 2 we present preliminaries on some structures in any braided monoidal category C. In the next section we study the braided monoidal category of left-right Yetter-Drinfel d modules YD(C) op (assuming that is transparent). We point out that the categories YD(C) and YD(C) are braided monoidal without any São Paulo J.ath.Sci. 8, 1 (2014), 33 82

5 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 37 symmetricity conditions on the braiding. Section 4 recalls the bicrossproduct construction (with trivial coactions) in C. We use it to study the Drinfel d double D() ( op ) in C for a finite, when Φ, is symmetric. Section 5 is devoted to the braided monoidal isomorphism D()C YD(C) op. In Section 6 we compare different versions of the braided Yetter-Drinfel d categories in C, connecting them with the categories of left and right modules over the Drinfel d double in C. In the penultimate section we deal with the center construction and relate it to the Yetter-Drinfel d categories. The last section presents some examples. Acknowledgements. This work has partially been developed in the athematical Institute of the Serbian Academy of Sciences and Arts in elgrade (Serbia). The author wishes to thank to Facultad de Ciencias de la Universidad de la República in ontevideo for their worm hospitality and provision of the necessary facilities. y gratitude to Yuri espalov for clarifying me his proof of [1, Proposition 3.6.1], and to Alain ruguière for the discussions on the construction of the Drinfel d double via monads. 2. Preliminaries We assume the reader is familiar with the theory of braided monoidal categories as well as with the notation of braided diagrams. For the references we recommend [10] and [1]. We recall that a opf algebra in a braided monoidal category C was introduced by ajid in [15]. In the same paper it was proved that the categories of modules and comodules over a bialgebra in C are monoidal. We only outline some basic conventions. In view of ac Lane s Coherence Theorem we will assume that our braided monoidal category C is strict. Our braided diagrams are read from top to bottom, the braiding Φ X Y Y X and its inverse in C we denote by: Φ X,Y X Y Y X and Φ 1 Y,X Y X X Y. For an algebra A C and a coalgebra C C the multiplication in the opposite algebra A op of A and the comultiplication in the co-opposite coalgebra C cop of C we denote by: A op A A A and C cop C respectively. The antipode S of a opf algebra in C is a bialgebra map S op,cop. Its compatibility with multiplication and comultiplication C C São Paulo J.ath.Sci. 8, 1 (2014), 33 82

6 38. Femić is written as: S S S and S S S respectively. oreover, S is the antipode for op,cop. ote that for a bialgebra C, neither op nor cop is a bialgebra, unless the braiding Φ fulfills Φ, Φ 1,. We recall some basic facts A monoidal category C is called right closed if the functor C C has a right adjoint, denoted by [, ], for all C. For C, the object [, ] is called inner hom-object. The counit of the adjunction evaluated at is denoted by ev, [, ]. It satisfies the following universal property: for any morphism f T there is a unique morphism g T [, ] such that f ev, (g ). If f is a morphism in C, then [, f] [, ] [, ] is the unique morphism such that ev, ([, f ] ) f ev,. The unit of the adjunction α [, ] is induced by ev, (α ) id. A monoidal category C is called left closed if the functor C C has a right adjoint {, } for all C. The counit of this adjunction evaluated at C is denoted by ev, {, } and the unit by α {, }. It obeys ev, ( α) id. When C is braided, there is a natural equivalence of functors [, ] {, } and C is right closed if and only if it is left closed. Throughout we will write [, ] for both types of inner hom-bifunctors, the difference will be clear from the context. The object [, ] is an algebra for all C Let P be an object in C. An object P C together with a morphism e P P P I is called a left dual object for P if there exists a morphism d P I P P in C such that (P e P )(d P P ) id P and (e P P )(P d P ) id P. The morphisms e P and d P are called evaluation and dual basis, respectively. In braided diagrams the evaluation e P and dual basis d P are denoted by: e P P P and d P P P and the two identities they satisfy by: P id P (2.1) P P id P. (2.2) P São Paulo J.ath.Sci. 8, 1 (2014), 33 82

7 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 39 Symmetrically, one defines a right dual object P for P with morphisms e P P P I and d P I P P. Left and right dual objects are unique up to isomorphism. In a braided monoidal category the left and the right dual for P coincide. The corresponding evaluation and dual basis morphisms are related via: e P e P Φ P,P (2.3) d P Φ 1 P,P d P (2.4) see e.g. [25, Prop. 2.13, b)] (we take here the opposite sign of the first power of the braiding) An object P C is called right finite, if [P, I] and [P, P ] exist and the morphism db P [P, I ] [P, P], called the dual basis morphism as well, defined via the universal property of [P, P ] by ev P,P (db P) P ev P,I is an isomorphism. One may easily prove that if P is right finite, then ([P, I], e P ev) is its left dual. The dual basis morphism is d P db 1 η [P,P ], where η [P,P ] is the unit for the algebra [P, P ]. A similar claim holds for a left finite object, which is defined similarly as a right finite object. In a braided monoidal category an object is left finite if and only if it is right finite. If P is a finite object, then so is P and there is a natural isomorphism P P In the following we collect some facts about duality of opf algebras from [25, 2.5, 2.14 and 2.16]. Let C be a closed braided monoidal category. (i) If is a coalgebra in C, then [, I] is an algebra. (ii) If is a finite algebra in C, then is a coalgebra. (iii) If is a finite opf algebra in C, then so is. We give here the structure morphisms. The finiteness condition in ii) and iii) is needed in order to be able to consider ( ), which allows to define a comultiplication on using the universal property of [, I]. The multiplication, comultiplication, antipode S, unit and counit of are given by: S S (2.7) (2.5) (2.8) (2.6) (2.9) respectively (one uses the universal property of [, I]). It is easy to see that a finite algebra A in C is commutative if and only if A is a cocommutative coalgebra. For an algebra A C and a coalgebra C C we denote by A C and C C the categories of left A-modules and right C-comodules, respectively. The São Paulo J.ath.Sci. 8, 1 (2014), 33 82

8 40. Femić proof of the following proposition is not difficult. The first statement is proved in [25, Proposition 2.7]. Proposition 2.5. Let C be a finite coalgebra. If C, then C with the structure morphism given in (2.10). If C, then C with the structure morphism given in (2.11). These assignments make the categories C and C isomorphic. (2.10) (2.11) Throughout the paper C will be a braided monoidal category with braiding Φ and C a opf algebra having a bijective antipode. 3. Some braided monoidal categories of Yetter-Drinfel d modules A left -module and left -comodule C and a right -module and right -comodule L C are called respectively left-left and right-right Yetter-Drinfel d modules over in C if they obey the compatibility conditions: (3.1) and L L L L (3.2) respectively. A left-right Yetter-Drinfel d module over is a left -module and right -comodule C whose -structures are related via the relation:. (3.3) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

9 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 41 In all the cases we will shorten the term Yetter-Drinfel d module to YDmodule. The categories of left-left YD-modules and left -linear and left -colinear morphisms in C (which we denote by YD(C)) and that of rightright YD-modules and right -linear and right -colinear morphisms in C (denoted by YD(C) ) respectively, are known to be braided monoidal categories with braidings: Φ L X,Y X Y and Φ R W,Z W Z (3.4) Y X Z W for objects X, Y YD(C) and W, Z YD(C) respectively, (see e.g. [1]). owever, in order that the category of left-right YD-modules be braided monoidal, some symmetricity conditions on the braiding in C should be assumed, as we will see further below. Like in [1, Thm ] espalov has that the category YD(C) op is braided monoidal, but there he considers the tensor product of two left-right YD-modules a right op -comodule via the codiagonal structure in the category C, whereas the -module structure he considers in C (as in [1, Lemma 3.3.2]). Thus for two objects, of this category, the object has the -comodule structure: ρ ( op)( Φ 1, )(ρ ρ ). espalov considers op Φ 1, (instead, we regard here the positive sign of the braiding) in order that op be a bialgebra in C. In the present paper we prefer to consider all the structures in C. Accordingly, we will have that the categories YD(C) op and copyd(c) are braided monoidal if the braiding Φ in C fulfills Φ,X Φ 1 X, for every corresponding YD-module X C. We will say that Φ,X is symmetric. As a matter of fact, if Φ, and Φ,X are symmetric (indeed itself is a YD-module over itself), then the upper structure coincides with the usual codiagonal comodule structure on in C. evertheless, we will prove explicitly the claims by our approach as this is the general setting of our work and we will prove also other results in this manner. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

10 42. Femić efore proving that the category YD(C) op will note some important facts. Observe that: is braided monoidal, we (3.5) since: + +. From this point on we will assume that the antipode of is bijective (which is fulfilled for example if is finite and C has equalizers, [25, Theorem 4.1]). The sign + stands for the antipode whereas stands for the inverse of the antipode. Furthermore, we have that the condition (3.3) is equivalent to: (3.6) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

11 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 43 To prove this assume that (3.3) holds. Then: (3.3) Conversely, (3.6) implies: (3.6) coass. ass. (3.5) unit counit (3.5) coass. unit ass. counit.. Remark 3.1. If Φ, is symmetric, (3.5) can be considered with Φ, instead of Φ 1, ; then one proves that (3.7) is equivalent to (versions of the relations (3.3) and (3.6)). (3.8) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

12 44. Femić It is important to note that itself is a YD-module over itself with suitable structures. For example, it is a left-right YD-module with the regular action and the adjoint coaction:. For the other versions of a YD-module (see Section 6) can be equipped with similar structures - regular (co)actions and adjoint (co)actions. The last convention before the promissed proof is that throughout, by abuse of notation, we will write Φ, is symmetric for all YD(C) op, and similarly for other versions of the YD-categories, when strictly speaking we should say for all C. Indeed, via the forgetful functor U YD(C) op C every YD(C) op is an object in C, and every C can be equipped with trivial -(co)module structures to form a YD-module. Proposition 3.2. Assume that Φ, is symmetric for every left-right YDmodule over in C. The category YD(C) op is braided monoidal with braiding and its inverse given by: Φ, and (Φ,) 1 for, YD(C) op. Proof. ecause of the symmetricity assumption on Φ we will consider the YD-compatibility condition from the above Remark. Let and be two left-right YD-modules over. We consider their tensor product as a left -module and right op -comodule with the (co)diagonal structures. We now prove that the YD-compatibility of these -structures holds for São Paulo J.ath.Sci. 8, 1 (2014), 33 82

13 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 45 : coass. ass. (3.7) coass. ass. Φ, São Paulo J.ath.Sci. 8, 1 (2014), 33 82

14 46. Femić (3.7) Φ, Φ, coass. ass.. The check that Φ satisfies the braiding axioms we leave to the reader. We prove here the -linearity of Φ : Φ, (3.6) coass. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

15 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 47 mod. ass. Φ, Φ, (3.5) mod.. The op -colinearity of Φ follows from: comod. (3.8) coass. ass. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

16 48. Femić Φ, (3.5) comod. Φ, Φ, The proof that the inverse of Φ is given as in the announcement of the claim is straightforward. Remark 3.3. ote that because of the assumption that Φ, is symmetric, instead of Φ 1+ Φ in Proposition 3.2 we can also consider the braiding:. Φ 1,. Remark 3.4. With the same conditions as in Proposition 3.2 one has that the category copyd(c) is braided monoidal with braiding and its inverse given by: Φ 2+, and (Φ 2+,) 1 for, copyd(c). Analogously as in Remark 3.3, the braiding Φ 2+ can be taken in the form Φ 2. ote that copyd(c) is not braided by Φ 1±, since Φ 1± is not left cop -linear even if CV ec, the category of vector spaces. Thus the identity functor Id YD(C) op copyd(c) is not an isomorphism of braided monoidal categories although it is monoidal. 4. icrossproducts in braided monoidal categories icrossproducts in braided monoidal categories (also called cross product bialgebras) were treated in [29, 3]. We recall here bicrossproducts with São Paulo J.ath.Sci. 8, 1 (2014), 33 82

17 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 49 trivial coactions. Let and be bialgebras in C, where is a left - module coalgebra and is a right -module coalgebra. Assume further that the following conditions are fulfilled: ; and ; ialgebras and described above are called a matched pair of bialgebras in C. We define as the tensor product endowed with the codiagonal comultiplication, usual unit η and counit ε (that is, η η and ε ε respectively), and associative multiplication given by:. In [29, Theorem 1.4] it is proved that is a bialgebra. oreover, if both and are opf algebras, by [29, Theorem 1.5] we know that so is with the antipode given by: S S S + + S +. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

18 50. Femić From here it follows: S (η ) η S. (4.1) For a module over in C we will consider:. (4.2) Lemma 4.1. Let and be a matched pair of bialgebras. An object is a module over in C if and only if it is an - and a -module satisfying the compatibility condition: Proof. An object is a module over if and only if:.. (4.3) Applying this to η η, we obtain (4.3). For the converse observe that the above equality follows from (4.3) and the - and -module properties of. We now want to consider a particular case of a bicrossproduct - the Drinfel d double of. A tedious direct check, which we omit here for practical reasons, shows: São Paulo J.ath.Sci. 8, 1 (2014), 33 82

19 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 51 Proposition 4.1. Let C be a finite opf algebra with a bijective antipode and the braiding such that Φ, and Φ, are symmetric. Then is a bicrossproduct with ( op ) and the actions: op ev op ev and The bialgebra is called the Drinfel d double of and is denoted by D(). Throughout, apart from assuming that our opf algebras have a bijective antipode, when we deal with D() we will also assume that is finite. As we mentioned before, the antipode of a finite opf algebra is bijective if e.g. C has equalizers. ote that is a bialgebra since Φ, is symmetric (we commented this before 2.1). We only point out that in the proof of the above claim one uses the identity that we next present. earing in mind that ( op ), we have: (2.6) ev (4.4) Composing this from above (in the braided diagram orientation) with Φ, and applying ev evφ due to (2.3), by naturality we obtain: ev Φ, ev ev (4.5) As a matter of fact the two symmetricity conditions for Φ, and Φ, in Proposition 4.1 are equivalent (in the next Lemma we add the last condition): Lemma 4.2. [28, Lemma 1.1] The following conditions are equivalent: (1) Φ,, Φ, and Φ, are symmetric; (2) Φ, is symmetric; (3) Φ, is symmetric; São Paulo J.ath.Sci. 8, 1 (2014), 33 82

20 52. Femić (4) ( ev)(φ, ) (ev )( Φ, ); (5) ( ev)(φ, ) (ev )( Φ, ); (6) the conditions 4) and 5) hold true; (7) Φ, is symmetric. One proves similarly: Lemma 4.3. Let C be any object. Then Φ, only if Φ, is symmetric. is symmetric if and Remark 4.2. We remark that ( op ) ( ) cop as coalgebras: ( op ) op op ( op ) op op Φ, ( ) cop The claim follows by the universal property of ( ). If Φ, is symmetric, then ( op ) and ( ) cop are bialgebras and they are isomorphic as opf algebras. Remark 4.3. There are several ways to construct a Drinfel d double. In [3, Prop. 3.6] one can find a construction of a matched pair of bialgebras, and hence a bicrossproduct A. With (A op ) and the pairing.,. ev it is given a different construction than the one in our Proposition 4.1. Taking A ( cop ) and.,. ev, one obtains a Drinfel d double of the form ( cop ) ( ) op. The authors proved that if A and are opf algebras where the antipode of A is invertible, than A and are a matched pair of bialgebras if and only if Φ A, is symmetric. In [28, Theorem 3.2] a result similar to our Proposition 4.1 is proved, but the - and op -actions are given differently. The quasitriangularity of D() we will discus in the next section. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

21 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 53 Developing the right hand-side of the expression (4.3) applied to the Drinfel d double and using the actions given in Proposition 4.1, yields: ev (4.6). Taking and applying the above equality to η, one gets: (4.7) ev The following result generalizes [21, Proposition 4.6] to the braided case. Lemma 4.4. Assume that Φ, is symmetric. Then the following are equivalent: (i) D() is commutative, (ii) and are commutative; (iii) and are cocommutative; (iv) D() is cocommutative. Proof. In view of 2.4 it suffices to prove the equivalence of (i) and (ii). We omit to type the whole proof, we only give a sketch of it. First observe that we have identities: São Paulo J.ath.Sci. 8, 1 (2014), 33 82

22 54. Femić op ev ev op Φ, op (2.5) op op ev (4.8) and op op op id (4.9) op op Suppose that D() is commutative. Using ev evφ and evaluating the product in D() at op, we obtain: op op op D() (4.9) op D() (4.9) (4.10) Apply this to op η η and compose the obtained identity with ε to obtain: op op y (4.8) one gets that, and hence, is commutative. Applying (4.10) to η η η, one obtains that is commutative. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

23 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 55 Conversely, assuming (ii), using (4.8) and that Φ, is symmetric, one may prove that (4.10) - which expresses commutativity of D() - holds true. 5. Yetter-Drinfel d modules as modules over the Drinfel d double Since D() is a bialgebra in C, the category of its left (and right) modules is monoidal. In this and the next section we study the isomorphism between these categories and the appropriate categories of YD-modules. The functors we will consider will act as identity functors on objects and morphisms, we will only have to define the new (co)module structures. Let us regard the pair of functors F ( op ) C YD(C) op G. For ( op ) C and K YD(C) op we define: F() F() and G(K) G(K) K K. Regard F() as a left -module by the action of η on, and consider G(K) K as a left -module. y Proposition 2.5 we know that F() is a right -comodule and G(K) a left -module. Assume that Φ, and Φ, are symmetric, where ( op ) C. Let ( op ). We have: F() F F() (4.4) Φ, (2.2) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

24 56. Femić (4.6) Φ, (2.3) ev ev ev ev (4.5) ev coass. ass. ev Φ, (3.5) (4.2) ev Σ. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

25 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 57 On the other hand, it is: F() F() F (2.4) Φ, ev ev ev Φ, Φ, Σ From the universal property of [, I] the obtained identity implies that F() obeys (3.7), thus F is well defined. For the converse assume that moreover Φ,K is symmetric for K YD(C) op. We will need: (2.1) id (5.1) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

26 58. Femić ow we compute: G(K) K K ev (4.2) G ev (5.1) G(K) K ev K Φ, Φ, K K Φ, K ev K K ev K ev São Paulo J.ath.Sci. 8, 1 (2014), 33 82

27 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 59 K K Φ,K Y D K ev coass. ass. K ev Φ, (3.5) K K ev (2.3) Φ, K K (4.2) G G(K) G(K) y (4.3) and (4.6) this proves that G(K) is a module over. From Proposition 2.5 we then know that F and G make an isomorphism of categories. Let us show that F is a monoidal functor. Take, ( op ) C, then: F( ) F( ) F() F() F()F() op São Paulo J.ath.Sci. 8, 1 (2014), 33 82

28 60. Femić Finally, for, ( op ) C consider: Ψ, D() D() F Φ,. ote that the right hand-side is Φ 1+,. Then we have that Ψ becomes the braiding in ( op ) C. Its inverse is given by: Ψ 1,. Proposition 5.1. Assume C is a finite opf algebra with a bijective antipode. Suppose that Φ, is symmetric for all YD(C) op. The categories YD(C) op and D() C are isomorphic as braided monoidal categories. In [16, Definition 1.2] ajid defined an opposite comultiplication op for a bialgebra. Let O(, op ) denote the subcategory of those - modules with respect to which op is an opposite comultiplication. If R I is a quasitriangular structure for, [16, Definition 1.3], then by [16, Proposition 3.2] the subcategory O(, op ) is braided by R µ µ. Φ, Φ, São Paulo J.ath.Sci. 8, 1 (2014), 33 82

29 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 61 We denote this composition by Φ(R). It is straightforward to check that if Φ, is symmetric for all -modules in C, then op Φ, is an opposite comultiplication for with respect to the whole category C, i.e. O(, op ) C. The same is true for C. In particular, the above holds for D(). The morphism: R is a quasitriangular structure for D() and it induces a braiding Φ(R) on D()C. ote that it equals to our Ψ from above. As it is the case in the category of modules over a commutative ring and a usual quasitriangular opf algebra, the axioms of quasitriangularity of D() are equivalent to the two braiding axioms for Ψ, its left D()-linearity and invertibility, if Φ, is symmetric for all D() C osonization and an isomorphism of categories. espalov proved in [1, Lemma and Section 5.4] that a left (right) module over a quasitriangular bialgebra (, R) can be equipped with a left (right) comodule structure over so that the subcategory O(, op ) becomes a full braided subcategory of YD(C) (YD(C) ). This is a braided version of the classical result from [12]. Assume that is a quasitriangular opf algebra with respect to the whole category C (e.g. if Φ, is symmetric for all C ). Then C is braided. Let be a opf algebra in C. Equipped with a right -comodule structure: ρ R becomes a right-right YD-module. The structure morphisms of are right -linear. Since is quasitriangular, they turn out to be also right São Paulo J.ath.Sci. 8, 1 (2014), 33 82

30 62. Femić -colinear. We show this for the multiplication: R R R R Φ, R R where at the place we applied the quasitriangular axiom ( op )R R 23 R 13. Since C is a braided subcategory of YD(C), we have that the braiding in C induced by R (the right hand-side version of Φ(R)) equals Φ R from (3.4) and is indeed a opf algebra in YD(C). y [1, Theorem 4.1.2] the cross product algebra is then a opf algebra in C, the bosonization of the braided opf algebra. Its multiplication and comultiplication are given by: and which are the tensor product algebra and coalgebra respectively in the category YD(C). The antipode of is given by S Φ R, (S S )Φ R,. Similarly as in Lemma 4.1 one proves that the categories C and (C ) are isomorphic. An object of the latter category is a right - and a right -module satisfying the compatibility condition: São Paulo J.ath.Sci. 8, 1 (2014), 33 82

31 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 63 oreover, the isomorphism F (C ) C is monoidal, since for, (C ) it is: F( ) F( ) Φ(R) Φ R F() F() F() F(). Thus we have proved: Proposition 5.2. Let be a quasitriangular opf algebra such that Φ, is symmetric for all C. Let be a opf algebra in C. Then is a opf algebra in C and there is a monoidal isomorphism of categories C (C ). 6. Other versions of Yetter-Drinfel d categories We start this section by giving equivalent conditions for the left-left and the right-right YD-compatibility conditions and relating the corresponding categories with that of modules over the Drinfel d double. Subsequently, we will study two versions of left-right, as well as two versions of right-left YD-categories. At the end we will relate all the categories we have studied Left-left and right-right YD-modules as modules over the Drinfel d double. At the beginning of Section 3 we noted that the categories YD(C) of left-left YD-modules and YD(C), of right-right YDmodules, are braided monoidal categories without any further conditions. owever, in order to prove that these categories are isomorphic to that of left (respectively right) D()-modules in C for a finite opf algebra with a bijective antipode, one has to require the same symmetricity conditions on the braiding as in Proposition 5.1. efore supporting this claim, we note São Paulo J.ath.Sci. 8, 1 (2014), 33 82

32 64. Femić that the expressions (3.1) and (3.2) are equivalent to: (6.1) and L L L L (6.2) respectively, if Φ, (Φ,L ) is symmetric for YD(C) and L YD(C). The same symmetricity conditions are necessary to prove that YD(C) and YD(C), characterized by (6.1) and (6.2) respectively, are monoidal categories. Consider the functors F l D() C YD(C) G l defined by F l () F l () and G l() G l () for ( op ) C and YD(C), where F l() is a left -module by the action of η on, and G l () as a left -module. Even though one uses (3.1) as the defining relation for the category YD(C), one has that F l and G l define an isomorphism of categories if Φ, is symmetric. We show here only that this is a monoidal isomorphism. Observe first: (4.4) ow for, D() C we have: (6.3) F l ( ) F l ( ) + (6.3) + + São Paulo J.ath.Sci. 8, 1 (2014), 33 82

33 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 65 Φ, + + Φ, + F l () F l () F l ()F l (). It is easily shown that the functor L YD(C) YD(C) given by L() L() and L() L() for, YD(C) is an isomorphism of categories. It is even monoidal if Φ, is symmetric for every YD(C). We show only the compatibility of the -module structures on the tensor products: L( ) L( ) Φ, L() L() L() L() Φ, L() L() owever, L does not respect the braidings. L() L() Left-right and right-left YD-modules. In Section 3 we studied the categories of left-right YD-modules YD(C) op and copyd(c), Remark 3.4. Symmetrically, we may consider the category YD(C) of rightleft YD-modules. These are right -modules and left -comodules which São Paulo J.ath.Sci. 8, 1 (2014), 33 82

34 66. Femić satisfy the compatibility condition (6.4). If Φ, is symmetric, this condition is equivalent to (6.5). (6.4) (6.5) The category YD(C) cop is monoidal if Φ, is symmetric for all YD(C) cop. This is a braided monoidal category with braiding: Φ 3+, for, YD(C) cop. Another possibility for the braiding is Φ 3 (similarly as in Remark 3.3). Using the fact that Φ, is symmetric, one may show that the functor A YD(C) cop YD(C) op given by: A() A() and A() A() for, YD(C) cop is an isomorphism of categories. We show that it is monoidal. For the right -comodule structures we have: A( ) A( ) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

35 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 67 Φ, Φ, A() A() For the left -module structures we find: A( ) A( ) A()A(). A()A() + + Φ, A()A(). Analogously to the two versions of left-right YD-categories, we have two versions of righ-left YD-categories, where the second one is: op YD(C). It is monoidal if Φ, is symmetric for all op YD(C). This is a braided monoidal category with braiding: for, op YD(C). braiding for op YD(C). Φ 4+, As in Remark 3.4 we have that Φ 3± is not a São Paulo J.ath.Sci. 8, 1 (2014), 33 82

36 68. Femić Let us next examine the relation between the categories YD(C) op and copyd(c), on the one hand, and op YD(C) and YD(C) cop, on the other hand. First of all recall that the corresponding identity functors are not isomorphisms of braided monoidal categories (Remark 3.4). Take YD(C). The object () with structures: () and () + () () is a right cop -comodule and a left op -module. This defines a (bijective) functor YD(C) op,copyd(c) op,cop (the objects of the latter category are left-right YD-modules over the opf algebra op,cop ). Indeed, YD(C) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

37 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 69 is equivalent to () () () (). The functor restricts to monoidal functors 1 C copc and 2 C C op. (For, YD(C), the module structures of i ( ) and i () i ( ), for i 1, are compatible since the antipode is a coalgebra anti-morphism and since Φ, is symmetric, while the corresponding comodule structures for i 2 are compatible since the antipode of is an algebra anti-morphism.) ence induces a monoidal functor from YD(C) op to the category D with objects in op,copyd(c) op,cop, whose monoidal structure and the braiding are like the ones in copyd(c). The category D is shown to be indeed a braided monoidal category, however the functor does not respect the braidings. It is easily seen that (Φ 1+ ) Φ 1+ / Φ 2+. Thus we will not consider that induces a braided monoidal functor YD(C) op copyd(c). (Id,Ω) op In [1, Lemma 3.5.4] it is proved that YD(C) ( copyd(c) ) cop is an isomorphism of braided monoidal categories, where (Id, Ω) is the extension of the braided monoidal isomorphism functor C C cop. As announced in the introduction of our paper, we do not make this kind of identifications, we stick to the original category C. Similarly, there is a functor YD(C) op,cop YD(C) op,cop defined via: () () and () () for YD(C) cop. It induces monoidal functors 1 C op C and 2 C C cop, but not a braided monoidal functor YD(C) cop op YD(C). São Paulo J.ath.Sci. 8, 1 (2014), 33 82

38 70. Femić 6.3. Comparing all the categories. To sum up the results of this section consider the following diagram: F l D()C YD(C) 1 F F 1 F 2 2 YD(C) op YD(C) cop A 1 (6.6) We define the functors F 1 and F 2 so that the triangles 1 and 2 commute. We write out the functor F 1 explicitly: F 1 () with inverse F 1 1 () F 1 () F 1 1 (). We saw that the functors F l, F and A are monoidal isomorphisms, so we have four mutually isomorphic monoidal categories. We now compare their braidings. We have: and Φ 3+, Φ L, A 1 F 1 1 Φ, (Φ 1,) 1 Φ 1+,. This proves that F 1 YD(C) YD(C) op and A YD(C) cop YD(C) op are isomorphisms of braided monoidal categories. y Proposition 5.1, F D() C YD(C) op is also such a functor. Then São Paulo J.ath.Sci. 8, 1 (2014), 33 82

39 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 71 by commutativity of 1 and 2 in (6.6) we have four mutually isomorphic braided monoidal categories. Symmetrically as in (6.6), we may consider: C S D() YD(C) 4 T F 3 F 4 3 copyd(c) op YD(C) E (6.7) The functors S C D() YD(C), T C D() copyd(c) and E copyd(c) op YD(C) are given by: S() S() + with S 1 () S 1 () and with T () T () E(K) E(K) E 1 (L) E 1 (L) + K K L L ; ; ; E(K) E(K) E 1 (L) E 1 (L) T () T () (in the definitions of S and T the symbols and stand for the morphisms d I and e I, recall 2.2). The functors F 3 and F 4 are defined so that the triangles 3 and 4 in (6.7) commute. The proofs that S, F 3 and E are monoidal functors are analogous K K L L São Paulo J.ath.Sci. 8, 1 (2014), 33 82

40 72. Femić to the corresponding proofs for the functors F l, F and A, respectively. Then clearly also T and F 4 are monoidal. The braiding in C D() is given by: Ψ R, D() D() and we have: Ψ R, S 1 (Φ R,) 1 Ψ R, T 1 Φ, Φ 2, and Φ 2+, E 1 Φ, Φ, Φ 4+,. (The braiding Φ 2+, is the one from Remark 3.4.) This proves that the functors S, T and E respect the braidings. ote that our result that E copyd(c) op YD(C) is an isomorphism of braided monoidal categories generalizes [1, Lemma 3.5.2], where a São Paulo J.ath.Sci. 8, 1 (2014), 33 82

41 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 73 braided monoidal isomorphism functor op YD(C) ( copyd(c) ) op,cop is given if the objects of C have right duals. It sends an object from the source category to its dual object. Finally, let us record that we do not find any braided monoidal functor which would connect the two groups of categories from (6.6) and (6.7). At the end of Subsection 6.2 we showed that a natural candidate for a monoidal functor from YD(C) op to copyd(c) is not a braided functor. Likewise, at the end of Subsection 6.1 we showed that L YD(C) YD(C) is a monoidal but not a braided functor. In the relation (3.5.1) after [1, Corollary 3.5.5] two (mutually isomorphic) isomorphism functors G 1, G 2 YD(C) YD(C) are given. For YD(C) with right module and comodule structure morphisms µ and ρ respectively, the functors G 1 and G 2 are defined by G 1 (, ν, ρ) (, µ 1 νφ 1 (S 1 ), λ 1 (S )Φρ) and G 2 (, ν, ρ) (, µ 2 νφ(s ), λ 2 (S 1 )Φ 1 ρ), respectively. ere µ i and λ i denote the left module and comodule structure morphisms of G i (), respectively, for i 1, 2. That these functors are well-defined one can check directly applying (6.2). owever, that they are not monoidal we can see even when C V ec, the category of vector spaces. Let us see this for G 1 : h (m n) (m n) S 1 (h) m S 1 (h (2) ) n S 1 (h (1) ) h (2) m h (1) n, which shows that G 1 restricts to a monoidal functor cop. oreover, a direct check shows that if Φ, is symmetric for any YD(C), the functor G 1 is a braided monoidal isomorphism YD(C) op op copyd(c), where is a braided monoidal category with braiding: copyd(c). Thus, we can complete (6.7), and symmetrically (6.6), into commutative diagrams of isomorphic braided monoidal categories: D()C YD(C) C D() YD(C) YD(C) op YD(C) cop YD(C) op cop and copyd(c) op YD(C) op copyd(c) São Paulo J.ath.Sci. 8, 1 (2014), 33 82

42 74. Femić There are further monoidal isomorphisms for YD-categories. In [1, Lemma 3.5.6] there is given a monoidal isomorphism YD(C) op YD(C) A, where A is a further bialgebra with a bialgebra pairing ρ A I. ere the latter is a monoidal category without any symmetricity conditions, but the former requires some. On the other hand, we checked that there is a monoidal isomorphism YD(C),A copyd(c), where the latter does require some symmetricity conditions whereas the former does not. The objects of,a copyd(c) satisfy the condition: A ρ A ρ where ρ A I is a bialgebra pairing. This is another example of the apeearance that a (braided) monoidal isomorphism functor from a YD-category in C necessarily requires that the braiding in C be symmetric between and any object of the category. 7. Center construction The center construction for monoidal categories has been introduced independently by Drinfel d 1 and Joyal and Street [9]. It consists of assigning a braided monoidal category called center of C to a monoidal category C. We will differ the left Z l (C) and the right Z r (C) center of C. We recall here the definition of the (right) center from [10, Definition XIII.4.1]. Proposition and Definition For a monoidal category C the objects of Z r (C) are pairs (V, c,v ) with V C, where c,v is a family of natural isomorphisms c X,V X V V X for X C such that for all Y C it is c X Y,V (c X,V Y )(X c Y,V ). (7.1) A morphism between (V, c,v ) and (W, c,w ) is a morphism f V W in C such that for all X C it is (f X)c X,V c X,W (X f). (7.2) 1 Private communication to ajid in response to the preprint of [13], February São Paulo J.ath.Sci. 8, 1 (2014), 33 82

43 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 75 The identity morphism in C is the identity morphism in Z r (C) and the composition of two morphisms in C is a morphisms in Z r (C). Thus Z r (C) is a category, called the right center of C. From the definition it is clear that c, is a transformation natural in both arguments. In [10, Theorem XIII.4.2] it is proved that Z r (C) is a braided monoidal category. The unit object is (I, Id), the tensor product of (V, c,v ) and (W, c,w ) is (V W, c,v W ), where c X,V W X V W V W X is a morphism in C defined for all X C by The braiding in Z r (C) is given by: c X,V W (V c X,W )(c X,V W ). (7.3) c V,W (V, c,v ) (W, c,w ) (W, c,w ) (V, c,v ). The left center Z l (C) of C is defined analogously an object in Z l (C) has the form (V, c V, ) with V C. For a opf algebra over a field the left center of the category of left modules over is isomorphic to YD [15, Example 1.3], and the right center of the category of left modules over is isomorphic to YD [10, Theorem XIII.5.1]. Generalizing these results to a braided monoidal category C, espalov indicated in [1, Proposition 3.6.1] that YD(C) is isomorphic as a braided monoidal category to a subcategory Zl C( C) of the (left) center of C. The condition that the objects (V, c V, ) of Zl C( C) fulfill is that for every X C with trivial -action (via the counit) the morphism c V,X coincides with the braiding Φ V,X in C. In other words, with the forgetful functor U C C one has that c V,U(X) Φ V,U(X) for every X C. For completeness we present below the proof for an analogous statement. Proposition 7.1. The categories Z C r ( C) and YD(C) op are isomorphic as braided monoidal categories. Proof. First of all, note that for (V, c,v ) Z C r ( C) we have: c,v V (7.1) c,v V c,v c,v V c,v (7.4) V V V. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

44 76. Femić The morphism ρ c,v (η V ) V V defines a right -comodule structure on V : V V c,v V V c,v (7.1) V c,v V V c,v V V c,v V V V. The counit property follows from c I,V id V (see (7.1)). With this - comodule and the existing -module structure V is a left-right YD-module: V V c,v V V (7.4) V c,v V V c,v V c,v C V c,v V V V. A morphism f V W in Z C r ( C) becomes a morphism of left-right YDmodules it is right -colinear because of (7.2). This defines a functor K from Z C r ( C) to the category of left-right YD-modules. We now prove that K Z C r ( C) YD(C) op is monoidal. Let (V, c,v ) and (W, c,w ) be in Z C r ( C). Then we have: K(V W ) V W c,v W (7.3) V W c,v c,w (7.4) V W c,v V c,v W c,w K(V ) K(W ) K(V W ) V W V W V W c,w V W K(V )K(W ) If (V, c,v ) Zr C ( C), then Φ 1+,V c,v because of (7.4). On the other hand, for YD(C) op its comodule structure morphism is obviously equal to Φ 1+, (η ). ence the inverse functor of K is given by sending a YDmodule into the pair (, Φ 1+, ). Consequently, K respects the braiding and this finishes the proof. São Paulo J.ath.Sci. 8, 1 (2014), 33 82

45 Transparency condition in the categories of Yetter-Drinfel d modules over opf algebras in braided categories 77 Similarly, one may prove that the following categories are braided monoidally isomorphic: Z C l ( C) YD(C) Z C r ( C), Z C r (C ) YD(C) Z C l (C ) YD(C) cop Z C l ( C), copyd(c) Z C r (C ), op YD(C) Z C l (C ). The above center subcategories are defined analogously to Z C r ( C). Adding to this list the categories ( op ) C and C ( op ), we may identify ( op ) C Z C l ( C) and C ( op ) Z C r (C ) (7.5) having in mind that the corresponding -module structures remain unchanged by the isomorphism functors. Then due to (6.6) and (6.7) we obtain the following diagrams of isomorphic braided monoidal categories: Zl C( C) Zr C ( C) Zr C (C ) Zl C(C ) Zr C ( C) Zl C( C) and Zr C (C ) Zl C(C ) Transparency and üger s centers Z 1 and Z 2. Throughout the paper we have used the condition that Φ, is symmetric for every C. This means that is transparent in C in terms of [4], or that belongs to üger s center Z 2 (C) {X C Φ Y,X Φ X,Y id X Y for all Y C}, [18, Definition 2.9]. ote that due to Lemma 4.3, is transparent if and only if so is. The center of a monoidal category D that we studied above is denoted by Z 1 (D) in [18] (neglecting the difference between the left and the right center). Then we may state: Proposition 7.2. Let be a finite opf algebra with a bijective antipode in a braided monoidal category C. If Z 2 (C), then there are embeddings of braided monoidal categories: YD(C) D() C Z 1,l ( C) and YD(C) C D() Z 1,r (C ) The whole center category and the coend. The center category of a monoidal category C is a particular case of the Pontryagin dual monoidal category introduced by ajid in [13, Section 3]. For C braided, rigid and cocomplete from [14, Theorem 3.2] one deduces that there is an isomorphism of monoidal categories: Z l (C) C Aut(C) (7.6) São Paulo J.ath.Sci. 8, 1 (2014), 33 82