Duality for infinite-dimensional braided bialgebras and their (co)modules
aa r X i v : . [ m a t h . QA ] A ug DUALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS ANDTHEIR (CO)MODULES
ELMAR WAGNERA
BSTRACT . The paper presents a detailed description of duality for braided algebras, coal-gebras, bialgebras, Hopf algebras and their modules and comodules in the infinite setting.Assuming that the dual objects exist, it is shown how a given braiding induces compatiblebraidings for the dual objects, and how actions (resp. coactions) can be turned into coactions(resp. actions) of the dual coalgebra (resp. algebra), with an emphasis on braided bialgebrasand their braided (co)module algebras.
1. I
NTRODUCTION
The objective of this paper is to give a detailed description of duality for braided algebras,coalgebras, bi- and Hopf algebras, and their modules and comodules in the infinite setting.The interest in duality for these structures goes back to Majid [11, 14], who did much ofthe pioneering work on braided Hopf algebras [9, 10, 12, 13, 15], and to Takeuchi [18, 19].Focusing on specific applications, duality for braided Hopf algebras has been studied byLyuvashenko [8] and by Guo and Zhang [5] in the context of integrals, and by Da Rocha, J.A.Guccione and J. J. Guccione [2] in connection with crossed products. The theory of braidedHopf crossed products reveals in particular the relevance of braided (co)module algebras[3, 4] which serve as a guiding principle for this paper. Duality for infinite braided Hopfalgebras, under the assumption that the braiding is symmetric, has been considered in theYetter-Drinfeld module category in the light of the Blattner-Montgomery duality theorem byHan and Zhang [6], and by Cheng, Xu and Zhang [1].In the finite setting, duality is most conveniently studied in a rigid monoidal category [14,18]. The problem in the infinite setting is the lack of a so-called coevaluation map. To avoidthis problem, we do not follow a categorical approach but define the dual objects by a setof conditions, similar to the path taken by Takeuchi in [19]. In particular, we will neitherprove the existence nor the uniqueness of a dual space with the desired properties, so wewill not define a functor into a dual category. This may be especially useful in a topologicalsetting, where it is not always practical to work with full dual space, for instance if thereare unbounded (braiding) operators involved. However, the entire paper is kept completelyalgebraic even though the more interesting examples arise in a topological framework. In thissense, one may consider all tensor products as algebraic tensor products of linear spaces overa field K .Our first aim is to establish a duality theory for infinite-dimensional braided bialgebrasand Hopf algebras. This will be done in Section 4. The guiding principle emanates fromthe definition of a dual pairing between braided bialgebras in Definition 4.2. To take intoaccount the braided setting, we include in this definition a braiding between dual spaces so Mathematics Subject Classification.
Primary 16T10; Secondary 16T15, 18D10.
Key words and phrases.
Duality, braided bialgebra, braided Hopf algebra, (co)module algebra. that the dual pairing of two-fold tensor products can be realized by evaluating simultaneouslyadjacent tensor factors. The fundamental idea of our approach is that all structures on dualspaces should be induced from the given ones, including the braiding appearing in the dualpairing. In this sense, our method is constructive, only the existence of a dual space with therequired properties will be assumed, wheras all algebraic properties will be deduced from theoriginal source.To develop the theory step by step, we start by elaborating a duality theory for infinite-dimensional algebras and coalgebras in Propositions 4.3 and 4.4. Before doing so, we showin Lemma 4.1 how a given braiding induces braidings on a dual space and between the spaceand its dual which are compatible with additional algebraic structures like multiplication orcomultiplication. Our definition of a product or coproduct on the dual space is intimatelyrelated to the compatibility conditions of a dual pairing. The construction of a dual braidedbialgebra will be achieved in Theorem 4.5 under certain assumptions on the chosen dualspace which guarantee that the induced braidings define bijective maps into the correct tensorproducts and that the product and coproduct are well-defined. Proposition 4.7 shows that theconstruction is reflexive in the sense that taking twice the dual gives back the same braidedbialgebra. The extension of these results to braided Hopf algebras requires only a minorcondition regarding the antipode.In Section 5, we address duality for braided modules and comodules. The starting point isagain to induce new braidings for dual spaces from a given braiding between a (co)algebraand a (co)module in such a way that the compatibility properties are maintained. This will bedone step by step in Lemmas 5.1-5.3, each time replacing one of the two involved spaces bya dual space. Since a left braided vector space induces the structure of a right braided vectorspace on duals, and vice versa, we will frequently use the inverse of a braiding to recover aleft-handed or right-handed version. In fact, one of the purposes of this paper is to single outthe correct braidings and formulas so that it may serve as a reference for others.Theorems 5.4 and 5.6 are the main results of Section 5. There it is shown how to transforma braided comodule into a braided module of a dual (bi)algebra, and a braided module into abraided comodule of a dual co- or bialgebra. Interestingly, in the latter case, it will not exactlyyield a comodule of a dual bialgebra but a version of it that corresponds to taking twice thebraided opposite and co-opposite bialgebra. These structures, where the product or coproductis flipped by a power of the braiding, are discussed at greater length in Section 3 since suchconsiderations do play a role in subsequent results. For instance, they justify to present onlyone version of a braided dual (co)algebra in Propositions 4.3 and 4.4, other versions can beobtained by combining the constructed (co)product with powers of the braiding. Moreover,twisting products, coproducts, actions or coactions with a braiding may give rise to wholefamilies of new structures as illustrated in Proposition 3.1 and Corollary 5.7. In Section 3, wealso review the significance of the antipode in the braided setting for turning left (co)actionsinto right (co)actions and vice versa.In Proposition 5.8, we dualize a coaction on a comodule to an action of a dual algebraon a dual of the comodule, and in Proposition 5.9, we dualize an action on a module to acoaction of a dual coalgebra on a dual of the module. Despide the fact that our main interestlies in module and comodule algebras of braided bialgebras, Propositions 5.8 and 5.9 do notconsider these topics since we would then have to introduce the dual objects, namely module
UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 3 and comodule coalgebras. To keep the length of the paper reasonable, we refrain from intro-ducing (co)module coalgebras. With a detailed description of braided (co)module algebras athand, it should be clear how to dualize these notions to braided (co)module coalgebras.Also for brevity, we do not elaborate an example in full detail but sketch only that gradedbraided (co)algebras, bi- and Hopf algebras and their graded (co)modules provide infinite-dimensional examples. For the same reason, we completely avoid braid diagrams since theproofs presented by braid diagrams would occupy considerably more space. Instead of braiddiagrams, we introduce a Sweedler-type notation and annotate the employed relations overthe equality signs. Once the reader gets used to this notation, it shouldn’t be a problem todraw the corresponding braid diagrams, one only has to be careful with the chosen cross-ings. For instance, the crossings of the induced braidings have to be compatible with thegiven braidings, and the braidings obtained from an inverse braiding should be denoted dif-ferently than those induced from a given one. Although the proofs are rather straightforward,we present most of them in order to show where the involved braidings and compatibilityrelations are used.2. P
RELIMINARIES ON BRAIDED BIALGEBRAS AND THEIR ( CO ) MODULES
In this section, we give a working definition for braided bi- and Hopf algebras withoutusing braided tensor categories. The reason is that we want to establish a duality theoryfor infinite-dimensional braided bialgebras. The categorical approach works well in rigidmonoidal categories, but for infinite-dimensional examples, there is a problem with the rig-orous definition of a co-evaluation map, see e.g. [2, 19]. Moreover, a dual braided bialgebramay not exist, and if it exists, it may not be unique, therefore we do not aim at defining afunctor into a dual category.Throughout this paper, the letter K stands for an arbitrary field. A braiding for a vectorspace V is a bijective linear map Ψ V V : V ⊗ V → V ⊗ V fulfilling the Yang–Baxter equation(1) (Ψ V V ⊗ id) ◦ (id ⊗ Ψ V V ) ◦ (Ψ V V ⊗ id) = (id ⊗ Ψ V V ) ◦ (Ψ V V ⊗ id) ◦ (id ⊗ Ψ V V ) . Let V be a braided vector space. A left V -braided vector space is a vector space W togetherwith a bijective linear map Ψ V W : V ⊗ W → W ⊗ V such that(2) (Ψ V W ⊗ id) ◦ (id ⊗ Ψ V W ) ◦ (Ψ V V ⊗ id) = (id ⊗ Ψ V V ) ◦ (Ψ V W ⊗ id) ◦ (id ⊗ Ψ V W ) . Similarly, V is called a right W -braided vector space if W is a braided vector space and thebijective linear map Ψ V W : V ⊗ W → W ⊗ V satisfies(3) (id ⊗ Ψ V W ) ◦ (Ψ V W ⊗ id) ◦ (id ⊗ Ψ W W ) = (Ψ
W W ⊗ id) ◦ (id ⊗ Ψ V W ) ◦ (Ψ V W ⊗ id) . The archetypal example, also in the case V = W , is given by the flip:(4) τ : V ⊗ W −→ W ⊗ V, τ ( v ⊗ w ) := w ⊗ v. Let V be an algebra with multiplication m V : V ⊗ V → V . If W is a left V -braidedvector space, or if V is a right W -braided vector space, then we say that the braiding Ψ V W iscompatible with the multiplication if(5) Ψ V W ◦ ( m V ⊗ id) = (id ⊗ m V ) ◦ (Ψ V W ⊗ id) ◦ (id ⊗ Ψ V W ) and, if ∈ V ,(6) Ψ V W (1 ⊗ w ) = w ⊗ , w ∈ W. E. WAGNER
Assume now that W is an algebra with multiplication m W : W ⊗ W → W and that W is aleft V -braided vector space or that V is a right W -braided vector space. Then we say that thebraiding Ψ V W is compatible with the multiplication if(7) Ψ V W ◦ (id ⊗ m W ) = ( m W ⊗ id) ◦ (id ⊗ Ψ V W ) ◦ (Ψ V W ⊗ id) and(8) Ψ V W ( v ⊗
1) = 1 ⊗ v, v ∈ V. A braided algebra is an algebra ( A, m ) which is a braided vector space such that the braid-ing is compatible with the multiplication. In this case, it follows from (5) and (7) that(9) Ψ AA ◦ ( m ⊗ m ) = ( m ⊗ m ) ◦ (id ⊗ Ψ AA ⊗ id) ◦ (Ψ AA ⊗ Ψ AA ) ◦ (id ⊗ Ψ AA ⊗ id) . Clearly, each algebra A becomes a braided algebra with the usual flip defined in (4) as braid-ing isomorphism.Suppose now that ( V, ∆ , ε ) is a coalgebra and W is a left V -braided vector space or V is a right W -braided vector space. We say that the braiding Ψ V W : V ⊗ W → W ⊗ V iscompatible with the comultiplication of V if(10) (id ⊗ ∆) ◦ Ψ V W = (Ψ
V W ⊗ id) ◦ (id ⊗ Ψ V W ) ◦ (∆ ⊗ id) , (id ⊗ ε ) ◦ Ψ V W = ε ⊗ id . If ( W, ∆ , ε ) is a coalgebra and V is a right W -braided vector space or W is a left V -braidedvector space, then the analogous definitions read(11) (∆ ⊗ id) ◦ Ψ V W = (id ⊗ Ψ V W ) ◦ (Ψ V W ⊗ id) ◦ (id ⊗ ∆) , ( ε ⊗ id) ◦ Ψ V W = id ⊗ ε. A braided coalgebra is a coalgebra ( H, ∆ , ε ) which is a braided vector space such that thebraiding is compatible with the comultiplication. In this case, combining (10) and (11) yields(12) (∆ ⊗ ∆) ◦ Ψ HH = (id ⊗ Ψ HH ⊗ id) ◦ (Ψ HH ⊗ Ψ HH ) ◦ (id ⊗ Ψ HH ⊗ id) ◦ (∆ ⊗ ∆) . As for algebras, each coalgebra becomes a braided coalgebra with the braiding defined by theflip τ given in (4).The compatibility conditions permit to extend the (co)algebra structures to tensor products.If ( A, m A ) and ( B, m B ) are algebras such that A is a right B -braided vector space, B is aleft A -braided vector space, and the braiding Ψ BA : B ⊗ A → A ⊗ B is compatible with themultiplications of A and B , then(13) mmm : ( A ⊗ B ) ⊗ ( A ⊗ B ) −→ A ⊗ B, mmm := ( m A ⊗ m B ) ◦ (id ⊗ Ψ BA ⊗ id) , defines a product on A ⊗ B turning it into an associative algebra denoted by A ⊗ B . If A and B are unital, then ⊗ yields the unit of A ⊗ B .If H and G are coalgebras such that H is a right G -braided vector space, G is a left H -braided vector space, and the braiding Ψ HG : H ⊗ G → G ⊗ H is compatible with thecomultiplications of H and G , then the coproduct(14) ∆∆∆ : H ⊗ G −→ ( H ⊗ G ) ⊗ ( H ⊗ G ) , ∆∆∆ := (id ⊗ Ψ HG ⊗ id) ◦ (∆ ⊗ ∆) , turns H ⊗ G into a coalgebra with counit εεε := ε ⊗ ε .Recall that, for a coalgebra ( H, ∆ , ε ) and a unital algebra ( A, m ) , the space L ( H, A ) of alllinear mappings from H to A becomes an associative unital algebra under the convolutionproduct(15) ( φ ∗ ψ )( h ) := m ◦ ( φ ⊗ ψ ) ◦ ∆( h ) , h ∈ H, φ, ψ ∈ L ( H, A ) , UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 5 and with unit h ε ( h )1 . In this picture, the antipode of a Hopf algebra can be viewed as theconvolution inverse of the identity map id : H → H .The central objects of this paper are braided bialgebras which will be defined in the defini-tion below. We include there the definition of a braided Hopf algebra although the existenceof an antipode will play rather a minor role in our presentation. Definition 2.1 ([9, 19]) . A braided bialgebra is a unital algebra ( H, m ) together with a coal-gebra structure ( H, ∆ , ε ) and a braiding Ψ HH : H ⊗ H → H such that the following compa-tibility conditions hold: H is a braided vector space, the braiding Ψ HH is compatible with themultiplication and the comultiplication of H , and the coproduct is an algebra homomorphism ∆ : H → H ⊗ H , i.e.,(16) ∆ ◦ m = ( m ⊗ m ) ◦ (id ⊗ Ψ HH ⊗ id) ◦ (∆ ⊗ ∆) . A braided Hopf algebra is a braided bialgebra H such that the identity map has a convolutioninverse S : H → H called the antipode.For a braided Hopf algebra H , it can be shown that Ψ HH ◦ ( S ⊗ id) = (id ⊗ S ) ◦ Ψ HH , Ψ HH ◦ (id ⊗ S ) = ( S ⊗ id) ◦ Ψ HH ,S ◦ m = m ◦ Ψ HH ◦ ( S ⊗ S ) , ∆ ◦ S = Ψ HH ◦ ( S ⊗ S ) ◦ ∆ , ε ◦ S = ε, (17)see e.g. [14, 19].If A and B are braided bialgebras with a braiding Ψ AB satisfying the compatibility con-ditions on the multiplication and comultiplication, then A ⊗ B becomes a braided bialgebrawith braiding(18) Ψ A ⊗ B, A ⊗ B := (id ⊗ Ψ AB ⊗ id) ◦ (id ⊗ id ⊗ Ψ BB ) ◦ (Ψ AA ⊗ id ⊗ id) ◦ (id ⊗ Ψ − AB ⊗ id) , and multiplication and comultiplication defined in (13) and (14), respectively.The following presentation of actions and coactions in the braided setting is taken from [2].Let ( A, m ) be a braided (unital) algebra, and let W be a left A -braided vector space such thatthe braiding is compatible with the multiplication. We say that W is a braided left A -moduleif there is a map ν L : A ⊗ W → W satisfying ν L ◦ (id ⊗ ν L ) = ν L ◦ ( m ⊗ id) , ν L (1 ⊗ w ) = w, w ∈ W, (19) Ψ AW ◦ (id ⊗ ν L ) = ( ν L ⊗ id) ◦ (id ⊗ Ψ AW ) ◦ (Ψ AA ⊗ id) . (20)Equation (19) says that ν L is an algebra action in the usual sense, and (20) means that ν L iscompatible with the braiding. A braided right A -module V is defined analogously, i.e., V is aleft A -braided vector space, the braiding is compatible with the multiplication, and the rightaction ν R : V ⊗ A → V satisfies ν R ◦ ( ν R ⊗ id) = ν R ◦ (id ⊗ m ) , ν R ( v ⊗
1) = v, v ∈ V, (21) Ψ V A ◦ ( ν R ⊗ id) = (id ⊗ ν R ) ◦ (Ψ V A ⊗ id) ◦ (id ⊗ Ψ AA ) . (22)Applying the inverse braidings to the compatibility relations yields (id ⊗ ν L ) ◦ (Ψ − AA ⊗ id) ◦ (id ⊗ Ψ − AW ) = Ψ − AW ◦ ( ν L ⊗ id) , (23) ( ν R ⊗ id) ◦ (id ⊗ Ψ − AA ) ◦ (Ψ − V A ⊗ id) = Ψ − V A ◦ (id ⊗ ν R ) . (24)To shorten notation, we often write(25) ν L ( a ⊗ w ) := a ⊲ w, ν R ( v ⊗ a ) := v ⊳ a, w ∈ W, v ∈ V, a ∈ A. E. WAGNER
Let ( H, ∆ , ε ) be a braided coalgebra. Let V be a right H braided vector space such thatthe braiding is compatible with the comultiplication. Recall that a left coaction on a vectorspace V is a linear map ρ L : V → H ⊗ V satisfying(26) (id ⊗ ρ L ) ◦ ρ L = (∆ ⊗ id) ◦ ρ L , ( ε ⊗ id) ◦ ρ L = id . We say that the coaction is compatible with the braiding, if(27) (id ⊗ ρ L ) ◦ Ψ V H = (Ψ HH ⊗ id) ◦ (id ⊗ Ψ V H ) ◦ ( ρ L ⊗ id) . In this case, V is called a braided left H -comodule.Analogously, for a braided right H -comodule W with a right coaction ρ R : W → W ⊗ H ,we require that W is a left H braided vector space and that ( ρ R ⊗ id) ◦ ρ R = (id ⊗ ∆) ◦ ρ R , (id ⊗ ε ) ◦ ρ R = id , (28) ( ρ R ⊗ id) ◦ Ψ HW = (id ⊗ Ψ HH ) ◦ (Ψ HW ⊗ id) ◦ (id ⊗ ρ R ) . (29)Note that we defined a braided left H -comodule for a right H braided vector space andvice versa. By Lemma 2.4 below, the inverse braidings turn a right (resp. left) H braidedvector space into a left (resp. right) H braided vector space. For the inverse braidings, thecompatibility conditions read (id ⊗ Ψ − V H ) ◦ (Ψ − HH ⊗ id) ◦ (id ⊗ ρ L ) = ( ρ L ⊗ id) ◦ Ψ − V H , (30) (Ψ − HW ⊗ id) ◦ (id ⊗ Ψ − HH ) ◦ ( ρ R ⊗ id) = (id ⊗ ρ R ) ◦ Ψ − HW . (31)The objects of our interest are H -module algebras and H -comodule algebras for a braidedbialgebra H , so we will highlight them in a separate definition. Definition 2.2.
Let H be a braided bialgebra and let B be a braided algebra such that B is aleft H -braided vector space and the braiding Ψ HB is compatible with the multiplications of H and B and with the comultiplication of H .Assume that B is a braided left H -module with left action ν L : H ⊗ B → B . We say that B is a braided left H -module algebra if the left action ν L and the multiplication m B of B satisfy the compatibility condition(32) ν L ◦ (id ⊗ m B ) = m B ◦ ( ν L ⊗ ν L ) ◦ (id ⊗ Ψ HB ⊗ id) ◦ (∆ ⊗ id ⊗ id) . If ∈ B , then it is additionally required that(33) ν L ( f ⊗
1) = ε ( f )1 , f ∈ H. Assume that B is a braided right H -comodule with right coaction ρ R : B → B ⊗ H . Then B is called a braided right H -comodule algebra, if(34) ρ R ◦ m B = ( m B ⊗ m H ) ◦ (id ⊗ Ψ HB ⊗ id) ◦ ( ρ R ⊗ ρ R ) , If ∈ B , then it is additionally required that(35) ρ R (1) = 1 ⊗ . A braided right H -module algebra and braided left H -comodule algebra are defined anal-ogously by flipping the tensor products and replacing Ψ HB by Ψ BH . Remark . The dual notions of H -module algebra and H -comodule algebra are H -modulecoalgebra and H -comodule coalgebra, respectively. The compatibility conditions are the dualversions of those in Definition 2.2. To keep the size of the paper in reasonable limits, we willnot discuss these structures here. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 7
Throughout our presentation of duality, we will make frequent use of the inverse of a givenbraiding. For later reference, we finish this section with a lemma that summarizes someproperties of inverse braidings. It is proven by applying repeatedly the corresponding inversemorphism on both sides of the defining relations.
Lemma 2.4.
Let H be a braided vector space with braiding Ψ HH . Then Ψ − HH defines abraiding on H . If V carries the structure of a left (resp. right) H -braided vector space withrespect to Ψ HH , then it does so with respect to Ψ − HH , and it becomes a right (resp. left) H -braided vector space with respect to Ψ − HV (resp. Ψ − V H ). If H or V is an algebra and Ψ HV (resp. Ψ V H ) is compatible with the multiplication, then Ψ − HV (resp. Ψ − V H ) is also compatiblewith the multiplication. If H or V is a coalgebra and Ψ HV (resp. Ψ V H ) is compatible withthe comultiplication, then Ψ − HV (resp. Ψ − V H ) is also compatible with the comultiplication. Inparticular, if H is a braided (co)algebra with respect to Ψ HH , then it is also one with respectto Ψ − HH .
3. B
RAIDED PRODUCTS , COPRODUCTS AND ( CO ) ACTIONS
The purpose of this section is to discuss generalizations of opposite algebras and coalgebrasin the braiding setting. It is also shown how to use the braiding, and possibly the inverse ofthe antipode, to turn right (co)actions into left (co)actions and vice versa.Given a braided bialgebra H , there exists a different braided bialgebra structure on H withthe multiplication m and the coproduct ∆ replaced by(36) m := m ◦ Ψ HH : H ⊗ H −→ H, ∆ − := Ψ − HH ◦ ∆ : H −→ H ⊗ H. This bialgebra will be denoted by H (1 , − , differing from the standard notation H op , cop forreasons that will become clear below. If H is a braided Hopf algebra, then it follows from(17) that H (1 , − is also one with the same antipode S .The opposite algebra, say H ( − , , with m replaced by m − := m ◦ Ψ − HH and the oppositecoalgebra, say H (0 , − , with ∆ replaced by ∆ − := Ψ − HH ◦ ∆ will also yield braided bial-gebras, but with respect to the inverse braiding. These constructions may yield an infinitefamily of braided bialgebras provided that Ψ HH is infinite cyclic. For later use, and since wedid not find it in the literature, we will state the result in the following proposition. In thesubsequent corollary, it is shown that, for a braided Hopf algebra H with bijective antipode, H is isomorphic to its H op , cop -version. Proposition 3.1.
Let ( H, m ) be a braided (unital) algebra and let k ∈ Z . Then (37) m k := m ◦ Ψ kHH : H ⊗ H −→ H, defines a product on H turning it again into a braided (unital) algebra (with the same unit ele-ment). If V is a left or right H -braided vector space such that the braiding is compatible withthe multiplication m on H , then the braiding is also compatible with the multiplication m k .Let ( H, ∆ , ε ) be a braided coalgebra and let n ∈ Z . Then (38) ∆ n := Ψ nHH ◦ ∆ : H −→ H ⊗ H, defines a coproduct on H turning it again into a braided coalgebra with the same counit.If V is a left or right H -braided vector space such that the braiding is compatible with thecomultiplication of H , then the braiding is also is compatible with the coproduct ∆ n .Assume that H is a braided bialgebra and let H ( k,n ) denote the linear space H equippedwith the product m k , the unmodified unit element, the coproduct ∆ n and the unmodified E. WAGNER counit. Then, for all n ∈ Z , H ( n, − n ) is a braided bialgebra with respect to the braiding Ψ HH ,and H ( n − , − n ) is a braided bialgebra with respect to the braiding Ψ − HH .If H is a braided Hopf algebra with antipode S , then all H ( n, − n ) are braided Hopf alge-bras with the unmodified antipode S . If S is invertible, then all H ( n − , − n ) are braided Hopfalgebras with antipode S − .Proof. We begin by showing that m defines an associative product: m ◦ (id ⊗ m ) (37) = m ◦ Ψ HH ◦ (id ⊗ m ) ◦ (id ⊗ Ψ HH ) (7) = m ◦ ( m ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) (1) = m ◦ (id ⊗ m ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) (5) = m ◦ Ψ HH ◦ ( m ⊗ id) ◦ (Ψ HH ⊗ id) = m ◦ ( m ⊗ id) . If ∈ H , then clearly m (1 ⊗ h ) = h = m ( h ⊗ for all h ∈ H by (6) and (8).To show the compatibility of Ψ HH with m , we compute that Ψ HH ◦ ( m ⊗ id) (5) , (37) = (id ⊗ m ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) (1) = (id ⊗ m ) ◦ (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) (37) = (id ⊗ m ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) , which proves (5). The proof of (7) is completely analogous. Moreover, if ∈ H , then (6)and (8) are trivially satisfied. Therefore H is again a braided algebra with respect to themultiplication m and the braiding Ψ HH .Since m k +1 = m k ◦ Ψ HH , we conclude by induction that the same holds for all k ∈ N .From Lemma 2.4 and the previous computations, it follows that m − turns H also into abraided algebra, and again by induction, we obtain the result for all m − k , k ∈ N .Next we prove that ( H, ∆ , ε ) yields a braided coalgebra. The coassociativity follows from ((Ψ HH ◦ ∆) ⊗ id) ◦ (Ψ HH ◦ ∆) (11) = (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ ∆) ◦ ∆ (1) = (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (∆ ⊗ id) ◦ ∆ (10) = (id ⊗ (Ψ HH ◦ ∆)) ◦ (Ψ HH ◦ ∆) . Furthermore, (id ⊗ ε ) ◦ (Ψ HH ◦ ∆) = id = ( ε ⊗ id) ◦ (Ψ HH ◦ ∆) by the second relations in(10) and (11). To verify the compatibility with the braiding, we compute that (id ⊗ ∆ ) ◦ Ψ HH (10) , (38) = (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (∆ ⊗ id) (1) = (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (Ψ HH ⊗ id) ◦ (∆ ⊗ id) (38) = (Ψ HH ⊗ id) ◦ (id ⊗ Ψ HH ) ◦ (∆ ⊗ id) . This shows the first relation of (10). The first relation of (11) is proven analogously, andsecond relations in (10) and (11) are trivially satisfied. Therefore ( H, ∆ , ε ) is a braidedcoalgebra.By Lemma 2.4, the same arguments show that ( H, ∆ − , ε ) yields also a braided coalgebra.Similar to the above, since ∆ k ± = ∆ k ◦ Ψ ± HH , we can now proceed by induction to concludethat ( H, ∆ k , ε ) is a braided coalgebra for all k ∈ Z . UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 9
Let H be a braided bialgebra. We first show that H ( − , is a braided bialgebra with respectto Ψ − HH . From the first part of the proof and Lemma 2.4, we know that Ψ − HH is compatiblewith the multiplication m − and the comultiplication ∆ , so it remains to verify (16). Againby Lemma 2.4, we conclude that Ψ − HH satisfies (12). Therefore, ∆ ◦ m − (16) = ( m ⊗ m ) ◦ (id ⊗ Ψ HH ⊗ id) ◦ (∆ ⊗ ∆) ◦ Ψ − HH (12) = ( m ⊗ m ) ◦ (Ψ − HH ⊗ Ψ − HH ) ◦ (id ⊗ Ψ − HH ⊗ id) ◦ (∆ ⊗ ∆) (37) = ( m − ⊗ m − ) ◦ (id ⊗ Ψ − HH ⊗ id) ◦ (∆ ⊗ ∆) . This finishes the proof that H ( − , is a braided bialgebra with respect to Ψ − HH .To conclude the same for H (0 , − , note that (9) remains valid if we replace Ψ HH by Ψ − HH .Thus ∆ − ◦ m (16) = Ψ − HH ◦ ( m ⊗ m ) ◦ (id ⊗ Ψ HH ⊗ id) ◦ (∆ ⊗ ∆) (9) = ( m ⊗ m ) ◦ (id ⊗ Ψ − HH ⊗ id) ◦ (Ψ − HH ⊗ Ψ − HH ) ◦ (∆ ⊗ ∆) (38) = ( m ⊗ m ) ◦ (id ⊗ Ψ − HH ⊗ id) ◦ (∆ − ⊗ ∆ − ) , hence (16) is satisfied. Together with the previous results, it follows that H (0 , − is a braidedbialgebra with respect to Ψ − HH .Now we proceed by induction. Let n ∈ N and assume that H ( n − , − n ) is a braided bialgebrawith respect to Ψ − HH . From what has already been shown and since Ψ HH = (Ψ − HH ) − , weconclude that H ( n, − n ) with m n = m n − ◦ Ψ HH and ∆ − n is a braided bialgebra with respectto Ψ HH . Likewise, if H ( − n,n − is a braided bialgebra with respect to Ψ − HH , then H ( − n,n ) with m − n and ∆ n = ∆ n − ◦ Ψ HH is a braided bialgebra with respect to Ψ HH . Continuingin this way, if H ( − n,n ) is a braided bialgebra with respect to Ψ HH , we can replace in abovecalculations H by H ( − n,n ) and see that H ( − ( n +1) ,n ) with m − ( n +1) = m − n ◦ Ψ − HH and ∆ − n is abraided bialgebra with respect to Ψ − HH . Finally, if H ( n, − n ) is a braided bialgebra with respectto Ψ HH , it follows that H ( n, − ( n +1)) with m n and ∆ − ( n +1) = ∆ − n ◦ Ψ − HH is braided bialgebraswith respect to Ψ − HH . Thus the usual induction argument yields the result.If H is braided Hopf algebra, then, by (17), m n ◦ ( S ⊗ id) ◦ ∆ − n = m ◦ Ψ nHH ◦ ( S ⊗ id) ◦ Ψ − nHH ◦ ∆ = (cid:26) m ◦ ( S ⊗ id) ◦ ∆ , n ∈ Z ,m ◦ (id ⊗ S ) ◦ ∆ , n ∈ Z + 1 . Thus m n ◦ ( S ⊗ id) ◦ ∆ − n = 1 ε , and similarly, m n ◦ (id ⊗ S ) ◦ ∆ − n = 1 ε . Therefore H ( n, − n ) is a Hopf algebra with antipode S . If S − exists, we obtain from (17) for n ∈ Z that m n − ◦ (id ⊗ S − ) ◦ ∆ − n = m ◦ Ψ n − HH ◦ (id ⊗ S − ) ◦ Ψ − nHH ◦ ∆= m ◦ ( S − ⊗ id) ◦ Ψ − HH ◦ ∆ = m ◦ (id ⊗ S ) ◦ ( S − ⊗ S − ) ◦ Ψ − HH ◦ ∆= m ◦ (id ⊗ S ) ◦ ∆ ◦ S − = 1 ε ◦ S − = 1 ε. The remaining cases, which prove that S − yields an antipode for H ( n − , − n ) , are shownanalogously. Finally, let V be a left H -braided vector space. If Ψ HV is compatible with the multiplica-tion on H , then Ψ HV ◦ ( m k ⊗ id) (5) , (37) = (id ⊗ m ) ◦ (Ψ HV ⊗ id) ◦ (id ⊗ Ψ HV ) ◦ (Ψ kHH ⊗ id) (2) = (id ⊗ m ) ◦ (id ⊗ Ψ kHH ) ◦ (Ψ HV ⊗ id) ◦ (id ⊗ Ψ HV ) (37) = (id ⊗ m k ) ◦ (Ψ HV ⊗ id) ◦ (id ⊗ Ψ HV ) , which shows the compatibility of Ψ HV with the multiplication m k . Likewise, if Ψ HV iscompatible with the comultiplication on H , then (id ⊗ ∆ n ) ◦ Ψ HV (10) , (38) = (id ⊗ Ψ nHH ) ◦ (Ψ HV ⊗ id) ◦ (id ⊗ Ψ HV ) ◦ (∆ ⊗ id) (2) = (Ψ HV ⊗ id) ◦ (id ⊗ Ψ HV ) ◦ (Ψ nHH ⊗ id) ◦ (∆ ⊗ id) (38) = (Ψ HV ⊗ id) ◦ (id ⊗ Ψ HV ) ◦ (∆ n ⊗ id) proves the compatibility of Ψ HV with the comultiplication ∆ n . The relations regarding theunmodified unit or counit remain trivially true. The proof for a right H -braided vector spaceis completely analogous. (cid:3) Since, by (17), ( S ⊗ S ) ◦ Ψ HH = Ψ HH ◦ ( S ⊗ S ) and S k ◦ m = m ◦ Ψ kHH ◦ ( S k ⊗ S k ) = m k ◦ ( S k ⊗ S k ) , ( S k ⊗ S k ) ◦ ∆ = Ψ − kHH ◦ ∆ ◦ S k = ∆ − k ◦ S k , we obtain immediately the following corollary. Corollary 3.2.
Let H be a braided Hopf algebra and n ∈ Z . Then the antipode S definesbraided Hopf algebra homomorphisms S k : H ( n, − n ) → H ( n + k, − ( n + k )) and braided bialgebrahomomorphisms S k : H ( n − , − n ) → H ( n + k − , − ( n + k )) , where k ∈ Z if S is invertible and k ∈ N otherwise. For invertible S , all these homomorphisms are isomorphisms of braidedHopf algebras. In the unbraided case, a left action of an algebra yields a right action of the opposite algebrawith flipped multiplication, and a left coaction of a coalgebra defines a right coaction of theopposite coalgebra with the flipped coproduct. Evidently, the same holds if left and rightare interchanged. However, the usual flip is in general not compatible with the braiding. Aproper version in the braided setting is given in the next proposition.
Proposition 3.3.
Let ( H, m ) be a braided algebra and V a braided left H -module. Then (39) ν ◦ R : V ⊗ H −→ H, ν ◦ R := ν L ◦ Ψ − HV turns V into a braided right H -module with respect to the multiplication m − := m ◦ Ψ − HH and the braiding Ψ − HH on H . If H is a braided bialgebra and V is a braided left H -modulealgebra, then ν ◦ R transforms V into a right H ( − , -module algebra.Analogously, if V is a braided right H -module, then ν ◦ L : H ⊗ V −→ H, ν ◦ L := ν R ◦ Ψ − V H turns V into a braided left H -module with respect to the multiplication m − and the braiding Ψ − HH on H . Furthermore, a braided right H -module algebra becomes a left H ( − , -modulealgebra. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 11
Given a coalgebra H with coproduct ∆ and a braided right H -comodule V , the left coac-tion ρ ◦ L : V −→ H ⊗ V, ρ ◦ L := Ψ − HV ◦ ρ R turns V into a braided left H -comodule with respect to the coproduct ∆ − := ∆ ◦ Ψ − HH andthe braiding Ψ − HH on H . If H is a braided bialgebra and V is a braided right H -comodulealgebra, then ρ ◦ L transforms V into a left H (0 , − -comodule algebra.For a braided left H -comodule V , the right coaction ρ ◦ R : V −→ V ⊗ H, ρ ◦ R := Ψ − V H ◦ ρ L turns V into a braided left H -comodule with respect to the coproduct ∆ − and the braid-ing Ψ − HH on H , and a braided left H -comodule algebra becomes a right H (0 , − -comodulealgebra.Proof. The compatibility of the inverse braidings with algebraic structures can be deducedfrom Lemma 2.4. In particular, the braiding Ψ − HV turns a V into right H -braided vector spaceand, by Proposition 3.1, is compatible with the multiplication m − . Since ν ◦ R ◦ ( ν ◦ R ⊗ id) (39) = ν L ◦ Ψ − HV ◦ ( ν L ⊗ id) ◦ (Ψ − HV ⊗ id) (23) = ν L ◦ (id ⊗ ν L ) ◦ (Ψ − HH ⊗ id) ◦ (id ⊗ Ψ − HV ) ◦ (Ψ − HV ⊗ id) (2) , (19) = ν L ◦ ( m ⊗ id) ◦ (id ⊗ Ψ − HV ) ◦ (Ψ − HV ⊗ id) ◦ (id ⊗ Ψ − HH ) (5) = ν L ◦ Ψ − HV ◦ (id ⊗ m ) ◦ (id ⊗ Ψ − HH ) (39) = ν ◦ R ◦ (id ⊗ m − ) , and ν ◦ R ( v ⊗
1) = v by (6) and (19), it follows that ν ◦ R defines a right H -action with respect tothe multiplication m − . Moreover, Ψ − HV ◦ ( ν ◦ R ⊗ id) (39) = Ψ − HV ◦ ( ν L ⊗ id) ◦ (Ψ − HV ⊗ id) (23) = (id ⊗ ν L ) ◦ (Ψ − HH ⊗ id) ◦ (id ⊗ Ψ − HV ) ◦ (Ψ − HV ⊗ id) (2) , (39) = (id ⊗ ν ◦ R ) ◦ (Ψ − HV ⊗ id) ◦ (id ⊗ Ψ − HH ) proves (22). Therefore ν ◦ R equips V with the structure of a braided right H -module withrespect to the multiplication m − and the braidings Ψ − HH and Ψ − HV .If H is a braided bialgebra and V is a braided left H -module algebra, then ν ◦ R ◦ ( m V ⊗ id) (7) , (39) = ν L ◦ (id ⊗ m V ) ◦ (Ψ − HV ⊗ id) ◦ (id ⊗ Ψ − HV ) (32) = m V ◦ ( ν L ⊗ ν L ) ◦ (id ⊗ Ψ HV ⊗ id) ◦ (∆ ⊗ id ⊗ id) ◦ (Ψ − HV ⊗ id) ◦ (id ⊗ Ψ − HV ) (10) = m V ◦ ( ν L ⊗ ν L ) ◦ (Ψ − HV ⊗ Ψ − HV ) ◦ (id ⊗ Ψ − HV ⊗ id) ◦ (id ⊗ id ⊗ ∆) (39) = m V ◦ ( ν ◦ R ⊗ ν ◦ R ) ◦ (id ⊗ Ψ − HV ⊗ id) ◦ (id ⊗ id ⊗ ∆) . This shows that ν ◦ R satisfies the compatibility condition of a braided right H -module algebrawith respect to the unmodified coproduct ∆ and the braiding Ψ − HV , i.e., ν ◦ R equips V with thestructure of a braided right H ( − , -module algebra.The proof of the opposite version and the proofs for the coactions are similar and left tothe reader. (cid:3) In the last proposition, we had to replace the product of the braided algebra by the oppositeone in order to interchange left and right actions. If H is a braided Hopf algebra with bijectiveantipode, we can use the antipode to turn a left action into a right action of the same algebra,but with a modified coproduct. To see this, it suffices to observe that, if V is a braided left H -module and ϕ : H → H is a Hopf algebra homomorphism, then V becomes a braided left H -module in the obvious way. Thus, setting H := H (1 , − and ϕ := S − : H (1 , − → H ,we obtain from Corollary 3.2 and Proposition 3.3 a right action of the Hopf algebra H (0 , − with the unmodified product, the coproduct ∆ − and the braiding Ψ − HH . Similar argumentscan be applied to right actions and left or right coactions. We summarize these observationsin the next corollary for (co)module algebras. Corollary 3.4.
Let H be a braided Hopf algebra with invertible antipode S . If V is a left H -module algebra, then the right action ν R,S : V ⊗ H −→ V, ν
R,S := ν L ◦ Ψ − HV ◦ (id ⊗ S − ) , turns V into a right H (0 , − -module algebra. Analogously, a right H -module algebra V becomes a left H (0 , − -module algebra for the left action defined by ν L,S : V ⊗ H −→ V, ν
L,S := ν R ◦ Ψ − V H ◦ ( S − ⊗ id) . Given a right H -comodule algebra V , the left coaction ρ L,S : V −→ H ⊗ V, ρ
L,S := ( S − ⊗ id) ◦ Ψ − HV ◦ ρ R , turns V into a left H ( − , -comodule algebra, and a left H -comodule algebra V becomes aright H ( − , -comodule algebra for the right coaction defined by ρ R,S : V −→ V ⊗ H, ρ
R,S := (id ⊗ S − ) ◦ Ψ − V H ◦ ρ L .
4. D
UALITY FOR INFINITE - DIMENSIONAL BRAIDED ALGEBRAS , COALGEBRAS , BIALGEBRAS AND H OPF ALGEBRAS
This section provides a detailed description of duality for braided algebras, braided coalge-bras, and both structures together, i.e., braided bialgebras and Hopf algebras. As dual objectsmay not exist in a braided monoidal category, specifically in the infinite-dimensional setting(cf. [18]), we continue with our non-categorical approach. That is, we assume the existenceof a dual space with certain properties without proving its existence or uniqueness, whichmeans that we will not define a functor into a dual category. Furthermore, our definitions willbe rather constructive in the sense that they are expressed by explicit formulas derived fromthe given structures.A dual pairing between two vector spaces U and H is a linear map h· , ·i : U ⊗ H −→ K .Let H ′ denote the dual space of H . Given a subspace U ⊂ H ′ , we define a dual pairingbetween U and H by(40) h· , ·i : U ⊗ H −→ K , h f, a i := f ( a ) . Identifying by a slight abuse of notation H with its image ι ( H ) ⊂ H ′′ under the cannonicalembedding ι : H → H ′′ , ι ( a )( f ) := f ( a ) , the dual pairing (40) becomes symmetric in thesense that h f, a i = h a, f i . A subspace U ⊂ H ′ is called non-degenerate, or synonymouslythe dual pairing is called non-degenerate, if the associated bilinear map h· , ·i : U × H → K is non-degenerate. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 13
The dual pairing h· , ·i defined in (40) is actually the restriction of the fundamental evalua-tion map ev : H ′ ⊗ H → K , ev( f ⊗ a ) := f ( a ) . The problem in the infinite setting is thatthe so called coevaluation map coev : K → H ⊗ H ′ may not exist, see e.g. [18]. In otherwords, the braided monoidal category may not be rigid. Nevertheless, the evaluation map,or rather its restriction h· , ·i , will play a fundamental role in the dual pairing between tensorspaces. In particular, the dual pairing between n -fold tensor product spaces will entirely betraced back to the evaluation map on adjacent tensor factors. That is, given linear spaces H j and subspaces U j ⊂ H ′ j , j = 1 , . . . , n , we define hh · , · ii : ( U n ⊗ . . . ⊗ U ) ⊗ ( H ⊗ . . . ⊗ H n ) −→ K , hh · , · ii := h · , · i ◦ (cid:0) id ⊗ h · , · i ⊗ id (cid:1) ◦ · · · ◦ (cid:0) id ⊗ . . . ⊗ h · , · i ⊗ . . . ⊗ id (cid:1) . (41)This definition is consistent with the representation of braided monoidal categories by braidedstrings. Once a convention for a dual pairing between tensor product spaces is agreed upon,it should be avoided to use isomorphisms between tensor spaces in the dual pairing that donot arise from braidings. For instance, to pair the second leg in H ′ ⊗ H ′ with the second legin H ⊗ H , it is more appropriate to apply first a braiding Ψ H ′ H and to consider ( h· , ·i ⊗ h· , ·i ) ◦ (id ⊗ Ψ H ′ H ⊗ id) : H ′ ⊗ H ′ ⊗ H ⊗ H −→ K . On the other hand, we will also make use of the embedding U ⊗ · · · ⊗ U n ⊂ ( H ⊗ . . . ⊗ H n ) ′ (mind the order). In this case, we write ( f ⊗ · · · ⊗ f n )( a ⊗ · · · ⊗ a n ) := f ( a ) · · · f n ( a n ) . Now let H be a braided vector space with braiding Ψ HH . Our first aim is to show that,for appropriate subspaces U ⊂ H ′ , Ψ HH induces braidings on U ⊗ U and between U and H . Moreover, the braidings between U and H will be compatible with the multiplicationand comultiplication on H if these structures are compatible with Ψ HH . According to ournon-categorical approach, we will not assume that U is unique nor prove that it always exists.To begin, consider the linear map(42) Ψ H ′ H ′ : H ′ ⊗ H ′ → ( H ⊗ H ) ′ , Ψ H ′ H ′ ( f ⊗ g )( b ⊗ a ) := hh f ⊗ g, Ψ HH ( a ⊗ b ) ii , where a, b ∈ H and f, g ∈ H ′ . Note that we do not assume that Ψ H ′ H ′ ( H ′ ⊗ H ′ ) ⊂ H ′ ⊗ H ′ .Similarly, using the fact that the canonical pairing h· , ·i : H ′ ⊗ H → K is non-degenerate,we define Ψ H ′ H : H ′ ⊗ H −→ ( H ′ ⊗ H ) ′ , Ψ H ′ H ( g ⊗ a )( f ⊗ b ) := hh f ⊗ g, Ψ HH ( a ⊗ b ) ii , (43) Ψ ◦ H ′ H : H ′ ⊗ H −→ ( H ′ ⊗ H ) ′ , Ψ ◦ H ′ H ( g ⊗ a )( f ⊗ b ) := hh f ⊗ g, Ψ − HH ( a ⊗ b ) ii , (44) Ψ HH ′ : H ⊗ H ′ −→ ( H ⊗ H ′ ) ′ , Ψ HH ′ ( b ⊗ f )( a ⊗ g ) := hh f ⊗ g, Ψ HH ( a ⊗ b ) ii , (45) Ψ ◦ HH ′ : H ⊗ H ′ −→ ( H ⊗ H ′ ) ′ , Ψ ◦ HH ′ ( b ⊗ f )( a ⊗ g ) := hh f ⊗ g, Ψ − HH ( a ⊗ b ) ii . (46)It will be convenient to introduce some Sweedler-type notation. As usual, a coproduct iswritten ∆( a ) = a (1) ⊗ a (2) with increasing numbers for multiple coproducts. Analogously,left and right coactions are written ρ L ( v ) = v ( − ⊗ v (0) and ρ R ( v ) = v (0) ⊗ v (1) , respectively.For a given braiding on a vector space H , we employ the notation Ψ HH ( a ⊗ b ) = b h i ⊗ a h i , a, b ∈ H. For multiple braidings, an index will be used to indicate the chronological order. As anexample, a combination of (1) and (10) gives for a and b from a braided coalgebra(47) b h i ⊗ a h i (2) h i ⊗ a h i (1) h i = b h i h i ⊗ a (2) h i h i ⊗ a (1) h i h i . We use a back-prime to denote the inverse of Ψ HH , i.e.,(48) Ψ − HH ( a ⊗ b ) = b h i ⊗ a h i , a, b ∈ H. Then clearly(49) a h ih i ⊗ b h ih i = a ⊗ b = a h i h i ⊗ b h i h i , a, b ∈ H. In a similar vein, (1) and (49) yield the identity(50) c h ih i ⊗ b h ih i ⊗ a h i h i = c h i h i ⊗ b h i h i ⊗ a h i h i , a, b, c ∈ H. The same notations will be used for subspaces U ⊂ H ′ such that Ψ H ′ H ′ ( U ⊗ U ) ⊂ U ⊗ U .Then (42) reads for instance(51) f h i ( a ) g h i ( b ) = g ( b h i ) f ( a h i ) , f, g ∈ U, a, b ∈ H. If Ψ H ′ H ( g ⊗ a ) , Ψ ◦ H ′ H ( g ⊗ a ) ∈ H ⊗ U ⊂ ( H ′ ⊗ H ) ′ for g ⊗ a ∈ U ⊗ H , we write(52) Ψ H ′ H ( g ⊗ a ) := a { } ⊗ g { } , Ψ ◦ H ′ H ( g ⊗ a ) := a { } ◦ ⊗ g { } ◦ , and a similar notation will be employed for Ψ HH ′ and Ψ ◦ HH ′ . Under the assumption that allmaps belong to the tensor products of the corresponding spaces, (42)–(46) yield in Sweedler-type notation g ( b h i ) f ( a h i ) = g h i ( b ) f h i ( a ) = f ( a { } ) g { } ( b ) = f { } ( a ) g ( b { } ) , (53) g ( b h i ) f ( a h i ) = g h i ( b ) f h i ( a ) = f ( a { } ◦ ) g { } ◦ ( b ) = f { } ◦ ( a ) g ( b { } ◦ ) . (54)Furthermore, if U ⊂ H ′ is non-degenerate, we conclude from (53) and (54) that f ( a { } ) g { } = f h i ( a ) g h i , a { } g { } ( b ) = g ( b h i ) a h i , (55) f ( a { } ◦ ) g { } ◦ = f h i ( a ) g h i , a { } ◦ g { } ◦ ( b ) = g ( b h i ) a h i , (56)for all f, g ∈ U and a, b ∈ H .The next lemma shows that, under suitable conditions on U ⊂ H ′ , the braiding Ψ HH induces braidings on U ⊗ U and between U and H which are compatible with possiblyadditional structures on H . Lemma 4.1.
Let H be a braided vector space and U ⊂ H ′ a non-degenerate subspace.Assume that (57) Ψ UU := Ψ H ′ H ′ ↾ U ⊗ U : U ⊗ U −→ U ⊗ U is bijective. Then Ψ UU defines a braiding on U . If (58) Ψ UH := Ψ H ′ H ↾ U ⊗ H : U ⊗ H −→ H ⊗ U is bijective, then Ψ UH turns H into a left U -braided vector space and U into a right H -braided vector space with respect to the braidings Ψ HH and Ψ − HH on H , and Ψ UU and Ψ − UU on U .In case (59) Ψ ◦ UH := Ψ ◦ H ′ H ↾ U ⊗ H : U ⊗ H −→ H ⊗ U UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 15 is bijective, it also turns H into a left U -braided vector space and U into a right H -braidedvector space with respect to the braidings Ψ HH and Ψ − HH on H , and Ψ UU and Ψ − UU on U .If H is a braided (unital) algebra, then the braidings Ψ UH and Ψ ◦ UH are compatible withthe multiplication on H . If H is a braided coalgebra, then the braidings Ψ UH and Ψ ◦ UH arecompatible with the comultiplication on H .The analogous statements hold for the opposite versions with respect to the braidings Ψ HU := Ψ HH ′ ↾ H ⊗ U : H ⊗ U −→ U ⊗ H, and Ψ ◦ HU := Ψ ◦ HH ′ ↾ H ⊗ U : H ⊗ U −→ U ⊗ H. Proof.
Let f ⊗ g ⊗ h ∈ U ⊗ U ⊗ U . Since H ⊗ H ⊗ H separates the points of U ⊗ U ⊗ U ⊂ H ′ ⊗ H ′ ⊗ H ′ , we can prove (1) by evaluating both sides on all x ⊗ y ⊗ z ∈ H ⊗ H ⊗ H . InSweedler-type notation, we get h h i h i ( x ) g h i h i ( y ) f h i h i ( z ) (51) = h ( x h i h i ) g ( y h i h i ) f ( z h i h i ) (1) = h ( x h i h i ) g ( y h i h i ) f ( z h i h i ) (51) = h h i h i ( x ) g h i h i ( y ) f h i h i ( z ) . (60)As h· , ·i : U ⊗ H → K is non-degenerate, we conclude that Ψ UU satisfies (1), and so does Ψ − UU by Lemma 2.4.To prove (2), we use again the non-degeneracy of the pairing h· , ·i : U ⊗ H → K andevaluate a { } { } ⊗ g h i{ } ⊗ f h i{ } ∈ H ⊗ U ⊗ U on all h ⊗ y ⊗ z ∈ U ⊗ H ⊗ H , where a ∈ H and f, g ∈ U . This gives h ( a { } { } ) g h i{ } ( y ) f h i{ } ( z ) (53) = h h i ( a h i ) g h i h i ( y ) f h i ( z h i ) (61) (51) = h ( a h i h i ) g ( y h i h i ) f ( z h i h i ) (1) = h ( a h i h i ) g ( y h i h i ) f ( z h i h i ) (51) = h h i ( a h i ) g ( y h i h i ) f h i ( z h i ) (53) = h ( a { } { } ) g { } ( y h i ) f { } ( z h i ) (51) = h ( a { } { } ) g { } h i ( y ) f { } h i ( z ) . Since U ⊗ H ⊗ H separates the points of H ⊗ U ⊗ U , these calculations show that (2) issatisfied.Much in the same way, for all f ⊗ a ⊗ b ∈ U ⊗ H ⊗ H and g ⊗ h ⊗ z ∈ U ⊗ U ⊗ H , wecompute g ( b h i{ } ) h ( a h i{ } ) f { } { } ( z ) (53) = g ( b h i h i ) h ( a h i h i ) f ( z h i h i ) (1) = g ( b h i h i ) h ( a h i h i ) f ( z h i h i ) (53) = g h i ( b { } ) h h i ( a { } ) f { } { } ( z ) (51) = g ( b { } h i ) h ( a { } h i ) f { } { } ( z ) , which proves (3). This finishes the proof of first part of the lemma regarding the braidings Ψ HH , Ψ UU and Ψ UH . By Lemma 2.4, the same holds with respect to the braidings Ψ − HH , Ψ − UU and Ψ UH . Replacing in above calculations Ψ HH by Ψ − HH and Ψ UU by Ψ − UU shows theanalogous results for Ψ ◦ UH Let H be a braided algebra. Using the fact that Ψ HH satisfies (5) and (6), we get for all f, g ∈ U and a, b ∈ H , f (( ab ) { } ) g { } ( c ) (53) = f (( ab ) h i ) g ( c h i ) (5) = f ( a h i b h i ) g ( c h i h i ) (53) = f ( a { } b { } ) g { } { } ( c ) (62)and(63) f (1 { } ) g { } ( a ) (53) = f (1 h i ) g ( a h i ) (6) = f (1) g ( a ) . This implies the compatibilty of Ψ UH with the multiplication on H .By Lemma 2.4, Ψ − HH is also compatible with the multiplication and on H . Replacing { k } by { k } ◦ and h k i by h k i in (62) and (63) shows the compatibilty of Ψ ◦ UH with the multipli-cation on H . Similarly, if H is a braided coalgebra, we obtain for f, g, h ∈ U and a, b ∈ H that f ( a { } (1) ) g ( a { } (2) ) h { } ( b ) (55) = f ( a h i (1) ) g ( a h i (2) ) h ( b h i ) (10) = f ( a (1) h i ) g ( a (2) h i ) h ( b h i h i ) (53) = f ( a (1) { } ) g ( a (2) { } ) h { } { } ( b ) and ε ( a { } ) h { } ( b ) (55) = ε ( a h i ) h ( b h i ) (10) = ε ( a ) h ( b ) , which proves (11) for Ψ UH .The same proof with the notational changes mentioned above shows the compatibility of Ψ ◦ UH with the multiplication or the comultiplication (as applicable) on H . The statements ofthe opposite versions are proven analogously. (cid:3) The following definition of a dual pairing between braided bialgebras is the central def-inition of this section because it will also serve as a guiding principle for duality betweenbraided algebras and braided coalgebras. Similar definitions can be found in [2, 5, 6, 16].
Definition 4.2.
Let U and H be braided bialgebras and let Υ UH : U ⊗ H → H ⊗ U be abraiding such that H is a left U -braided vector space, U is a right H -braided vector space,and the braiding is compatible with the multiplications and comultiplications of U and H . Adual pairing between U and H is a linear map h· , ·i : U ⊗ H → K such that h· , ·i ◦ ( m ⊗ id) = ( h· , ·i ⊗ h· , ·i ) ◦ (id ⊗ Υ UH ⊗ id) ◦ (id ⊗ id ⊗ ∆) , (64) h· , ·i ◦ (id ⊗ m ) = ( h· , ·i ⊗ h· , ·i ) ◦ (id ⊗ Υ UH ⊗ id) ◦ (∆ ⊗ id ⊗ id) , (65) h , a i = ε ( a ) , h h, i = ε ( h ) , a ∈ H, h ∈ U. (66)For a dual pairing between braided Hopf algebras, it is additionally required that(67) h· , ·i ◦ ( S ⊗ id) = h· , ·i ◦ (id ⊗ S ) . If the dual pairing is non-degenerate, U is called a left dual of H , and H is called a right dualof U .Given a braided coalgebra H , the convolution product (15) turns H ′ into an associativealgebra. In general, this product will not be compatible with the dual pairing of Defini-tion 4.2. On the other hand, under the assumptions of Definition 4.2, the product on U ⊂ H ′ is uniquely determined by (64) since H separates the points of H ′ . For this reason, we willconsider an alternative convolution product on H ′ such that the equality in (64) is automati-cally met for subalgebras U ⊂ H ′ satisfying the assumptions of Lemma 4.1. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 17
Proposition 4.3.
Let H be a braided coalgebra and set ∗ : H ′ ⊗ H ′ −→ H ′ , f ∗ g ( a ) := hh f ⊗ g , Ψ HH ◦ ∆( a ) ii = f ( a (1) h i ) g ( a (2) h i ) , (68) where f, g ∈ H ′ and a ∈ H . Then (68) turns H ′ into an associative unital algebra with theunit element given by the counit of H .Suppose that U ⊂ H ′ is a (unital) subalgebra separating the points of H . If Ψ UU definedin (57) is bijective, then it turns U into a braided algebra. In case Ψ UH or Ψ ◦ UH satisfies theassumptions of Lemma 4.1, then it defines a braiding that is compatible with the multiplica-tion on U . The same remains true for the opposite version with Ψ UH and Ψ ◦ UH replaced by Ψ HU and Ψ ◦ HU , respectively.Proof. The associativity of the product ∗ is equivalent to coassociativity of ∆ := Ψ HH ◦ ∆ which was proven in Proposition 3.1. Moreover, by (68) and the second identity in (11), f ∗ ε ( a ) = f ◦ ( ε ⊗ id) ◦ Ψ HH ◦ ∆( a ) = f ◦ (id ⊗ ε ) ◦ ∆( a ) = f ( a ) and similarly ε ∗ f ( a ) = f ( a ) for all f ∈ H ′ and a ∈ H . Therefore ε yields the unit elementin H ′ with respect to the product ∗ .To show the compatibilty of the multiplication with the braiding, we have to verify Equa-tions (5)–(8) for Ψ UU but only (5) and (6) for Ψ UH . Since U separates the points of H andvice versa, we may again prove the required relations by evaluating both sides on elementsfrom U and H . Let a, b, c ∈ H and f, g, h ∈ U . In Sweedler-type notation, the proof of (5)reads as follows: h h i ( b )( f ∗ g ) h i ( a ) (51) , (68) = h ( b h i ) g ( a h i (2) h i ) f ( a h i (1) h i ) (47) = h ( b h i h i ) g ( a (2) h i h i ) f ( a (1) h i h i ) (51) , (68) = h h i h i ( b )( f h i ∗ g h i )( a ) . (69)Similarly, ( f ∗ g ) h i ( b ) h h i ( a ) (51) , (68) = g ( b h i (2) h i ) f ( b h i (1) h i ) h ( a h i ) (1) , (11) = g ( b (2) h i h i ) f ( b (1) h i h i ) h ( a h i h i ) (51) , (68) = ( f h i ∗ g h i )( b ) h h i h i ( a ) , (70)which yields (7) for Ψ UU . Moreover, f h i ( b ) ε h i ( a ) (51) = f ( b h i ) ε ( a h i ) (10) = ε ( a ) f ( b ) (11) = ε ( a h i ) f ( b h i ) (51) = ε h i ( a ) f h i ( b ) (71)shows that Ψ UU fulfills (6) and (8). This finishes the proof that Ψ UU is compatible with themultiplication of U .Applying (53) to both sides of (70) and the right side of (71) gives h ( a { } )( f ∗ g ) { } ( b ) = h ( a { } { } )( f { } ∗ g { } )( b ) , f ( b { } ) ε { } ( a ) = f ( b ) ε ( a ) , so that Ψ UH is also compatible with the multiplication on U .By Lemma 2.4, Ψ − HH is compatible with the comultiplication on H . Similar as above, h ( a { } ◦ )( f ∗ g ) { } ◦ ( b ) (54) , (68) = h ( a h i ) g ( b h i (2) h i ) f ( b h i (1) h i ) (11) = h ( a h i h i ) g ( b (2) h i h i ) f ( b (1) h i h i ) (50) = h ( a h i h i ) g ( b (2) h ih i ) f ( b (1) h ih i ) (54) = h h i ( a { } ◦ ) g { } ◦ ( b (2) h i ) f h i ( b (1) h i ) (54) = h ( a { } ◦ { } ◦ ) g { } ◦ ( b (2) h i ) f { } ◦ ( b (1) h i ) (68) = h ( a { } ◦ { } ◦ )( f { } ◦ ∗ g { } ◦ )( b ) , hence Ψ ◦ UH satisfies (5). By Lemma 2.4 and the second relation of (11), f ( a { } ◦ ) ε { } ◦ ( b ) = f ( a h i ) ε ( b h i ) = f ( a ) ε ( b ) , which shows that Ψ ◦ UH satisfies also (6). This proves the compatibility of Ψ ◦ UH with themultiplication on U .Is opposite version is shown in the same way. (cid:3) Recall that, for any unital algebra H , the dual space H ′ contains a largest coalgebra H ◦ such that ∆ : H ◦ → H ◦ ⊗ H ◦ , ∆( f )( a ⊗ b ) := f ( ab ) for a, b ∈ H , and ε ( f ) := f (1) (seee.g. [7]). However, the compatibility condition (65) of the dual pairing requires to considera modified coproduct, say ∆ , on suitable subspaces of H ′ . To state an explicit formula,assume that H is a braided algebra and suppose that U is a linear subspace of H ′ satisfyingthe assumptions of Lemma 4.1. Then, for ∆ on U and Υ UH = Ψ UH , (65) is equivalent to hh ∆ ( f ) , Ψ HH ( a ⊗ b ) ii = h f, m ( a ⊗ b ) i , f ∈ U, a, b ∈ H, which leads to(72) hh ∆ ( f ) , a ⊗ b ii = h f, m ◦ Ψ − HH ( a ⊗ b ) i f ∈ U, a, b ∈ H. If we replace Ψ UH by Ψ ◦ UH in (65), then Ψ − HH needs to be replaced by Ψ HH in (72).Note that (72) determines uniquely ∆ ( f ) ∈ ( H ⊗ H ) ′ but ∆ ( f ) defined by the righthand side of (72) may not belong to H ′ ⊗ H ′ . The next proposition shows that, similar to theunbraided case, H ′ contains a largest coalgebra such that the coproduct is given as in (72) andany subcoalgebra U satisfying the assumptions of Lemma 4.1 becomes a braided coalgebrasuch that the comultiplication is compatible with the braidings Ψ UH and Ψ ◦ UH . Proposition 4.4.
Let H be a braided unital algebra and consider (73) ∆ : H ′ −→ ( H ⊗ H ) ′ , hh ∆( f ) , a ⊗ b ii := h f, m ◦ Ψ HH ( a ⊗ b ) i = f ( b h i a h i ) , where f ∈ H ′ , a, b ∈ H . Then there exists a largest coalgebra H ◦ in H ′ such that thecoproduct is given by (73) and ε ( f ) := f (1) .Suppose that U ⊂ H ◦ is a subcoalgebra separating the points of H . If Ψ UU defined in (57) is bijective, then it turns U into a braided coalgebra. In case Ψ UH or Ψ ◦ UH satisfies theassumptions of Lemma 4.1, then it defines a braiding that is compatible with the comultipli-cation on U . The same remains true for the opposite version with Ψ UH and Ψ ◦ UH replaced by Ψ HU and Ψ ◦ HU , respectively.Proof. By Proposition 3.1, m := m ◦ Ψ HH defines an associative multiplication on H withunit element ∈ H . Therefore the existence of a largest coalgebra follows from the knownresult of the unbraided case, i.e., there exists a largest coalgebra, say H ◦ , in H ′ with thecoproduct given by (73) and ε ( f ) := f (1) , see e.g. [7, Section 1.2.8].Assume that the subcoalgebra U ⊂ H ◦ satisfies the assumptions of the proposition whichguarantee that Ψ UU and Ψ UH are well-defined. Then, for all f, g ∈ U and a, b, c ∈ H , wehave g h i (1) ( c ) g h i (2) ( b ) f h i ( a ) (73) = g h i ( c h i b h i ) f h i ( a ) (53) = g (( c h i b h i ) h i ) f ( a h i ) (7) = g ( c h i h i b h i h i ) f ( a h i h i ) (1) = g ( c h i h i b h i h i ) f ( a h i h i ) (74) (73) = g (1) ( c h i ) g (2) ( b h i ) f ( a h i h i ) (53) = g (1) h i ( c ) g (2) h i ( b ) f h i h i ( a ) UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 19 which proves that Ψ UU satisfies the first relation of (11). The second relation of (11) followsfrom ε ( g h i ) f h i ( a ) = g h i (1) f h i ( a ) (53) = g (1 h i ) f ( a h i ) (8) = g (1) f ( a ) = ε ( g ) f ( a ) . In exactly the same way, one shows that Ψ UU satisfies (10), hence U is a braided coalgebrawith respect to braiding Ψ UU .Applying ∆ to (55), evaluating on b ⊗ c ∈ H ⊗ H and using (74) gives f ( a { } ) g { } (1) ( c ) g { } (2) ( b ) (55) = f h i ( a ) g h i (1) ( c ) g h i (2) ( b ) (74) = f h i h i ( a ) g (1) h i ( c ) g (2) h i ( b ) (53) = f ( a { } { } ) g (1) { } ( c ) g (2) { } ( b ) . Moreover, f (cid:0) a { } ε ( g { } ) (cid:1) = g { } (1) f ( a { } ) (53) = g (1 h i ) f ( a h i ) (8) = g (1) f ( a ) = f (cid:0) ε ( g ) a (cid:1) . From the last two computations, we conclude that Ψ UH satisfies (10), i.e., the braiding Ψ UH is compatible with the comultiplication on U .Much in the same way, by applying Lemma 2.4 to Ψ HH , f ( a { } ◦ ) g { } ◦ (1) ( c ) g { } ◦ (2) ( b ) (56) = f h i ( a ) g h i (1) ( c ) g h i (2) ( b ) (73) = f h i ( a ) g h i ( c h i b h i ) (54) = f ( a h i ) g (( c h i b h i ) h i ) (7) = f ( a h i h i ) g ( c h ih i b h ih i ) (50) = f ( a h i h i ) g ( c h i h i b h i h i ) (73) = f ( a h i h i ) g (1) ( c h i ) g (2) ( b h i ) (54) = f h i ( a h i ) g (1) h i ( c ) g (2) ( b h i ) (54) = f ( a { } ◦ { } ◦ ) g (1) { } ◦ ( c ) g (2) { } ◦ ( b ) , and f (cid:0) a { } ε ( g { } ◦ ) (cid:1) = g { } ◦ (1) f ( a { } ◦ ) (54) = g (1 h i ) f ( a h i ) (6) = g (1) f ( a ) = f (cid:0) ε ( g ) a (cid:1) . There-fore, if the braiding Ψ ◦ UH exists, it is compatible with the comultiplication on U .The opposite version is proven analogously. (cid:3) Given a braided bialgebra H , Lemma 4.1, Proposition 4.3 and Proposition 4.4 tell us howto define braidings, a product and a coproduct, respectively, on appropriate subspaces of H ′ .The next theorem shows that these structures fit well together, i.e., they can be used to obtaina braided bialgebra U ⊂ H ′ such that the canonical pairing yields a dual pairing betweenbraided bialgebras. It is noteworthy to mention that the braiding Υ UH in Definition 4.2 willnot be implemented by Ψ UH , but by Ψ ◦ UH . Theorem 4.5.
Let H be a braided bialgebra and consider the product m : H ′ ⊗ H ′ → H ′ defined by (75) h m ( f ⊗ g ) , a i := hh f ⊗ g, Ψ − HH ◦ ∆( a ) ii = f ( a (1) h i ) g ( a (2) h i ) , a ∈ H, f, g ∈ H ′ . Assume that U ⊂ H ′ is a unital subalgebra which is also a subcoalgebra of H ◦ for thecoproduct ∆ defined in Proposition 4.4.Suppose that U satisfies the left-handed version of Lemma 4.1 such that Ψ UU and Ψ ◦ UH define braidings. Then U is a braided bialgebra with respect to the braiding Ψ UU , and thecanonical pairing h· , ·i : U ⊗ H → K defines a pairing between braided bialgebras withrespect to the braiding Ψ ◦ UH such that U becomes a left dual of H .In case U satisfies the right-handed version of Lemma 4.1 such that Ψ UU and Ψ ◦ HU yieldbraidings, the canonical pairing h· , ·i : H ⊗ U → K defines a pairing between braidedbialgebras with respect to the braiding Ψ ◦ HU and U becomes a right dual of H . If H is a braided Hopf algebra and f ◦ S ∈ U for all f ∈ U , then U is a braided Hopfalgebra with antipode S ( f ) := f ◦ S .Proof. By Lemma 2.4, Ψ − HH defines a braiding on H which is compatible with the comul-tiplication on H . Therefore, by Proposition 4.3, the product (75) is well-defined. Note thatthe braiding in (75) is the inverse of that in (68). Combining Proposition 4.3 with Lemma2.4 shows that the multiplication on U is compatible with Ψ UU so that U becomes a braidedalgebra. From Proposition 4.4, it follows directly that U is a braided coalgebra with respectto Ψ UU . Therefore, to complete the proof that U yields a braided bialgebra, it suffices toprove (16).As in the previous proofs, we will verify (16) by evaluating both sides on a ⊗ b ∈ H ⊗ H .It was shown in Proposition 3.1 that H (1 , − is a braided bialgebra with respect to Ψ HH .Therefore, for all f ⊗ g ∈ U ⊗ U , hh ∆ ◦ m ( f ⊗ g ) , a ⊗ b ii (73) , (75) = hh f ⊗ g, ∆ − ◦ m ( a ⊗ b ) ii (16) = hh f ⊗ g, ( m ⊗ m ) ◦ (id ⊗ Ψ HH ⊗ id) ◦ (∆ − ⊗ ∆ − )( a ⊗ b ) ii (73) = hh (∆ ⊗ ∆)( f ⊗ g ) , (id ⊗ Ψ HH ⊗ id) ◦ (∆ − ⊗ ∆ − )( a ⊗ b ) ii (42) , (57) = hh (id ⊗ Ψ UU ⊗ id) ◦ (∆ ⊗ ∆)( f ⊗ g ) , ∆ − ⊗ ∆ − ( a ⊗ b ) ii (75) = hh ( m ⊗ m ) ◦ (id ⊗ Ψ UU ⊗ id) ◦ (∆ ⊗ ∆)( f ⊗ g ) , a ⊗ b ii , which implies that the product and the coproduct on U satisfy the compatibility condition(16) with respect to the braiding Ψ UU . Thus we have proven that U is a braided bialgebra.Our next aim is to show that the cannonical pairing defines a pairing between braidedbialgebras with respect to the braiding Ψ ◦ UH . From Lemma 4.1, Proposition 4.3 and Proposi-tion 4.4, we conclude that Ψ ◦ UH is compatible with the multilplications and the comultiplica-tions on U and H . By Definitionen 4.2, it remains to prove Equations (64)–(66).Equation (66) is trivially satisfied by the stated unit element in Proposition 4.3 and thedefinition of the counit in Proposition 4.4. Let f, g ∈ U and a, b ∈ H . Equation (64) followsfrom m ( f ⊗ g )( a ) (75) = g ( a (2) h i ) f ( a (1) h i ) (54) = f ( a (1) { } ◦ ) g { } ◦ ( a (2) ) , and f ( ab ) (49) = f ( a h i h i b h i h i ) (73) = f (1) ( a h i ) f (2) ( b h i ) (54) = f (1) ( a { } ◦ ) f (2) { } ◦ ( b ) implies (65).Now assume that H is a braided Hopf algebra with antipode S and that f ◦ S ∈ U for all f ∈ U . With the definition S ( f ) := f ◦ S , we compute for all a ∈ H h m ◦ (id ⊗ S ) ◦ ∆( f ) , a i (75) = hh (id ⊗ S ) ◦ ∆( f ) , ∆ − ( a ) ii (41) = hh ∆( f ) , ( S ⊗ id) ◦ ∆ − ( a ) ii (73) = h f , m ◦ ( S ⊗ id) ◦ ∆ − ( a ) i = ε ( a ) h f, i = h ε ( f )1 , a i , where we used the fact from Proposition 3.1 that H (1 , − is a braided Hopf algebra withantipode S . This yields m ◦ (id ⊗ S ) ◦ ∆( f ) = ε ( f )1 . Interchanging the positions of S and id in above calculations shows that also m ◦ ( S ⊗ id) ◦ ∆( f ) = ε ( f )1 , therefore the linearmapping S : U → U defined above turns U into a braided Hopf algebra. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 21
Since f { } ◦ ( a ) g ( b { } ◦ ) = f ( a h i ) g ( b h i ) = f ( a { } ◦ ) g { } ◦ ( b ) by (54), and h· , ·i is symmet-ric in the sense that h a, f i = f ( a ) = h f, a i , the proof of the opposite version is essentiallythe same. (cid:3) Note that the same bialgebra U serves as a left and as a right dual of H by consideringeither the braiding Ψ ◦ UH or the braiding Ψ ◦ HU . From now on, we restrict ourselves mainly tothe left version, the right version differs essentially only in notation.Our definitions of the product and the coproduct on U ⊂ H ′ deviate from those given in[14] for rigid (="finite") braided monoidal categories. By evaluating elements of B ∗ ( ∼ U ) onelements of B ( ∼ H ) in [14], it can be seen that the main difference boils down to omittingthe braiding Υ UH in Definition 4.2 and using the hh· , ·ii instead. For a finite dimensionalbraided bialgebra, the dual braided bialgebra in the sense of [14, Proposition 2.4] is a specialcase of our construction and corresponds to U (1 , − . We present this result in the followingcorollary, its proof is straightforward by applying the corresponding definitions. Corollary 4.6.
Let H be a braided bialgebra and assume that U ⊂ H ′ fulfills the assumptionsof Theorem 4.5. Then the product m and the coproduct ∆ − of U (1 , − satisfy h m ( f ⊗ g ) , a i = hh f ⊗ g, ∆( a ) ii , hh ∆ − ( f ) , a ⊗ b ii = h f, m ( a ⊗ b ) i for all f, g ∈ U and a, b ∈ H . Note that, for a finite-dimensional braided bialgebra H , the non-degeneracy condition im-plies U = H ′ , and H ′ satisfies automatically the assumptions on U in above theorem. In thiscase, the left dual U = H ′ is unique. For an infinite-dimensional braided bialgebra, this neednot to be the case, see e.g. [7, Section 11.2.3] with all braidings given by the usual flip.Now let H be a possibly infinite-dimensional braided bialgebra and U ⊂ H ′ a left dualof H . If H ⊂ U ′ satisfies the conditions in Lemma 4.1, then we can apply Theorem 4.5 toconstruct a multiplication m and a coproduct ∆ on H such that H becomes a left dual of U .The next proposition shows that this left dual is isomorphic to the braided bialgebra H . Inthis sense the construction is reflexive, i.e., taking twice the left dual yields the same braidedbialgebra. Proposition 4.7.
Let H be a braided bialgebra and let U ⊂ H ′ be a left dual of H withbraiding Ψ UU satisying (57) . Assume that the map Ψ ◦ HU defined in Lemma 4.1 is a bijection.Consider the canonical embedding ι : H → ι ( H ) ⊂ U ′ given by ι ( a )( f ) := f ( a ) . Then ι ( H ) with the multiplication m and the comultiplication ∆ from Theorem 4.5 becomes a leftdual of U , and ι : H → ι ( H ) yields an isomorphism of braided bialgebras.Analogously, if U ⊂ H ′ is a right dual of H and Ψ ◦ UH in (59) is a bijection, then ι ( H ) ∼ = H becomes a right dual of U .If H is a braided Hopf algebra, then ι yields an isomorphism of braided Hopf algebras.Proof. Let a, b ∈ H and f, g ∈ U . Since Ψ ι ( H ) ι ( H ) ( ι ( a ) ⊗ ι ( b ))( g ⊗ f ) (42) = hh a ⊗ b, Ψ UU ( f ⊗ g ) ii (57) = hh ( ι ⊗ ι ) ◦ Ψ HH ( a ⊗ b ) , f ⊗ g ii , we conclude that Ψ ι ( H ) ι ( H ) : ι ( H ) ⊗ ι ( H ) → ι ( H ) ⊗ ι ( H ) yields a bijection and that ι intertwines the braidings on H and ι ( H ) . In the same manner, it follows from (42) and (46)that Ψ ◦ ι ( H ) U ◦ ( ι ⊗ id) = (id ⊗ ι ) ◦ Ψ ◦ HU which shows that Ψ ◦ ι ( H ) U is bijective. Thus, by Lemma4.1, it defines a braiding. By (42), (73) and (75), we have h m ( ι ( a ) ⊗ ι ( b )) , f i = hh a ⊗ b, Ψ − UU ◦ ∆( f ) ii = h m ( a ⊗ b ) , f i = h ι ◦ m ( a ⊗ b ) , f i , hence ι ◦ m = m ◦ ( ι ⊗ ι ) . This implies that ι ( H ) ⊂ U ′ is a unital subalgebra and that ι yieldsan algebra isomorphism.Next we prove that ι determines an coalgebra isomorphism. From(76) hh ∆( ι ( a )) , f ⊗ g ii (73) = h a , m ◦ Ψ UU ( f ⊗ g ) i (42) , (75) = hh ∆( a ) , f ⊗ g ii = hh ( ι ⊗ ι ) ◦ ∆( a ) , f ⊗ g ii , we conclude that ( ι ⊗ ι ) ◦ ∆ = ∆ ◦ ι . As a consequence, ∆( ι ( H )) ⊂ ι ( H ) ⊗ ι ( H ) . Moreover, ε ( ι ( a )) = h a, i = ε ( a ) . Therefore ι ( H ) ⊂ U ◦ is a subcoalgebra isomorphic to U .Summarizing, we have shown that ι ( H ) satisfies the assumption of Theorem 4.5 so that itbecomes a left dual of U with multiplication and comultiplication given in (75) and (73), andthat ι : H −→ ι ( H ) yields an isomorphism of braided bialgebras. If H is a braided Hopfalgebra, then ι lifts to a braided Hopf algebra isomorphism with S ◦ ι = ι ◦ S . The oppositeversion is proven analogously. (cid:3) Given a braided bialgebra H , Proposition 3.1 allows us to construct a countable familyof braided bialgebras. The next proposition gives a description of the corresponding dualbialgebras obtained from a left (or right) dual of H . Proposition 4.8.
Let H be a braided bialgebra and let U be a left dual of H with respect tothe braidings Ψ UU and Ψ ◦ UH given in (57) and (59) , respectively. For n ∈ Z , U ( − n,n ) is a leftdual of H ( n, − n ) with respect to the braiding Ψ ◦ UH , and if Ψ UH is bijective, then U ( − n,n − isa left dual of H ( n − ,n ) with respect to the braiding Ψ UH . The analogous statements are truefor Hopf algebras, and for the opposite versions with Ψ ◦ UH and Ψ UH replaced by Ψ ◦ HU and Ψ HU , respectively.Proof. Let n ∈ Z . As h m − n ( f ⊗ g ) , a i (37) = h m ◦ Ψ − nUU ( f ⊗ g ) , a i (42) , (75) = hh f ⊗ g, Ψ − nHH ◦ Ψ − HH ◦ ∆( a ) ii (38) = hh f ⊗ g, Ψ − HH ◦ ∆ − n ( a ) ii and hh ∆ n ( f ) , a ⊗ b ii (38) = hh Ψ nUU ◦ ∆( f ) , a ⊗ b ii (42) , (73) = h f, m ◦ Ψ HH ◦ Ψ nHH ( a ⊗ b ) i (37) = h f, m n ◦ Ψ HH ( a ⊗ b ) i for f, g ∈ U and a, b ∈ H , it follows from Proposition 3.1 and Theorem 4.5 that U ( − n,n ) is aleft dual of H ( n, − n ) .In the case of H ( n − ,n ) , we have to replace the braiding Ψ HH on H by Ψ − HH and therefore Ψ ◦ UH by Ψ UH . A careful look at Proposition 4.8, Lemma 4.1 and Theorem 4.5 reveals thatthe assumption of Theorem 4.5 are still satisfied after these substitutions. Now the samecalculations, but with with ∆ n replaced by ∆ n − , show the result.The opposite versions are proven similarly. The statement about Hopf algebras followsfrom Proposition 3.1 and the definition of the antipode on the dual Hopf algebras. (cid:3) UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 23
5. M
ODULES AND COMODULES IN THE BRAIDED SETTING
A typical application of duality is to turn a comodule of a coalgebra into a module ofa dual algebra. Adding more structure, a braided comodule algebra of a braided bialgebrashould become a braided module algebra of a dual braided bialgebra. This will be discussedin Theorem 5.4. The dual version, i.e., turning a braided module into a braided comodule ofa dual coalgebra, will be presented in Theorem 5.6. Propositions 5.8 and 5.9 elaborate thesame idea on duals of (co)modules. Similarly to Lemma 4.1, we start by lifting braidings onvector spaces to dual spaces.As in the previous section, we will make frequent use of the Sweedler-type notation: Ψ V W ( v ⊗ w ) := w { } ⊗ v { } , Ψ − V W ( w ⊗ v ) := v { } ⊗ w { } , Ψ ◦ V W ( v ⊗ w ) := w { } ◦ ⊗ v { } ◦ , Ψ ◦− V W ( w ⊗ v ) := v { } ◦ ⊗ w { } ◦ , where v ∈ V , w ∈ W , and Ψ ◦ V W denotes a braiding that is constructed from the inverse of agiven one. In this notation, we have(77) w { } { } ⊗ v { } { } = w ⊗ v, v { }{ } ⊗ w { }{ } = v ⊗ w, and the same holds for Ψ ◦ V W . It turns out that the braidings of the type Ψ ◦ V W will be thecorrect ones for obtaining our desired result as it happened in Theorem 4.5. For this reason,we will give four versions of the auxiliary results on braidings, a left version, a right versionand the corresponding versions arising from the inverse braidings.
Lemma 5.1.
Let V be a right H -braided vector space. Consider the linear map (78) Ψ HV ′ : H ⊗ V ′ −→ ( V ⊗ H ′ ) ′ , Ψ HV ′ ( a ⊗ e )( v ⊗ f ) := hh e ⊗ f , Ψ V H ( v ⊗ a ) ii , for a ∈ H , v ∈ V , f ∈ H ′ and e ∈ V ′ . Let W ⊂ V ′ be a non-degenerate subspace such that (79) Ψ HW := Ψ HV ′ ↾ H ⊗ W : H ⊗ W −→ W ⊗ H ⊂ ( V ⊗ H ′ ) ′ is bijective. Then Ψ HW turns W into a left H -braided vector space.If H is a braided coalgebra and Ψ V H is compatible with the comultiplication, then Ψ HW isalso compatible with the comultiplication of H . If H is a braided (unital) algebra and Ψ V H is compatible with the multiplication, then Ψ HW is compatible with the multiplication of H .For a left H -braided vector space V , it is required that (80) Ψ W H := Ψ V ′ H ↾ W ⊗ H : W ⊗ H −→ H ⊗ W ⊂ ( H ′ ⊗ V ) ′ , is bijective, where (81) Ψ V ′ H : V ′ ⊗ H −→ ( H ′ ⊗ V ) ′ , Ψ V ′ H ( e ⊗ a )( f ⊗ v ) := hh f ⊗ e, Ψ HV ( a ⊗ v ) ii . In this case, Ψ W H turns W into a right H -braided vector space and the other implicationsremain the same under identical assumptions.The analogous statements hold for Ψ ◦ W H and Ψ ◦ HW if (82) Ψ ◦ V ′ H : V ′ ⊗ H −→ ( H ′ ⊗ V ) ′ , Ψ ◦ V ′ H ( e ⊗ a )( f ⊗ v ) := hh f ⊗ e, Ψ − V H ( a ⊗ v ) ii , yields a bijective map (83) Ψ ◦ W H := Ψ ◦ V ′ H ↾ W ⊗ H : W ⊗ H −→ H ⊗ W ⊂ ( H ′ ⊗ V ) ′ , and if (84) Ψ ◦ HV ′ : H ⊗ V ′ −→ ( V ⊗ H ′ ) ′ , Ψ ◦ HV ′ ( a ⊗ e )( v ⊗ f ) := hh e ⊗ f , Ψ − HV ( v ⊗ a ) ii yields a bijective map (85) Ψ ◦ HW := Ψ ◦ HV ′ ↾ H ⊗ W : H ⊗ W −→ W ⊗ H ⊂ ( V ⊗ H ′ ) ′ . Proof.
Let a, b ∈ H , f ∈ H ′ , v ∈ V and e ∈ W . In Sweedler-type notation, we can write(78) in the form e { } ( v ) f ( a { } ) = e ( v { } ) f ( a { } ) , which is equivalent to(86) e { } ( v ) a { } = e ( v { } ) a { } . The proof is now straightforward, nevertheless we give parts of the proof in order to showhow (86) enables us to move the action of the new braiding to the spaces where the givenbraiding is defined. For instance, e { } { } ( v ) b h i{ } ⊗ a h i{ } (86) = e ( v { } { } ) b h i{ } ⊗ a h i{ } (3) = e ( v { } { } ) b { } h i ⊗ a { } h i (86) = e { } { } ( v ) b { } h i ⊗ a { } h i proves (2). Furthermore, e { } { } ( v ) a (1) { } ⊗ a (2) { } (86) = e ( v { } { } ) a (1) { } ⊗ a (2) { } (11) = e ( v { } ) a { } (1) ⊗ a { } (2) (86) = e { } ( v ) a { } (1) ⊗ a { } (2) implies the first relation of (10). The second relation follows from ε ( a { } ) e { } ( v ) (86) = ε ( a { } ) e ( v { } ) (11) = ε ( a ) e ( v ) . Likewise, e { } ( v ) ( ab ) { } (86) = e ( v { } ) ( ab ) { } (7) = e ( v { } { } ) a { } b { } (86) = e { } { } ( v ) a { } b { } , which yields (5). If ∈ H , then (6) follows from (8) and (86) with a = 1 .The proof of the opposite version with the braiding Ψ W H uses similar arguments. Com-bining the obtained results with Lemmas 2.4 proves the statements for Ψ ◦ W H and Ψ ◦ HW . (cid:3) Given an H -braided vector space V , the previous lemma showed how to define braidingsbetween H and appropriate subspaces of V ′ . The next lemma fixes V and shows how toinduce braidings between V and appropriate subspaces U ⊂ H ′ . Moreover, if U inheritsa (co)multiplication from H , then the braidings between U and V inherit the compatibilityproperties from the corresponding braiding between H and V . Lemma 5.2.
Let V be a right H -braided vector space and consider the linear map (87) Ψ H ′ V : H ′ ⊗ V −→ ( V ′ ⊗ H ) ′ , Ψ H ′ V ( f ⊗ v )( e ⊗ a ) := hh e ⊗ f , Ψ V H ( v ⊗ a ) ii , for a ∈ H , v ∈ V , f ∈ H ′ and e ∈ V ′ . Assume that U ⊂ H ′ is a non-degenerate subspacesuch that Ψ UU given in (57) defines a braiding. If (88) Ψ UV := Ψ H ′ V ↾ U ⊗ V : U ⊗ V −→ V ⊗ U ⊂ ( V ′ ⊗ H ) ′ , is bijective, then Ψ UV turns V into a left U -braided vector space.If H is a braided coalgebra, Ψ V H is compatible with the comultiplication on H , and U satisfies the assumptions of Proposition 4.3, then Ψ UV is compatible with the multiplication ∗ from (68) on U . If H is a braided unital algebra, Ψ V H is compatible with the multiplicationon H , and U satisfies the assumptions of Proposition 4.4, then Ψ UV is compatible with thecomultiplication ∆ from (73) on U . UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 25
Assume that V is an algebra and Ψ V H is compatible with the multiplication on V . Then Ψ UV is compatible with the multiplication on V . If V is a coalgebra and Ψ V H is compatiblewith the comultiplication on V , then Ψ UV is also compatible with the comultiplication on V .Given a left H -braided vector space V such that Ψ V U := Ψ
V H ′ ↾ V ⊗ U : V ⊗ U −→ U ⊗ V ⊂ ( H ⊗ V ′ ) ′ is bijective, where (89) Ψ V H ′ : V ⊗ H ′ −→ ( H ⊗ V ′ ) ′ , Ψ V H ′ ( v ⊗ f )( a ⊗ e ) := hh f ⊗ e, Ψ HV ( a ⊗ v ) ii , the map Ψ V U defines a braiding that turns V into a right U -braided vector space and theopposite versions of above compatibility statements remain true.The analogous assertions hold if (90) Ψ ◦ V H ′ : V ⊗ H ′ −→ ( H ⊗ V ′ ) ′ , Ψ ◦ V H ′ ( v ⊗ f )( a ⊗ e ) := hh f ⊗ e, Ψ − V H ( a ⊗ v ) ii , yields a bijective map (91) Ψ ◦ V U := Ψ ◦ V H ′ ↾ V ⊗ U : V ⊗ U −→ U ⊗ V ⊂ ( H ⊗ V ′ ) ′ , and if (92) Ψ ◦ H ′ V : H ′ ⊗ V −→ ( V ′ ⊗ H ) ′ , Ψ ◦ H ′ V ( f ⊗ v )( e ⊗ a ) := hh e ⊗ f , Ψ − HV ( v ⊗ a ) ii , yields a bijective map (93) Ψ ◦ UV := Ψ ◦ H ′ V ↾ U ⊗ V : U ⊗ V −→ V ⊗ U ⊂ ( V ′ ⊗ H ) ′ . Proof.
Although the lemma is proven along the lines of the previous ones, we will state theproof in order to demonstrate where the duality between H and U is used. Let f, g ∈ U , u, v ∈ V , a, b ∈ H and e ∈ W . First note that e ( v { } ) f { } ( a ) (87) = e ( v { } ) f ( a { } ) implies(94) f { } ( a ) v { } = f ( a { } ) v { } and(95) g { } ( b ) f { } ( a ) v { } { } (94) = g { } ( b ) f ( a { } ) v { } { } (94) = g ( b { } ) f ( a { } ) v { } { } . Thus f h i{ } ( a ) g h i{ } ( b ) v { } { } (53) , (95) = f ( a { } h i ) g ( b { } h i ) v { } { } (3) = f ( a h i{ } ) g ( b h i{ } ) v { } { } (53) , (95) = f { } h i ( a ) g { } h i ( b ) v { } { } , which proves (2), so V becomes a left U -braided vector space with respect to the braidings Ψ UU and Ψ UV .To prove compatibility with the multiplication ∗ from (68) on U , we compute that ( f ∗ g ) { } ( a ) v { } (94) = ( f ∗ g )( a { } ) v { } (68) = f ( a { } (1) h i ) g ( a { } (2) h i ) v { } (3) , (11) = f ( a (1) h i{ } ) g ( a (2) h i{ } ) v { } { } (95) = f { } ( a (1) h i ) g { } ( a (2) h i ) v { } { } (68) = ( f { } ∗ g { } )( a ) v { } { } . This implies (5), and (6) follows from ε { } ( a ) v { } (94) = ε ( a { } ) v { } (11) = ε ( a ) v since ε yields theunit element in dual algebra U ⊂ H ′ . To prove the compatibility with the comultiplication of U , we proceed in the same manner. f { } (1) ( b ) f { } (2) ( a ) v { } (73) = f { } ( b h i a h i ) v { } (94) = f (( b h i a h i ) { } ) v { } (7) = f ( b h i{ } a h i{ } ) v { } { } (3) = f ( b { } h i a { } h i ) v { } { } (73) = f (1) ( b { } ) f (2) ( a { } ) v { } { } (95) = f (1) { } ( b ) f (2) { } ( a ) v { } { } shows the first relation of (10). The second relation of (10) follows from ε ( f { } ) v { } = f { } (1) v { } (94) = f (1 { } ) v { } (8) = f (1) v = ε ( f ) v. If V is an algebra and Ψ V H is compatible with the multiplication, then f { } ( a )( vu ) { } (94) = f ( a { } )( vu ) { } (5) = f ( a { } { } ) v { } u { } (94) = f { } { } ( a ) v { } u { } . Furthermore, if ∈ V , we have f { } ( a )1 { } (94) = f ( a { } )1 { } (6) = f ( a )1 . This and the previouscomputation show the compatibility of Ψ UV with the multiplication of V .Assume now that V is a coalgebra and that Ψ V H is compatible with the comultiplication.Then f { } { } ( a ) v (1) { } ⊗ v (2) { } (94) = f ( a { } { } ) v (1) { } ⊗ v (2) { } (10) = f ( a { } ) v { } (1) ⊗ v { } (2) (94) = f { } ( a ) v { } (1) ⊗ v { } (2) , and f { } ( a ) ε ( v { } ) (94) = f ( a { } ) ε ( v { } ) (10) = f ( a ) ε ( v ) . Hence Ψ UV is also compatible with thecomultiplication of V .The opposite versions are proven analogously, and the last part of the lemma follows fromthe first part by applying Lemma 2.4. (cid:3) Given a left H -braided vector space V and subspaces U ⊂ H ′ and W ⊂ V ′ satisfyingthe assumptions of Lemmas 5.1 and 5.2, there are two ways of constructing a braiding Ψ UW on U ⊗ W , either by the restriction of Ψ UV ′ from (78) or by the restriction of Ψ H ′ W from(87). The next lemma shows that both constructions coincide whenever one of them can berealized. Equally, we can use either Ψ ◦ V U or Ψ ◦ W H to construct a braiding on U ⊗ W . Inthis case, the resulting braiding will be denoted by Ψ • UW . Analogous results hold for right H -braided vector spaces. Lemma 5.3.
Let V be a left H -braided vector space. Assume that U ⊂ H ′ and W ⊂ V ′ satisfy the conditions of Lemmas 5.1 and 5.2 such that the braidings Ψ W H and Ψ V U are well-defined. If either Ψ H ′ W ↾ U ⊗ W : U ⊗ W → W ⊗ U or Ψ UV ′ ↾ U ⊗ W : U ⊗ W → W ⊗ U isbijective, then so is the other and Ψ H ′ W ↾ U ⊗ W = Ψ UV ′ ↾ U ⊗ W =: Ψ UW .Similarly, if Ψ ◦ HW and Ψ ◦ UV are well-defined and if either Ψ • V ′ U ↾ W ⊗ U or Ψ • W H ′ ↾ W ⊗ U yieldsa bijective map between W ⊗ U and U ⊗ W , then Ψ • V ′ U ↾ W ⊗ U = Ψ • W H ′ ↾ W ⊗ U =: Ψ • W U , where Ψ • V ′ U : V ′ ⊗ U → ( H ⊗ V ) ′ , Ψ • V ′ U ( e ⊗ f )( a ⊗ v ) := hh a ⊗ e, Ψ ◦ UV ( f ⊗ v ) ii , (96) Ψ • W H ′ : W ⊗ H ′ → ( H ⊗ V ) ′ , Ψ • W H ′ ( e ⊗ f )( a ⊗ v ) := hh f ⊗ v, Ψ ◦ HW ( a ⊗ e ) ii . (97) If V is a right H -braided vector space, then the analogous statements holds for Ψ W U and Ψ • UW under homologous assumptions, where (98) Ψ • UW ( f ⊗ e )( a ⊗ v ) = hh e ⊗ a, Ψ ◦ V U ( v ⊗ f ) ii = hh v ⊗ f , Ψ ◦ W H ( e ⊗ a ) ii . UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 27
Proof.
The claim Ψ H ′ W ↾ U ⊗ W = Ψ UV ′ ↾ U ⊗ W follows from the equation Ψ UV ′ ( f ⊗ e )( v ⊗ a ) (78) = e ( v { } ) f { } ( a ) (89) = e ( v { } ) f ( a { } ) (81) = e { } ( v ) f ( a { } ) (87) = Ψ H ′ W ( f ⊗ e )( v ⊗ a ) for a ∈ H , v ∈ V , f ∈ U and e ∈ W . Similarly, Ψ • V ′ U ( e ⊗ f )( a ⊗ v ) (96) = e ( v { } ◦ ) f { } ◦ ( a ) (92) = e ( v { } ) f ( a { } ) (84) = e { } ◦ ( v ) f ( a { } ◦ ) (97) = Ψ • W H ′ ( e ⊗ f )( a ⊗ v ) implies that Ψ • V ′ U ↾ W ⊗ U = Ψ • W H ′ ↾ W ⊗ U . The opposite versions are proven analogously. (cid:3) The next theorem shows how to transform a comodule V of a coalgebra H into a moduleof a dual algebra of H . Note that we will use again a braiding that is constructed from theinverse of the given one as it happened in Theorem 4.5. The same observation can be madefor succeeding results. Theorem 5.4.
Let H be a braided coalgebra and V a braided right H -comodule with co-action ρ R : V → V ⊗ H . Let U ⊂ H ′ be a non-degenerate (unital) subalgebra with productgiven by (75) such that Ψ UU and Ψ ◦ UV introduced in (57) and (93) , respectively, are bijective.Then V becomes a braided left U -module with respect to the braiding Ψ ◦ UV and the left action ν L : U ⊗ V → V given by (99) ν L ( f ⊗ v ) := h f, v (1) { } i v (0) { } . If H is a braided bialgebra, V is a braided right H -comodule algebra, and U is a left dualof H as in Theorem 4.5, then the action ν L turns V into a braided left U -module algebra.In case V is a braided left H -comodule, the analogous statements for the opposite versionshold under homologous assumptions for the right action ν R : V ⊗ U → V given by (100) ν R ( v ⊗ f ) := h f, v ( − { } i v (0) { } , and with respect to the braiding Ψ ◦ V U .Proof.
From Proposition 4.3 and Lemma 5.2, we conclude that U is a braided algebra withrespect to braiding Ψ UU and V is a left U -braided vector space such that the braiding Ψ ◦ UV is compatible with the multiplication of U . To show that the action ν L equips V with thestructure of a left braided U -module, we need to prove (19) and (20).Let f, g ∈ U and v ∈ V . Using the notation (25), we compute that f ⊲ ( g ⊲ v ) (99) = g ( v (1) { } ) f ( v (0) { } (1) { } ) v (0) { } (0) { } (28) , (31) = g ( v (2) h i { } ) f ( v (1) h i { } ) v (0) { } { } (3) = g ( v (2) { } h i ) f ( v (1) { } h i ) v (0) { } { } (11) = g ( v (1) { } (2) h i ) f ( v (1) { } (1) h i ) v (0) { } (75) = f ∗ g ( v (1) { } ) v (0) { } (99) = ( f ∗ g ) ⊲ v. This yields (19).Note that, since V ′ separates the points of V , Equations (92) and (93) give(101) f { } ◦ ( a ) v { } ◦ = f ( a { } ) v { } , f ∈ U, v ∈ V, a ∈ H. Therefore, for f, g ∈ U , v ∈ V and a ∈ H , f h i{ } ◦ ( a )( g h i ⊲ v { } ◦ ) (99) = f h i{ } ◦ ( a ) g h i ( v { } ◦ (1) { } ) v { } ◦ (0) { } (101) = f h i ( a { } ) g h i ( v { } (1) { } ) v { } (0) { } (31) = f h i ( a h i { } ) g h i ( v (1) h i { } ) v (0) { } { } (3) = f h i ( a { } h i ) g h i ( v (1) { } h i ) v (0) { } { } (49) , (51) = f ( a { } ) g ( v (1) { } ) v (0) { } { } (99) = f ( a { } ) ( g ⊲ v ) { } (101) = f { } ◦ ( a ) ( g ⊲ v ) { } ◦ , which implies (20).Now assume that H is a braided bialgebra, V is a braided right H -comodule algebra, and U is a left dual of H . Then, for u, v ∈ V and f ∈ U , ( f (1) ⊲ u { } ◦ )( f (2) { } ◦ ⊲ v ) (99) = f (1) ( u { } ◦ (1) { } ) f (2) { } ◦ ( v (1) { } ) u { } ◦ (0) { } v (0) { } (101) = f (1) ( u { } (1) { } ) f (2) ( v (1) { } { } ) u { } (0) { } v (0) { } (31) = f (1) ( u (1) h i { } ) f (2) ( v (1) { } h i { } ) u (0) { } { } v (0) { } (3) = f (1) ( u (1) { } h i ) f (2) ( v (1) { } { } h i ) u (0) { } { } v (0) { } (49) , (73) = f ( u (1) { } v (1) { } { } ) u (0) { } { } v (0) { } (7) = f (cid:0) ( u (1) v (1) { } ) { } (cid:1) u (0) { } v (0) { } (77) = f (cid:0) ( u (1) { }{ } v (1) { } ) { } (cid:1) u (0) { } v (0) { }{ } { } (7) = f (cid:0) ( u (1) { } v (1) ) { } { } (cid:1) u (0) { } v (0) { }{ } (5) = f (cid:0) ( u (1) { } v (1) ) { } (cid:1) ( u (0) v (0) { } ) { } (34) = f (cid:0) ( uv ) (1) { } (cid:1) ( uv ) (0) { } (99) = f ⊲ ( uv ) . This proves (32). Furthermore, if ∈ V , then f ⊲ (35) = f (1 { } ) 1 { } (6) = f (1) 1 = ε ( f ) 1 . Since,by Lemma 5.2, Ψ ◦ UV is compatible with the comultiplication of U and with the multiplicationsof U and V , we conclude that V is a braided left U -module algebra.The opposite versions are shown analogously. (cid:3) Let H be a braided bialgebra and U a left dual of H . Since U ⊂ H ′ is an algebra, thereis a natural left (resp. right) U -action on H given by right (resp. left) multiplication on U .On the other hand, the coproduct on H equips H trivially with the structure of a right (resp.left) H -comodule so that we may consider the left (resp. right) U -action on H described inTheorem 5.4. The next corollary shows that these actions coincide and turn H into a left(resp. right) U -module algebra. Similar results hold for U and H interchanged. Corollary 5.5.
Let H be a braided bialgebra and let U be a left dual of H as in Theorem 4.5.Then the natural left U -action ⊲ : U ⊗ H → H defined by f ( g ⊲ a ) := ( f g )( a ) , a ∈ H , f, g ∈ U , satisfies (102) g ⊲ a = h g, a (2) h i i a (1) h i and turns H into a left U -module algebra. The natural right U -action ⊳ : H ⊗ U → H defined by g ( a ⊳ f ) := ( f g )( a ) satisfies (103) a ⊳ f = h f, a (1) h i i a (2) h i and turns H into a right U -module algebra.The same formulas with a ∈ U , f, g ∈ H hold for the natural left (resp. right) H -actionon U and turn U into a left (resp. right) H -module algebra. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 29
Proof.
Setting V := H in Theorem 5.4 shows that the right-hand side in (102) (resp. (103))defines a left (resp. right) U -action on H such that H becomes a left (resp. right) U -modulealgebra. The equalities in (102) and (103) follow from (64) for the braiding Ψ ◦ UH by applying(54). The proof for the H -actions on U is identical but uses (65) instead of (64). (cid:3) Theorem 5.4 shows how to turn a comodule of a braided coalgebra into a module of anappropiate dual algebra. The dual construction corresponds to turning a module of a braidedalgebra into a comodule of a dual coalgebra. This will be done in the next theorem. Similarto Proposition 4.4, we will need an additional condition to ensure that the coaction belongsto the correct (algebraic) tensor product.
Theorem 5.6.
Let H be a braided unital algebra and V a braided left H -module with action ν L : H ⊗ V → V denoted by ν L ( a ⊗ v ) := a ⊲ v . Let U be a non-degenerate subcoalgebra of H ◦ as defined in Proposition 4.4 such that Ψ UU and Ψ ◦ UV given in (57) and (93) , respectively,are bijective. Consider the linear map (104) ρ R : V −→ ( V ′ ⊗ H ) ′ , ρ R ( v )( e ⊗ a ) := e ( a { } ⊲ v { } ) , where v ∈ V , a ∈ H and e ∈ V ′ . If (105) ρ R : V −→ V ⊗ U ⊂ ( V ′ ⊗ H ) ′ , then ρ R yields a right U -coaction on V such that V becomes a braided right U -comodulewith respect to the braiding Ψ ◦ UV and the coproduct ∆ − := Ψ − UU ◦ ∆ on U , where ∆ denotesthe coproduct introduced in (73) .If H is a braided bialgebra, V is a braided left H -module algebra, and U is a left dualof H as in Theorem 4.5, then the coaction ρ R turns V into a braided right U (2 , − -comodulealgebra.In case V is a braided right H -module, then the analogous statements hold under ho-mologous assumptions for the opposite versions with respect to the left coaction determinedby (106) ρ L : V −→ U ⊗ V ⊂ ( H ⊗ V ′ ) ′ , ρ L ( v )( a ⊗ e ) := e ( v { } ⊳ a { } ) , and again for the braided coalgebra ( U, ∆ − , ε ) and the braided bialgebra U (2 , − .Proof. As shown in Proposition 4.4, the braiding Ψ UU equips ( U, ∆ , ε ) with the structure of abraided coalgebra. From Lemma 5.2, we know that Ψ ◦ UV turns V into a left U -braided vectorspace such that the braiding is compatible with the comultiplication of U and, if defined, withthe multiplications of U and V . Furthermore, by Proposition 3.1, the compatibility conditionsare also fulfilled with respect to the modified coproduct ∆ − = Ψ − UU ◦ ∆ and the modifiedproduct m = m ◦ Ψ UU .Throughout this proof, let u, v ∈ V , a, b ∈ H and f, g ∈ U . To show that ρ R turns V into abraided right U -module, it remains to prove (28) and (29). First note that, since V ′ separatesthe points of V , (104) is equivalent to(107) v (0) v (1) ( a ) = a { } ⊲ v { } . Moreover, to distinguish between ∆ − and coproduct ∆ on U defined in Proposition 4.4, wewill employ the Sweedler notation ∆ − ( f ) := f (1) ⊗ f (2) . Then it follows from (42), (57)and (73) that(108) f (1) ( b ) f (2) ( a ) = f ( b h i a h i ) . Using Lemma 2.4 for the relations concerning the inverse braiding, we obtain ( v (0) ) (0) v (1) ( a ) ( v (0) ) (1) ( b ) (107) = b { } ⊲ ( a { } ⊲ v { } ) { } (23) = b { } h i ⊲ ( a { } h i ⊲ v { } { } ) (2) = b h i { } ⊲ ( a h i { } ⊲ v { } { } ) (19) = ( b h i { } a h i { } ) ⊲ v { } { } (5) = ( b h i a h i ) { } ⊲ v { } (107) = v (0) v (1) ( b h i a h i ) (108) = v (0) ( v (1) ) (1) ( b ) ( v (1) ) (2) ( a ) . Therefore v (0) ⊗ ( v (1) ) (1) ⊗ ( v (1) ) (2) = ( v (0) ) (0) ⊗ ( v (0) ) (1) ⊗ v (1) =: v (0) ⊗ v (1) ⊗ v (2) , which shows the first relation of (28). The second relation of (28) follows from the definitionof ε in Proposition 4.4 and v (0) ε ( v (1) ) = v (0) v (1) (1) (107) = 1 { } ⊲ v { } (6) = 1 ⊲ v (19) = v. Furthermore, v (0) { } ◦ v (1) h i ( b ) f { } ◦ h i ( a ) (53) = v (0) { } ◦ v (1) ( b h i ) f { } ◦ ( a h i ) (101) = v (0) { } v (1) ( b h i ) f ( a h i{ } ) (107) = ( b h i{ } ⊲ v { } ) { } f ( a h i{ } ) (23) = ( b h i{ } h i ⊲ v { } { } ) f ( a h i{ } h i ) (2) , (49) = ( b { } ⊲ v { } { } ) f ( a { } ) (107) = v { } (0) v { } (1) ( b ) f ( a { } ) (101) = v { } ◦ (0) v { } ◦ (1) ( b ) f { } ◦ ( a ) , which proves (29). Hence ρ R turns V into a braided right U -module.Now let H be a braided bialgebra, V a braided right H -module algebra, and U a leftdual of H as in Theorem 4.5. The compatibility of the braiding Ψ ◦ UV with the (modified)comultiplication on U and the (modified) multiplications on U and V has been discussedin the beginning of the proof. To show that V becomes a braided right U (2 , − -comodulealgebra, we need to verify (34).First note that m ( f ⊗ g )( a ) (37) , (75) = hh Ψ UU ( f ⊗ g ) , Ψ − HH ◦ ∆( a ) ii (42) , (57) = hh f ⊗ g, Ψ HH ◦ ∆( a ) ii (41) = f ( a (1) h i ) g ( a (2) h i ) (68) = f ∗ g ( a ) . (109)Starting with the right hand side of (34), we compute that u (0) v (0) { } ◦ ( u (1) { } ◦ ∗ v (1) )( a ) (109) = u (0) v (0) { } ◦ u (1) { } ◦ ( a (1) h i ) v (1) ( a (2) h i ) (101) = u (0) v (0) { } u (1) ( a (1) h i{ } ) v (1) ( a (2) h i ) (107) = u (0) (cid:0) a (2) h i{ } ⊲ v { } (cid:1) { } u (1) ( a (1) h i{ } ) (23) = u (0) (cid:0) a (2) h i{ } h i ⊲ v { } { } (cid:1) u (1) ( a (1) h i{ } h i ) (2) , (49) = u (0) (cid:0) a (2) { } ⊲ v { } { } (cid:1) u (1) ( a (1) { } ) (10) = u (0) (cid:0) a { } (2) ⊲ v { } (cid:1) u (1) ( a { } (1) ) (107) = ( a { } (1) { } ⊲ u { } )( a { } (2) ⊲ v { } ) (10) = ( a { } { } (1) ⊲ u { } { } )( a { } { } (2) { } ⊲ v { } ) (32) = a { } { } ⊲ ( u { } v { } ) (7) = a { } ⊲ ( u v ) { } (107) = ( u v ) (0) ( u v ) (1) ( a ) , which implies (34). Finally, if ∈ V , then (0) (1) ( a ) (107) = a { } ⊲ { } (8) = a ⊲ (33) = 1 ε ( a ) ,from which we conclude that ρ R (1) = 1 ⊗ . This finishes the proof that V is a braided right U (2 , − -comodule algebra.The opposite versions are proven analogously. (cid:3) UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 31
Taking V := H in the previous theorem and the multiplication of H as left action, the rightcoaction ρ R : H → H ⊗ U is determined by ρ R ( a )( f ⊗ b ) (104) = f ( b h i a h i ) (73) = hh ∆ − ( f ) , a ⊗ b ii , which is equivalent to a (0) ( f ) a (1) = f (2) ( a ) f (1) , ∆ − ( f ) := f (1) ⊗ f (2) . However, if we consider H as a braided algebra with respect to the inverse braiding Ψ − HH ,then ρ R ( a )( f ⊗ b ) = f ( b h i a h i ) and thus f ( a (0) ) a (1) = f (2) ( a ) f (1) , ∆( f ) := f (1) ⊗ f (2) . The corresponding left U -coaction ρ L ( a )( b ⊗ f ) = f ( a h i b h i ) satisfies f ( a (0) ) a ( − = f (1) ( a ) f (2) . These observations may be viewed as the dual version of Corollary 5.5.Note that the coproduct of the dual coalgebra ( U, ∆ , ε ) had to be changed to ∆ − inTheorem 5.6. Therefore, combining repeatedly Theorems 5.4 and 5.6 may turn V into a(co)module for a whole family of (co)algebras, each time with respect to a potentially differ-ent (co)action. The starting point for this observation is the next corollary. Corollary 5.7. (i) Let ( H, ∆ , ε ) be a braided coalgebra and V a braided right H -comodulewith coaction ρ R : V → V ⊗ H . Let U ⊂ H ′ be a non-degenerate unital subalgebrawith product given by (75) such that Ψ UU , Ψ ◦ UH and Ψ ◦ UV introduced in (57) , (59) and (93) ,respectively, are bijective. Assume that (110) Ψ ◦◦ HV : H ⊗ V −→ ( V ′ ⊗ U ) ′ , Ψ ◦◦ HV ( a ⊗ v )( e ⊗ f ) := hh e ⊗ a, Ψ ◦− UV ( v ⊗ f ) ii , yields a bijection Ψ ◦◦ HV : H ⊗ V → V ⊗ H ⊂ ( V ′ ⊗ U ) ′ . If the map (111) ρ R : V → ( V ′ ⊗ U ) ′ , ρ R ( v )( e ⊗ f ) := e ( v (0) ) f { } ◦ ( v (1) { } ◦ ) satisfies ρ R : V → V ⊗ H ⊂ ( V ′ ⊗ U ) ′ , then it defines a right H -coaction that turns V into a braided right H -comodule with respect to the braiding Ψ ◦◦ HV and the coproduct ∆ − := Ψ − HH ◦ ∆ on H .If H is a braided bialgebra, V is a braided right H -comodule algebra, and U is a left dualof H as in Theorem 4.5, then the right coaction defined in (111) turns V into a braided right H (2 , − -comodule algebra.(ii) Let ( H, m ) be a braided unital algebra and V a braided left H -module with leftaction ν L : H ⊗ V → V . Let U be a non-degenerate subcoalgebra of H ◦ as defined inProposition 4.4 such that Ψ UU and Ψ ◦ UV given in (57) and (93) , respectively, are bijective,and suppose that ρ R defined in (104) satisfies (105) . Assume that the map Ψ ◦◦ HV introducedin (110) yields a bijection Ψ ◦◦ HV : H ⊗ V → V ⊗ H , and that there exists a non-degeneratesubspace W ⊂ V ′ such that (112) Ψ ◦◦ W H : W ⊗ H −→ ( U ⊗ V ) ′ , Ψ ◦◦ W H ( e ⊗ a )( f ⊗ v ) := hh f ⊗ e, Ψ ◦◦ HV ( a ⊗ v ) ii , yields a bijection Ψ ◦◦ W H : W ⊗ H → H ⊗ W ⊂ ( U ⊗ V ) ′ . Then the map ν L : H ⊗ V → W ′ , (113) ν L ( a ⊗ v )( e ) := h· , ·i ◦ ( ν L ⊗ id) ◦ (Ψ − HV ⊗ id) ◦ (id ⊗ Ψ ◦◦ W H ) ◦ ( v ⊗ e ⊗ a ) , turns V into a braided left H -module with respect to the braiding Ψ ◦◦ HV and the multiplication m − := m ◦ Ψ − HH on H . If H is a braided bialgebra, V is a braided left H -module algebra, and U is a left dualof H as in Theorem 4.5, then the left action defined in (113) turns V into a braided left H ( − , -module algebra.Analogous statements hold for the opposite versions.Proof. ( i ) First note that the braided coalgebra ( H, ∆ , ε ) and the subalgebra U ⊂ H ′ in (i) satisfy the assumptions of Theorem 5.4. Hence the left action (99) turns V into a braidedleft U -module with respect to the braiding Ψ ◦ UV . From (76), it follows that H ∼ = ι ( H ) ⊂ U ◦ is a non-degenerate subcoalgebra. Furthermore, Ψ ◦◦ HV corresponds to Ψ ◦ UV in Theorem 5.6,where H and U swap the roles in the present situation. Applying Theorem 5.6, we concludethat the H -coaction defined in (104) turns V into a braided right H -comodule with respect tothe braiding Ψ ◦◦ HV and the coproduct ∆ − = Ψ − HH ◦ ∆ . But before we can apply Theorem 5.6,we need prove that the coaction resulting from this construction coincides with (111) so that(105) is fulfilled by assumption.Let v ∈ V . To distinguish the new coaction ρ R from the original one, we use the Sweedlernotation ρ R ( v ) := v (0) ⊗ v (1) . Evaluating ρ R ( v ) ∈ V ⊗ H on e ⊗ f ∈ W ⊗ U yields e ( v (0) ) f ( v (1) ) (104) = e ( f { } ◦ ⊲ v { } ◦ ) (99) = f { } ◦ ( v { } ◦ (1) { } ) e ( v { } ◦ (0) { } ) (92) = f { } ◦ { } ◦ ( v { } ◦ (1) ) e ( v { } ◦ (0) { } ◦ ) , which is equivalent to(114) v (0) f ( v (1) ) = f { } ◦ { } ◦ ( v { } ◦ (1) ) v { } ◦ (0) { } ◦ . Furthermore, f { } ◦ ( a ) v { } ◦ (0) ⊗ v { } ◦ (1) (101) = f ( a { } ) v { } (0) ⊗ v { } (1) (31) = f ( a h i { } ) v (0) { } ⊗ v (1) h i (56) , (101) = f { } ◦ { } ◦ ( a ) v (0) { } ◦ ⊗ v (1) { } ◦ gives ( ρ R ⊗ id) ◦ Ψ ◦ UV = (id ⊗ Ψ ◦ UH ) ◦ (Ψ ◦ UV ⊗ id) ◦ (id ⊗ ρ R ) . Applying on both sides the corresponding inverse braidings, we obtain(115) (id ⊗ Ψ ◦− UH ) ◦ ( ρ R ⊗ id) = (Ψ ◦ UV ⊗ id) ◦ (id ⊗ ρ R ) ◦ Ψ ◦− UV . Inserting (115) into (114) yields v (0) f ( v (1) ) (115) , (114) = f { } ◦ ( v (1) { } ◦ ) v (0) . This proves (111) for the given right H -coaction ρ R ( v ) = v (0) ⊗ v (1) on V and with thebraiding Ψ ◦ − UH ( a ⊗ f ) = f { } ◦ ⊗ a { } ◦ on H ⊗ U .Assume now that H is a braided bialgebra and V is a braided right H -comodule algebra.Then, by Theorem 5.4, V becomes a left U -module algebra. In Proposition 4.7, it has beenshown that H ∼ = ι ( H ) is a left dual of U . Applying Theorem 5.6 with the roles of U and H reversed shows that the new coaction ρ R turns V into a braided right H (2 , − -comodulealgebra. ( ii ) Consider a braided unital algebra ( H, m ) , a braided left H -module V and a non-degenerate subcoalgebra U ⊂ H ◦ as described in (ii) . From Theorem 5.6, we conclude that V becomes a braided right U -comodule with respect to the coproduct ∆ − := Ψ − UU ◦ ∆ on U , the right coaction defined in (104), and the braiding Ψ ◦ UV . As in the proof of (i) , Ψ ◦◦ HV corresponds to the braiding Ψ ◦ UV in Theorem 5.4 with the roles of H and U reversed. UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 33
Furthermore, the multiplication on H ∼ = ι ( H ) ⊂ U ′ corresponding to the coproduct ∆ − on U in (75) is given by m − = m ◦ Ψ − HH . Indeed, for all a, b ∈ H and f ∈ U , we have h m ( ι ( a ) ⊗ ι ( b )) , f i (75) = hh a ⊗ b, Ψ − UU ◦ ∆ − ( f ) ii (38) , (42) = hh Ψ − HH ( a ⊗ b ) , ∆( f ) ii (73) = h m ◦ Ψ − HH ( a ⊗ b ) , f i (37) = h m − ( a ⊗ b ) , f i . Applying now Theorem 5.4 shows that V becomes a braided left H -module with respect tothe left H -action defined in (99), the multiplication m − := m ◦ Ψ − HH on H , and the braiding Ψ ◦◦ HV .Let ν L denote the new left H -action on V . To show that ν L is given by (113), we use theSweedler-type notation Ψ ◦◦ HV ( a ⊗ v ) := v { } ◦◦ ⊗ a { } ◦◦ , Ψ ◦◦ W H ( e ⊗ a ) := a { } ◦◦ ⊗ e { } ◦◦ , for a ∈ H , v ∈ V and e ∈ W , and compute that e ( ν L ( a ⊗ v )) (99) = e ( v (0) { } ◦ ) v (1) { } ◦ ( a ) (110) = e ( v (0) { } ◦◦ ) v (1) ( a { } ◦◦ ) (112) = e { } ◦◦ ( v (0) ) v (1) ( a { } ◦◦ ) (107) = e { } ◦◦ ( a { } ◦◦ { } ⊲ v { } ) . (116)This shows that ν L is given by (113). In particular, we have ν L : H ⊗ V → V ⊂ W ′ .If H is a braided bialgebra, V is a braided left H -module algebra, and U is a left dualof H as in Theorem 4.5, then we conclude from Theorem 5.6 that V becomes a braidedright U (2 , − -comodule algebra. Since H ( − , is a left dual of U (2 , − by Proposition 4.8, wededuce from Theorem 5.4 that the left action ν L turns V into a braided left H ( − , -modulealgebra.The opposite versions versions are proven analogously. (cid:3) The results of the last corollary may be seen as an induction step. Starting with an H -(co)module V , Corollary 5.7 produces another H -(co)action on V but for the modified co-product ∆ − or product m − . If the assumptions of the corollary are again satisfied, we canrepeat the process, and so on, obtaining thus for all n ∈ N an H -(co)action on V for the co-product ∆ − n or the product m − n . If H is a braided bialgebra and V a braided (co)modulealgebra, then we get an coaction of the braided bialgebra H (2 n, − n ) or an action of H ( − n, n ) .Theorem 5.4 shows how to turn a braided comodule of a braided coalgebra into braidedmodule of the dual algebra. However, it is more natural to dualize a coaction in such a waythat a dual space of the comodule becomes a module of the dual algebra. This will be donein the next proposition. Unlike Theorem 5.4, a braided comodule algebra will not dualize tobraided module algebra. The correct way would be to dualize it to a braided module coalgebrabut, as mentioned in Remark 2.3, we do not discuss these structures here. Proposition 5.8.
Let H be a braided coalgebra and V a braided right H -comodule withcoaction ρ R : V → V ⊗ H . Let U ⊂ H ′ and W ⊂ V ′ satisfy the assumptions of Lemmas4.1, 5.1, 5.2 and 5.3 which guarantee that the braidings Ψ UU , Ψ ◦ HW , Ψ ◦ UV and Ψ • W U are well-defined. Assume that U ⊂ H ′ is a (unital) subalgebra with respect to the product defined in (75) . Consider (117) ν R : W ⊗ U −→ V ′ , ν R ( e ⊗ f )( v ) := f ( v (1) { } ) e ( v (0) { } ) If ν R : W ⊗ U → W ⊂ V ′ , then it defines a right U -action on W such that W becomes abraided right U -module with respect to the braidings Ψ − UU and Ψ • W U . For a braided left H -comodule V , it is required that U and W satisfy the assumptions ofLemmas 4.1, 5.1, 5.2 and 5.3 which guarantee that the braidings Ψ UU , Ψ ◦ W H , Ψ ◦ V U and Ψ • UW are well-defined. If (118) ν L : U ⊗ W −→ V ′ , ν L ( f ⊗ e )( v ) := f ( v ( − { } ) e ( v (0) { } ) yields a map ν L : U ⊗ W → W ⊂ V ′ , then it defines a left U -action on W such that W becomes a braided left U -module with respect to the braidings Ψ − UU and Ψ • UW .Proof. As customary, we denote the map defined in (117) by ν R ( e ⊗ f ) := e ⊳ f and the leftaction given in (99) by ν L ( f ⊗ v ) := f ⊲ v . Since ( e ⊳ f )( v ) (117) = f ( v (1) { } ) e ( v (0) { } ) (99) = e ( f ⊲ v ) , f ∈ U, v ∈ V, e ∈ V ′ , it follows immediately from Theorem 5.4 that (117) defines a right U -action on V ′ . Assumingthat ν R : W ⊗ U → W , we need to prove the compatibility with the braiding Ψ • W U .Although the proof of (22) goes along the lines of previous ones, we present the com-putations in order to show where all the listed braidings and Lemma 5.3 are needed. Letnow f, g ∈ U , a ∈ H , e ∈ W and v ∈ V . Employing the Sweedler-type notation Ψ • W U ( e ⊗ f ) := f { } • ⊗ e { } • , we have f { } • ( a ) ( e ⊳ g ) { } • ( v ) (96) = f { } ◦ ( a ) ( e ⊳ g )( v { } ◦ ) (92) = f ( a { } ) ( e ⊳ g )( v { } ) (117) = f ( a { } ) g ( v { } (1) { } ) e ( v { } (0) { } ) (2) , (31) = f ( a { } h i ) g ( v (1) { } h i ) e ( v (0) { } { } ) (54) , (84) = f h i ( a { } ◦ ) g h i ( v (1) { } ) e { } ◦ ( v (0) { } ) (97) = f h i { } • ( a ) g h i ( v (1) { } ) e { } • ( v (0) { } ) (117) = f h i { } • ( a ) ( e { } • ⊳ g h i )( v ) , which implies (22) for the braidings Ψ • W U and Ψ − UU . Furthermore, we know from Lemma 2.4that U is a braided algebra with respect to the braiding Ψ − UU , so this finishes the proof the firstpart. The second part is proven similarly. (cid:3) The analog of dualizing a coaction to an action as in the last proposition consists in dualiz-ing an action to obtain a coaction of the dual coalgebra on the dual space. This is the purposeof our last proposition.
Proposition 5.9.
Let H be a braided unital algebra and V a braided left H -module withaction ν L : H ⊗ V → V . Let U ⊂ H ′ and W ⊂ V ′ satisfy the assumptions of Lemmas 4.1,5.1, 5.2 and 5.3 which guarantee that the braidings Ψ UU , Ψ ◦ HW , Ψ ◦ UV and Ψ • W U are well-defined. Assume that U ⊂ H ◦ is a subcoalgebra as in Proposition 4.4 but with respect to thebraiding Ψ − HH on H , i.e., ∆ ◦ ( f ) ∈ U ⊗ U for all f ∈ U , where (119) hh ∆ ◦ ( f ) , a ⊗ b ii := h f, m ◦ Ψ − HH ( a ⊗ b ) i = f ( b h i a h i ) , a, b ∈ H. Consider the linear map (120) ρ L : W −→ ( H ⊗ V ) ′ , ρ L ( e )( a ⊗ v ) := e ( a { } ⊲ v { } ) . If ρ L : W → U ⊗ W ⊂ ( H ⊗ V ) ′ , then it defines a left U -coaction on W such that W becomesa braided left U -comodule with respect to the coalgebra ( U, ∆ ◦ , ε ) , the braiding Ψ − UU on U and the braiding Ψ • W U between W and U . UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 35
For a braided right H -comodule V , it is required that U and W satisfy the assumptions ofLemmas 4.1, 5.1, 5.2 and 5.3 which guarantee that the braidings Ψ UU , Ψ ◦ W H , Ψ ◦ V U and Ψ • UW are well-defined. If the linear map (121) ρ R : W −→ ( V ⊗ H ) ′ , ρ R ( e )( v ⊗ a ) := e ( v { } ⊳ a { } ) , fulfills ρ R : W → W ⊗ U ⊂ ( V ⊗ H ) ′ , then it defines a right U -action on W such that W becomes a braided right U -comodule with respect to the coalgebra ( U, ∆ ◦ , ε ) , the braiding Ψ − UU on U and the braiding Ψ • UW between U and W .Proof. As in the proof the previous proposition, we prove (26) and (27) in order to show thatthe correct braidings and the correct coproduct are used. Let f ∈ U , a, b ∈ H , e ∈ W and v ∈ V . Then e ( − ( a ) ( e (0) ) ( − ( b ) ( e (0) ) (0) ( v ) (120) = e ( a { } ⊲ ( b { } ⊲ v { } ) { } ) (2) , (23) = e ( a h i { } ⊲ ( b h i { } ⊲ v { } { } )) (19) = e (( a h i { } b h i { } ) ⊲ v { } { } ) (5) = e (( a h i b h i ) { } ⊲ v { } ) (120) = e ( − ( a h i b h i ) e (0) ( v ) (119) = ( e ( − ) (1) ( a ) ( e ( − ) (2) ( b ) e (0) ( v ) , so that the first relation of (26) holds for the coproduct given in (119). Moreover, ε ( e ( − ) e (0) ( v ) = e ( − (1) e (0) ( v ) (120) = e (1 { } ⊲ v { } ) (6) , (19) = e ( v ) . Hence ρ L defines a left U -coaction for the coalgebra ( U, ∆ ◦ , ε ) .Recall from (54) that Ψ − UU is the braiding on U which corresponds to Ψ − HH on H accordingto (42). Now, f { } • h i ( a ) e ( − h i ( b ) e (0) { } • ( v ) (54) = f { } • ( a h i ) e ( − ( b h i ) e (0) { } • ( v ) (92) , (96) = f ( a h i { } ) e ( − ( b h i ) e (0) ( v { } ) (120) = f ( a h i { } ) e ( b h i { } ⊲ v { } { } ) (2) , (23) = f ( a { } ) e (( b { } ⊲ v { } ) { } ) (92) , (96) = f { } • ( a ) e { } • ( b { } ⊲ v { } ) (120) = f { } • ( a ) e { } • ( − ( b ) e { } • (0) ( v ) , which proves (27) for the braidings Ψ − UU on U and Ψ • W U between W and U .The proof of the opposite version is similar. (cid:3) Note that we used all lemmas of this section in the proofs of Propositions 5.8 and 5.9. Asa closing remark, let us point out that the relevance of the inverse braidings is undeniablethroughout this paper, not only for turning right braided vector spaces into left braided vectorspaces and vice versa, but also in the definitions of the braidings of the type Ψ ◦ XY , in thedefinitions of actions and coactions in Theorems 5.4 and 5.6, and in our final propositions.6. E XAMPLES
Example 6.1.
Finite-dimensional braided (co)algebras, bialgebras and Hopf algebras andtheir finite-dimensional (co)modules yield examples of all presented structures [15, 19]. Inthe finite-dimensional situation, the non-degenerated dual spaces are obviously unique. Ex-plicit formulas can be deduced by using the coevaluation map(122) coev H : K −→ H ⊗ H ′ , coev H (1) = n P j =1 e j ⊗ e j , where { e , . . . , e n } ⊂ H is a linear basis and { e , . . . , e n } ⊂ H ′ its dual basis. Under theidentification K ⊗ H ∼ = H ∼ = H ⊗ K , (122) yields (id ⊗ ev) ◦ (coev H ⊗ id) = id : H → H and (ev ⊗ id) ◦ (id ⊗ coev H ′ ) = id : H → H . From the dual versions of these identities, weobtain for instances the following formulas for the induced braidings Ψ H ′ H ′ and Ψ H ′ H , g h i ⊗ f h i = n P j,k =1 g ( e h i j ) f ( e h i k ) e j ⊗ e k , a h i ⊗ f h i = n P j =1 f ( e h i j ) a h i ⊗ e j , and similar formulas for all other induced braidings. Furthermore, the coproduct (73) and theproduct (75) may be written in the form ∆( f ) = n P j,k =1 f ( e h i j e h i k ) e j ⊗ e k and m ( f ⊗ g ) = n P j =1 f ( e j (1) h i ) g ( e j (2) h i ) e j , respectively. Analogous expressions can be derived for actions and coactions. For example,the coaction ρ R : H → H ⊗ H ′ in (104) for V = H and with the multiplication as left actionis given by ρ R ( a ) = n P j =1 e h i j a h i ⊗ e j . Similarly, the left action ν L : H ⊗ H ′ → H ′ in (99) for U = H , V = H ′ , and with the right H ′ -coaction on H ′ given by the coproduct ρ R = ∆ : H ′ → H ′ ⊗ H ′ from (73), becomes ν L ( a ⊗ f ) = n P j =1 f ( e j a ) e j . Example 6.2.
Graded braided (co)algebras, bi- and Hopf algebras and their graded (co)-modules [17] provide examples if the spaces of homogeneous elements are finite-dimensionalfor all grades. The dual spaces may be given as the direct sum of the duals of the spaces ofhomogeneous elements so that the existence of the presented structures can be deduced gradeby grade from Example 6.1. In this way, we obtain a large class of infinite-dimensional exam-ples. With some care, these arguments can be generalized to filtered (co)algebras, bialgebras,and so on. A
CKNOWLEDGMENTS
It is a pleasure to thank Andrzej Sitarz for stimulating discussions that inspired this report.The author gratefully acknowledges partial financial support from the CONACyT projectA1-S-46784 and the CIC-UMSNH project "Grupos cuánticos y geometría no conmutativa".R
EFERENCES [1] J. Cheng, Y. Xu, S. Zhang,
Duality Theorem and Hom Functor in Braided Tensor Categories, arXiv:0712.1870.[2] M. Da Rocha, J.A. Guccione, .J. Guccione,
Braided module and comodule algebras, Galois extensionsand elements of trace 1.
J. Algebra 307 (2007), 727–768.[3] J.A. Guccione, J.J. Guccione,
Theory of braided Hopf crossed products.
J. Algebra 261 (2004), 54–101.[4] J.A. Guccione, J.J. Guccione, C. Valqui,
Universal deformation formulas and braided module algebras.
J.Algebra 330 (2011), 263–297.[5] X. Guo, S. Zhang,
Integrals of braided Hopf algebras.
J. Math. Res. Exposition 26 (2006), 3–17.[6] Y. Hang, S. Zhang,
Duality Theorems for Infinite Braided Hopf Algebras, arXiv:math/0309007v6.
UALITY FOR INFINITE-DIMENSIONAL BRAIDED BIALGEBRAS 37 [7] A.U. Klimyk, K. Schmüdgen,
Quantum Groups and their Representations.
Texts and Monographs inPhysics, Springer, Berlin, 1998.[8] V. Lyuvashenko,
Hopf algebras and vector symmetries.
Russian Math. Surveys 4 (1988), 153–154.[9] S. Majid,
Braided Groups and Algebraic Quantum Field Theories.
Lett. Math. Phys. 22 (1991), 167–176.[10] S. Majid,
Examples of braided groups and braided matrices.
J. Math. Phys. 32 (1991), 3246–3253.[11] S. Majid,
Braided groups and duals of monoidal categories.
Canad. Math. Soc. Conf. Proc. 13 (1992),329–343, .[12] S. Majid,
Braided Groups.
J. Pure Appl. Algebra 86 (1993), 187–221.[13] S. Majid,
Quantum and braided linear algebra.
J. Math. Phys. 34 (1993), 1176–1196.[14] S. Majid,
Algebras and Hopf algebras in braided categories.
Lec. Notes Pure and Applied Maths 158(1994), 55–105.[15] S. Majid,
Foundations of Quantum Group Theory.
Cambridge University Press, Cambridge, 1995.[16] S. Majid,
Majid, Introduction to braided geometry and q-Minkowski space.
IOS Press, Proc. School ‘En-rico Fermi’ CXXVII (1996), 267–348[17] C. Nastasescu, B. Torrecillas,
Graded coalgebras.
Tsukuba J. Math. 17 (1993), 461–479.[18] M. Takeuchi,
Finite Hopf algebras in braided tensor categories.
J. Pure Appl. Algebra 138 (1999), 59–82.[19] M. Takeuchi,
Survey of braided Hopf algebras.
Contemp. Math. 267 (2000), 301–323.I
NSTITUTO DE F ÍSICA Y M ATEMÁTICAS , U
NIVERSIDAD M ICHOACANA DE S AN N ICOLÁS DE H IDALGO ,E DIFICIO
C-3, C
IUDAD U NIVERSITARIA , 58040 M
ORELIA , M
ICHOACÁN , M
ÉXICO
E-mail address ::