aa r X i v : . [ m a t h . R A ] O c t On the Notion of a Ribbon Quasi-Hopf Algebra
Yorck Sommerh¨auser
To Susan Montgomery on the occasion of her 65th birthday
Abstract
We show that two competing definitions of a ribbon quasi-Hopf algebraare actually equivalent. Along the way, we look at the Drinfel’d elementfrom a new perspective and use this viewpoint to derive its fundamentalproperties.
Introduction
While quasi-Hopf algebras were introduced by V. G. Drinfel’d (cf. [4]), thefirst authors to contemplate the notion of a ribbon quasi-Hopf algebra wereD. Altsch¨uler and A. Coste (cf. [1], Par. 4.1, p. 89). They define them as qua-sitriangular quasi-Hopf algebras with an additional central element, the ribbonelement, that is subject to four axioms. However, as the authors point outthemselves, these axioms are not completely satisfactory, as they neither reducedirectly to the axioms of a ribbon Hopf algebra, in the case where the quasi-Hopfalgebra happens to be an ordinary Hopf algebra, nor are in complete analogy tothe axioms for a ribbon category. They therefore analyzed their notion furtherand explained that, in the case where the evaluation element α is invertible,their axioms are equivalent to a set of four different axioms which are consid-erably closer to the notion of a ribbon Hopf algebra and the notion of a ribboncategory.However, in the case of ribbon Hopf algebras, one of the four axioms is actuallya consequence of the remaining axioms. Therefore D. Bulacu, F. Panaite, andF. van Oystaeyen proposed a different definition of a ribbon quasi-Hopf algebra,leaving out this supposedly superfluous axiom (cf. [3], Def. 2.3, p. 6106). Again inthe case where the evaluation element is invertible, they showed that this axiomreally was superfluous, so that their definition was equivalent to the revisedversion of Altsch¨uler and Coste (cf. [3], Prop. 5.5, p. 6119).Of course, this raised the question whether the assumption on the invertibilityof the evaluation element is really necessary to establish these two equivalences,r whether this assumption was only made to simplify the argument. In thecase of the first equivalence, between the two versions of the definition alreadyproposed by Altsch¨uler and Coste, this question was addressed by D. Bulacuand E. Nauwelaerts, who showed that the assumption is not necessary (cf. [2],Thm. 3.1, p. 667). In a recent article, when using ribbon quasi-Hopf algebrasto exemplify certain properties of modular data, the authors have claimed thatthis assumption is also not necessary for the second equivalence between thedefinition of Altsch¨uler and Coste and the definition of Bulacu, Panaite, andvan Oystaeyen (cf. [9], Cor. 5.1, p. 50). The purpose of the present article is toprove this claim.To do this, we take a certain viewpoint, which is suitable not only for this proof,but also for similar questions: The R-matrix can be viewed as a twist that takesthe coproduct into the coopposite coproduct. However, while twisting leaves theantipode unchanged, the coopposite coproduct naturally comes endowed withthe inverse antipode. The so-called Drinfel’d element now appears as the elementthat connects these two choices for the antipode of the coopposite quasi-Hopfalgebra. Viewing the Drinfel’d element in this way enables us not only to givea relatively easy proof of our claim, but also allows us to give a new derivationof the fundamental properties of the Drinfel’d element in a comparatively shortand conceptual way.The article consists of two sections. The first, preliminary section contains a briefsummary of the basic facts about quasi-bialgebras, quasi-Hopf algebras, quasi-triangularity, and twisting. However, we trace more precisely than the availablereferences how some elements already introduced in Drinfel’d’s original articletransform under twisting and other modifications, as this turns out to be crucialfor our treatment.The second section contains our main result, Theorem 2.3. As explained above,we prove it by viewing the R-matrix as a twist, a viewpoint developed in Para-graph 2.1. The new proof of the fundamental properties of the Drinfel’d ele-ment also mentioned above is given in Paragraph 2.2. The article concludeswith Proposition 2.4, a formula for the image of the Drinfel’d element underthe antipode. Although this formula was needed in our earlier proofs of Theo-rem 2.3, it is not needed in the proof presented here. We include it nonetheless,because it is of independent interest and its proof nicely illustrates the ideasthat we have developed.In the following, we work over a base field that is denoted by K . All vectorspaces that we will consider will be defined over this base field K , and alltensor products will be taken over K . With respect to enumeration, we use theconvention that propositions, definitions, and similar items are referenced bythe paragraph in which they occur; an additional third digit indicates a part ofthe corresponding item. For example, a reference to Proposition 2.2.3 refers tothe third assertion of the unique proposition in Paragraph 2.2.2 Preliminaries
Recall that a quasi-bialgebra is a quadruple ( A, ∆ , ε, Φ), where A is anassociative algebra over our base field K , whose multiplication and unit elementwe have not explicitly listed as part of the structure elements. Out of the struc-ture elements that we have listed explicitly, two are algebra homomorphisms,namely ∆ : A → A ⊗ A , which we call the coproduct, and ε : A → K , which wecall the counit. The remaining structure element is the associator Φ ∈ A ⊗ A ⊗ A .These structure elements are required to satisfy several axioms: Besides that Φis required to be invertible, four equations have to be satisfied, which we nowlist. We give each equation a name that we will use in later references:1. Quasi-coassociativity: (id ⊗ ∆)∆( a )Φ = Φ(∆ ⊗ id)∆( a )2. Pentagon axiom:(id ⊗ id ⊗ ∆)(Φ)(∆ ⊗ id ⊗ id)(Φ) = (1 ⊗ Φ)(id ⊗ ∆ ⊗ id)(Φ)(Φ ⊗ ε ⊗ id)∆( a ) = a = (id ⊗ ε )∆( a )4. Counit-associator axiom: (id ⊗ ε ⊗ id)(Φ) = 1 ⊗ a ∈ A . These axiomsimply another property, which we call the counit-associator property: Proposition ( ε ⊗ id ⊗ id)(Φ) = (id ⊗ id ⊗ ε )(Φ) = 1 ⊗ Proof.
This is proved in [4], Remark on p. 1422. (cid:3)
We will use the version ∆( a ) = a (1) ⊗ a (2) of the Heyneman-Sweedler sigmanotation for the coproduct, and the notation ∆ cop ( a ) = a (2) ⊗ a (1) for thecoopposite coproduct. Also, it will frequently be necessary to write Φ and itsinverse as a sum of decomposable tensors, which we do in the formΦ = n X i =1 X i ⊗ Y i ⊗ Z i Φ − = m X j =1 ¯ X j ⊗ ¯ Y j ⊗ ¯ Z j Because the number of decomposable tensors in these sums is never importantin the sequel, we will also write such equations in slightly abbreviated forms,like Φ = P i X i ⊗ Y i ⊗ Z i . 3 .2 A quasi-bialgebra is a quasi-Hopf algebra if it is endowed with three ad-ditional structure elements: An algebra anti-automorphism S : A → A , calledthe antipode, an element α ∈ A , called the evaluation element, and an ele-ment β ∈ A , called the coevaluation element. The axioms that these structureelements have to satisfy are the following:1. Left antipode equation: S ( a (1) ) αa (2) = ε ( a ) α
2. Right antipode equation: a (1) βS ( a (2) ) = ε ( a ) β
3. Duality axiom: P i X i βS ( Y i ) αZ i = 1 = P j S ( ¯ X j ) α ¯ Y j βS ( ¯ Z j )These structure elements are compatible with the counit as follows: Lemma
We have ε ( S ( a )) = ε ( a ) and ε ( α ) ε ( β ) = 1. Proof.
The first assertion is proved in [4], Rem. 7, p. 1425. The second followsby applying the counit to the duality axiom. (cid:3)
The antipode is also compatible with the coproduct and the associator. Toformulate these compatibilities, we need to define two elements γ and δ in thesecond tensor power of A , which are in a sense analogues of the evaluationelement α and the coevaluation element β : γ := X i,j S ( ¯ X i Y j ) α ¯ Y i Z j (1) ⊗ S ( X j ) α ¯ Z i Z j (2) δ := X i,j X i (1) ¯ X j βS ( Z i ) ⊗ X i (2) ¯ Y j βS ( Y i ¯ Z j )From these elements, we derive the element F := X i ( S ( ¯ X i (2) ) ⊗ S ( ¯ X i (1) )) γ ∆( ¯ Y i βS ( ¯ Z i ))which appears in the compatibility conditions in the following way: Proposition F is invertible with inverse F − = X i ∆( S ( ¯ X i ) α ¯ Y i ) δ ( S ( ¯ Z i (2) ) ⊗ S ( ¯ Z i (1) ))and we have γ = F ∆( α ) and δ = ∆( β ) F − . The antipode is compatible withthe coproduct via ∆( S ( a )) = F − ( S ( a (2) ) ⊗ S ( a (1) )) F and with the associator via X i S ( Z i ) ⊗ S ( Y i ) ⊗ S ( X i ) = (1 ⊗ F )(id ⊗ ∆)( F )Φ(∆ ⊗ id)( F − )( F − ⊗ roof. This is proved in [4], Prop. 1.2, p. 1426. We note that it is also shownthere that the three properties ∆( S ( a )) = F − ( S ( a (2) ) ⊗ S ( a (1) )) F , γ = F ∆( α ),and δ = ∆( β ) F − characterize F uniquely; even stronger, it suffices to checkone of the two conditions γ = F ∆( α ) and δ = ∆( β ) F − . (cid:3) The antipode of a quasi-Hopf algebra is in general not unique; it can bemodified with the help of an invertible element x ∈ A by defining S x ( a ) := xS ( a ) x − α x := xα β x := βx − It is easy to check that S x is again an antipode for A with evaluation element α x and coevaluation element β x . However, this is the only possible modification:If S ′ is an arbitrary new antipode for the quasi-Hopf algebra A , with evaluationelement α ′ and coevaluation element β ′ , then the element x := X i S ′ ( ¯ X i ) α ′ ¯ Y i βS ( ¯ Z i )is invertible with inverse x − = P i S ( ¯ X i ) α ¯ Y i β ′ S ′ ( ¯ Z i ), and we have S ′ = S x , α ′ = α x , and β ′ = β x . This fact, which will be important in the sequel, is provedin [4], Prop. 1.1, p. 1425.By modifying the antipode as indicated by an invertible element x , we of courseindirectly modify all other elements derived from it; in particular the elements γ , δ , and F introduced in Paragraph 1.2. The modified elements, which we denoteby γ x , δ x , and F x , can be expressed in terms of the unmodified elements asfollows: Proposition γ x = ( x ⊗ x ) γ δ x = δ ( x − ⊗ x − ) F x = ( x ⊗ x ) F ∆( x − ) Proof.
The form of γ x follows directly from the definition: γ x = X i,j xS ( ¯ X i Y j ) x − ( xα ) ¯ Y i Z j (1) ⊗ xS ( X j ) x − ( xα ) ¯ Z i Z j (2) = ( x ⊗ x ) γ Similarly, the definition of δ x is δ x = X i,j X i (1) ¯ X j ( βx − ) xS ( Z i ) x − ⊗ X i (2) ¯ Y j ( βx − ) xS ( Y i ¯ Z j ) x − which immediately yields the second assertion. Finally, since F x = X i ( S x ( ¯ X i (2) ) ⊗ S x ( ¯ X i (1) )) γ x ∆( ¯ Y i β x S x ( ¯ Z i ))= X i ( xS ( ¯ X i (2) ) x − ⊗ xS ( ¯ X i (1) ) x − )( x ⊗ x ) γ ∆( ¯ Y i ( βx − ) xS ( ¯ Z i ) x − )= ( x ⊗ x ) X i ( S ( ¯ X i (2) ) ⊗ S ( ¯ X i (1) )) γ ∆( ¯ Y i βS ( ¯ Z i ) x − ) = ( x ⊗ x ) F ∆( x − )the third assertion also holds. (cid:3) .4 With every quasi-Hopf algebra A , one can associate another quasi-Hopfalgebra A cop , which has the same product as A , but the coopposite coproduct.For this quasi-Hopf algebra, the counit is unchanged, the associator is changed to P i ¯ Z i ⊗ ¯ Y i ⊗ ¯ X i , the antipode is changed to its inverse S − , the evaluation elementis changed to S − ( α ), and the coevaluation element is changed to S − ( β ) (cf. [4],Rem. 4, p. 1424; [6], Exerc. XV.6.2, p. 381).As in Paragraph 1.3, this modification of the defining structure elements alsoleads to a modification of the elements γ , δ , and F . In this case, however,we do not introduce a special notation for the new elements formed in A cop ,because their relation to the original elements is so simple: The new elementsare ( S − ⊗ S − )( γ ), ( S − ⊗ S − )( δ ), and ( S − ⊗ S − )( F ). To see this in thecase of γ , we use an alternative description of γ given in [4], Lem. 1, p. 1427,which yields( S − ⊗ S − )( γ ) = ( S − ⊗ S − )( X i,j S ( Y i ¯ X j (2) ) αZ i ¯ Y j ⊗ S ( X i ¯ X j (1) ) α ¯ Z j )= X i,j S − ( Z i ¯ Y j ) S − ( α ) Y i ¯ X j (2) ⊗ S − ( ¯ Z j ) S − ( α ) X i ¯ X j (1) But this last term is just what we get if we form γ in A cop according to theoriginal definition in Paragraph 1.2.In the case of δ , we argue similarly: An alternative formula given in [4], loc. cit.implies that( S − ⊗ S − )( δ ) = ( S − ⊗ S − )( X i,j ¯ X i βS ( ¯ Z i (2) Z j ) ⊗ ¯ Y i X j βS ( ¯ Z i (1) Y j ))= X i,j ¯ Z i (2) Z j S − ( β ) S − ( ¯ X i ) ⊗ ¯ Z i (1) Y j S − ( β ) S − ( ¯ Y i X j )which is again what we get if we form δ in A cop according to the original defi-nition in Paragraph 1.2.In the case of F , we argue differently: If we apply S − ⊗ S − to the equation( S ⊗ S )(∆ cop ( a )) = F ∆( S ( a )) F − in Proposition 1.2 and replace a by S − ( a ),we get∆ cop ( S − ( a )) = ( S − ⊗ S − )( F − )( S − ⊗ S − )(∆( a ))( S − ⊗ S − )( F )Similarly, if we apply S − ⊗ S − to the equation γ = F ∆( α ) in the sameproposition and use what we have just established, we get( S − ⊗ S − )( γ ) = ( S − ⊗ S − )(∆( α ))( S − ⊗ S − )( F )= ( S − ⊗ S − )( F )∆ cop ( S − ( α ))Finally, if we treat the equation δ = ∆( β ) F − in the same way, we get( S − ⊗ S − )( δ ) = ∆ cop ( S − ( β ))( S − ⊗ S − )( F − )But this establishes our assertion, since it shows that ( S − ⊗ S − )( F ) has thecharacteristic properties of the element F in A cop , as described in Paragraph 1.2.6 .5 A quasi-Hopf algebra is called quasitriangular if it is endowed with a so-called R-matrix, which is an invertible element R = P l s l ⊗ t l ∈ A ⊗ A thatsatisfies the following three conditions:1. Quasi-cocommutativity: ∆ cop ( a ) R = R ∆( a )2. Left hexagon axiom:(∆ ⊗ id)( R ) = X i,j,k,l,q Y i s l ¯ X j X k ⊗ Z i ¯ Z j s q Y k ⊗ X i t l ¯ Y j t q Z k
3. Right hexagon axiom:(id ⊗ ∆)( R ) = X i,j,k,l,q ¯ Z i s l Y j s q ¯ X k ⊗ ¯ X i X j t q ¯ Y k ⊗ ¯ Y i t l Z j ¯ Z k Note that the right-hand side in the hexagon axioms factors completely; forexample, the right-hand side in the left hexagon axiom is the product of thetensors P i Y i ⊗ Z i ⊗ X i , P l s l ⊗ ⊗ t l , P j ¯ X j ⊗ ¯ Z j ⊗ ¯ Y j , P q ⊗ s q ⊗ t q , and P k X k ⊗ Y k ⊗ Z k .The hexagon axioms obviously constitute a compatibility condition betweenthe R-matrix and the coproduct. But the R-matrix is also compatible with thecounit and the antipode: Denoting by F ′ the image of F under the interchangeof the two tensor factors, we have Lemma ( ε ⊗ id)( R ) = 1 (id ⊗ ε )( R ) = 1 ( S ⊗ S )( R ) = F ′ RF − Proof.
The equations involving the counit are proved in [4], Rem. 2, p. 1440;they are also stated in [1], Eq. (2.23), p. 87. The equation involving the antipodewas stated in [1], Eq. (4.22), p. 96 and proved in [5], Cor. 2.2, p. 559. A proofwithout the graphical calculus was given in [2], Lem. 2.3, p. 663. These refer-ences also list additional compatibility conditions between the R-matrix and theantipode. (cid:3)
From the R-matrix, we derive a special element u , called the Drinfel’d element.It is defined as u := X i,l S ( ¯ Y i βS ( ¯ Z i )) S ( t l ) αs l ¯ X i (cf. [1], Eq. (3.2), p. 87; [6], Exerc. XV.6.5, p. 381). This ad hoc definition mayappear unmotivated at this point; we will put it in its context in Paragraph 2.1.Although we could set down the fundamental properties of the Drinfel’d elementhere, as they appear in literature, we defer this to Paragraph 2.2, where we willactually reconfirm them from the viewpoint developed in Paragraph 2.1, as this7iewpoint allows for a proof that is in our opinion shorter and more conceptual.Here we only record how the Drinfel’d element changes if the antipode is modi-fied by an invertible element x as explained in Paragraph 1.3. The new Drinfel’delement u x relates to the old Drinfel’d element u as follows: Proposition u x = xS ( x − ) u Proof.
As we have u x = X i,l S x ( ¯ Y i β x S x ( ¯ Z i )) S x ( t l ) α x s l ¯ X i = X i,l xS ( ¯ Y i ( βx − )( xS ( ¯ Z i ) x − )) x − ( xS ( t l ) x − )( xα ) s l ¯ X i = X i,l xS ( ¯ Y i βS ( ¯ Z i ) x − ) S ( t l ) αs l ¯ X i = xS ( x − ) u we see that this follows directly from the definition. (cid:3) Quasi-Hopf algebras can be twisted to generate new quasi-Hopf algebras.The ingredient that we need for this is a twisting element; i.e., an invertibleelement T ∈ A ⊗ A in the second tensor power of our quasi-Hopf algebra A thatsatisfies the condition ( ε ⊗ id)( T ) = (id ⊗ ε )( T ) = 1. If we then introduce thenew coproduct ∆ T ( a ) := T ∆( a ) T − and the new associatorΦ T := (1 ⊗ T )(id ⊗ ∆)( T )Φ(∆ ⊗ id)( T − )( T − ⊗ α T and a new coeval-uation element β T via α T := X i S ( ¯ f i ) α ¯ g i β T := X i f i βS ( g i )where we have used the notation T = P i f i ⊗ g i and T − = P i ¯ f i ⊗ ¯ g i (cf. [4],Rem. 5, p. 1425; [6], Exerc. XV.6.4, p. 381).As a consequence of these modifications, we also get, according to our definitionsin Paragraph 1.2, new elements γ T , δ T , and F T . As we will show now, these newelements can be expressed in terms of the original elements γ , δ , and F . Ifwe denote, as for F , by T ′ the image of T under the interchange of the twotensor factors, the corresponding expressions look, in a slightly implicit form,as follows: 8 roposition
1. ( S ⊗ S )( T ′ ) γ T T = X i ( S ⊗ S )(∆ cop ( ¯ f i )) γ ∆(¯ g i )2. T − δ T ( S ⊗ S )( T ′− ) = X i ∆( f i ) δ ( S ⊗ S )(∆ cop ( g i ))3. F T = ( S ⊗ S )( T ′− ) F T − Proof. (1) We use the Sweedler notation ∆ T ( a ) = a [1] ⊗ a [2] for the twistedcoproduct, and primes for the twisted associator; i.e., we writeΦ T = X i X ′ i ⊗ Y ′ i ⊗ Z ′ i Φ − T = X j ¯ X ′ j ⊗ ¯ Y ′ j ⊗ ¯ Z ′ j With this notation, the definition of γ T reads γ T = X i,j S ( ¯ X ′ i Y ′ j ) α T ¯ Y ′ i Z ′ j [1] ⊗ S ( X ′ j ) α T ¯ Z ′ i Z ′ j [2] = X i,j,k,l S ( ¯ f k ¯ X ′ i Y ′ j ) α ¯ g k ¯ Y ′ i Z ′ j [1] ⊗ S ( ¯ f l X ′ j ) α ¯ g l ¯ Z ′ i Z ′ j [2] If we multiply this from the right by T = P q f q ⊗ g q and use the fact that∆ T ( a ) T = T ∆( a ), we get γ T T = X i,j,k,l,q S ( ¯ f k ¯ X ′ i Y ′ j ) α ¯ g k ¯ Y ′ i f q Z ′ j (1) ⊗ S ( ¯ f l X ′ j ) α ¯ g l ¯ Z ′ i g q Z ′ j (2) But from the definition of the twisted associator, we have X i,k,q ¯ f k ¯ X ′ i ⊗ ¯ g k ¯ Y ′ i f q ⊗ ¯ Z ′ i g q = ( T − ⊗ − T (1 ⊗ T )= (∆ ⊗ id)( T )Φ − (id ⊗ ∆)( T − )If we insert this into our expression, the term (∆ ⊗ id)( T ) cancels, and we get γ T T = X i,j,k,l S ( ¯ X i ¯ f k Y ′ j ) α ¯ Y i ¯ g k (1) Z ′ j (1) ⊗ S ( ¯ f l X ′ j ) α ¯ g l ¯ Z i ¯ g k (2) Z ′ j (2) Multiplying from the left by ( S ⊗ S )( T ′ ) yields( S ⊗ S )( T ′ ) γ T T = X i,j,k,l,q S ( ¯ X i ¯ f k Y ′ j g q ) α ¯ Y i ¯ g k (1) Z ′ j (1) ⊗ S ( ¯ f l X ′ j f q ) α ¯ g l ¯ Z i ¯ g k (2) Z ′ j (2) Now we have, again from the definition of the twisted associator, that X j,k,q X ′ j f q ⊗ ¯ f k Y ′ j g q ⊗ ¯ g k Z ′ j = (1 ⊗ T − )Φ T ( T ⊗ ⊗ ∆)( T )Φ(∆ ⊗ id)( T − ) = X j,k,q f k X j ¯ f q (1) ⊗ g k (1) Y j ¯ f q (2) ⊗ g k (2) Z j ¯ g q S ⊗ S )( T ′ ) γ T T = X i,j,k,l,q S ( ¯ X i g k (1) Y j ¯ f q (2) ) α ¯ Y i g k (2)(1) Z j (1) ¯ g q (1) ⊗ S ( ¯ f l f k X j ¯ f q (1) ) α ¯ g l ¯ Z i g k (2)(2) Z j (2) ¯ g q (2) Using quasi-coassociativity, we can write this as( S ⊗ S )( T ′ ) γ T T = X i,j,k,l,q S ( g k (1)(1) ¯ X i Y j ¯ f q (2) ) αg k (1)(2) ¯ Y i Z j (1) ¯ g q (1) ⊗ S ( ¯ f l f k X j ¯ f q (1) ) α ¯ g l g k (2) ¯ Z i Z j (2) ¯ g q (2) Here we can use the left antipode equation on the part S ( g k (1)(1) ) αg k (1)(2) , andafter that the summations over k and l cancel, so that we are left with( S ⊗ S )( T ′ ) γ T T = X i,j,q S ( ¯ X i Y j ¯ f q (2) ) α ¯ Y i Z j (1) ¯ g q (1) ⊗ S ( X j ¯ f q (1) ) α ¯ Z i Z j (2) ¯ g q (2) = X q ( S ( ¯ f q (2) ) ⊗ S ( ¯ f q (1) )) γ (¯ g q (1) ⊗ ¯ g q (2) )which is the first assertion.(2) The form of δ T can be established by a very similar computation. However,this computation can be avoided by using the argument that we present now. Inthis approach, we redefine F T to be what we claim it is according to the thirdassertion, i.e., we redefine it as F T := ( S ⊗ S )( T ′− ) F T − . By Proposition 1.2,the original element F satisfies ( S ⊗ S )(∆ cop ( a )) = F ∆( S ( a )) F − , so F T satisfies( S ⊗ S )(∆ cop T ( a )) = ( S ⊗ S )( T ′ ∆ cop ( a ) T ′− )= ( S ⊗ S )( T ′− ) F ∆( S ( a )) F − ( S ⊗ S )( T ′ ) = F T ∆ T ( S ( a )) F − T Furthermore, from the first assertion and the properties of F we have that( S ⊗ S )( T ′ ) γ T T = X i ( S ⊗ S )(∆ cop ( ¯ f i )) γ ∆(¯ g i ) = X i F ∆( S ( ¯ f i )) F − γ ∆(¯ g i )= F X i ∆( S ( ¯ f i ))∆( α )∆(¯ g i ) = F ∆( α T )so that γ T = ( S ⊗ S )( T ′− ) F ∆( α T ) T − = F T ∆ T ( α T ). But we know from Para-graph 1.2 that the two properties that we have just established characterize F T ,in other words, the third assertion of our proposition holds.(3) But then the equation δ T = ∆ T ( β T ) F − T holds by Proposition 1.2. Insertingthe form of F T , this says that δ T = T ∆( β T ) F − ( S ⊗ S )( T ′ ), so that the left-handside of the second assertion of our proposition is T − δ T ( S ⊗ S )( T ′− ) = ∆( β T ) F − = X i ∆( f i )∆( β )∆( S ( g i )) F − F again, we can rewrite this in the form T − δ T ( S ⊗ S )( T ′− ) = X i ∆( f i )∆( β ) F − ( S ⊗ S )(∆ cop ( g i ))= X i ∆( f i ) δ ( S ⊗ S )(∆ cop ( g i ))where the last step uses the original equation δ = ∆( β ) F − from Proposition 1.2.But this is exactly the second assertion of our proposition. (cid:3) If A is quasitriangular, its twist is also quasitriangular, with respect to the newR-matrix R T := T ′ RT − (cf. [4], Eq. (3.11), p. 1439; [6], Prop. XV.3.6, p. 376).This new R-matrix in principle also gives rise to a new Drinfel’d element u T .However, this new element coincides with the original one: Lemma u T = u Proof.
This is proved in [3], Lem. 4.2, p. 6115. (cid:3)
We can also relate quasitriangularity and twisting in another way: As thetwisting element T , we can choose the R-matrix R , because Lemma 1.5 assertsthat the R-matrix satisfies the conditions that a twist element should satisfy.By definition, the twisted coproduct is just the coopposite coproduct, which wehave discussed in Paragraph 1.4. However, not all of the other structure elementsmatch: Although the Yang-Baxter equation (cf. [6], Cor. XV.2.3, p. 372) yieldsthat the twisted associator is also P i ¯ Z i ⊗ ¯ Y i ⊗ ¯ X i , the antipode remains thesame, and is not changed to its inverse, and for the evaluation element ˆ α andthe coevaluation element ˆ β we find the expressionsˆ α = X l S (¯ s l ) α ¯ t l ˆ β = X l s l βS ( t l )where we have, as before, used the notation R = P l s l ⊗ t l and R − = P l ¯ s l ⊗ ¯ t l .This is, however, not a contradiction; we have already discussed in Paragraph 1.3that the antipode of a quasi-Hopf algebra is not unique, and we have also ex-plained there how the structures are related: The elementˆ u := X i S ( Z i )ˆ αY i S − ( β ) S − ( X i )is invertible with inverse ˆ u − = P i S − ( Z i ) S − ( α ) Y i ˆ βS ( X i ), and we have S ( a ) = ˆ uS − ( a )ˆ u − ˆ α = ˆ uS − ( α ) ˆ β = S − ( β )ˆ u − R ′− is also an R-matrix for A , where R ′ denotes, as for F and T before, the image of R under the interchange of thetwo tensor factors. We can therefore also use this R-matrix to twist the co-product into the coopposite coproduct. In this case, the twisted associator isagain P i ¯ Z i ⊗ ¯ Y i ⊗ ¯ X i , the antipode remains unchanged, and for the evaluationelement ˇ α and the coevaluation element ˇ β we find the expressionsˇ α = X l S ( t l ) αs l ˇ β = X l ¯ t l βS (¯ s l )Also the discussion in Paragraph 1.3 applies again to tell us the relation of thestructures: The elementˇ u := X i S ( Z i )ˇ αY i S − ( β ) S − ( X i )is invertible with inverse ˇ u − = P i S − ( Z i ) S − ( α ) Y i ˇ βS ( X i ), and we have S ( a ) = ˇ uS − ( a )ˇ u − ˇ α = ˇ uS − ( α ) ˇ β = S − ( β )ˇ u − It is to be expected that there is a connection between these two ways of twisting.A first connection involves the evaluation and the coevaluation elements:
Lemma
For the evaluation elements, we have S − (ˇ α ) = ˆ u − α S − (ˆ α ) = ˇ u − α For the coevaluation elements, we have S − ( ˇ β ) = β ˆ u S − ( ˆ β ) = β ˇ u Proof.
It is easy to solve the definitions of ˆ α and ˇ α for α ; we find α = X l S ( s l )ˆ αt l α = X l S (¯ t l )ˇ α ¯ s l If we apply the inverse antipode to the definition of ˇ α and use the precedingformulas, we therefore get S − (ˇ α ) = X l S − ( s l ) S − ( α ) t l = X l S − ( s l )ˆ u − ˆ αt l = ˆ u − X l S ( s l )ˆ αt l = ˆ u − α The formula S − (ˆ α ) = ˇ u − α can be established by a similar computation, buton the other hand, it also follows from the first equation by interchanging R and R ′− .The coevaluation elements can be treated similarly: Solving their definitionsfor β , we find β = X l ¯ s l ˆ βS (¯ t l ) β = X l t l ˇ βS ( s l )12f we apply the inverse antipode to the definition of ˇ β and use the precedingformulas, we therefore get S − ( ˇ β ) = X l ¯ s l S − ( β ) S − (¯ t l ) = X l ¯ s l ˆ β ˆ uS − (¯ t l ) = X l ¯ s l ˆ βS (¯ t l )ˆ u = β ˆ u Again, the formula S − ( ˆ β ) = β ˇ u can be established by a similar computation,or viewed as a consequence by interchanging R and R ′− . (cid:3) There is also a direct connection between the elements ˆ u and ˇ u , and, what isimportant for us, there is a connection to the Drinfel’d element u : Proposition u = ˇ u = S (ˆ u − ) Proof.
From the preceding lemma, we get thatˆ u − α = S − (ˇ α ) = S − ( α ) S − (ˇ u ) = ˆ u − α ˆ uS − (ˇ u )so that α = α ˆ uS − (ˇ u ). Now the square of the antipode is both conjugation withˆ u and conjugation with S − (ˇ u − ), so that ˆ uS − (ˇ u ) is a central element. Butthen the duality axiom implies thatˆ uS − (ˇ u ) = X i X i βS ( Y i ) α ˆ uS − (ˇ u ) Z i = X i X i βS ( Y i ) αZ i = 1This shows that ˆ u − = S − (ˇ u ) and therefore ˇ u = S (ˆ u − ).For the assertion about the Drinfel’d element, we first note that with our newterminology we can rewrite its definition, given in Paragraph 1.5, in the form u = X i S ( ¯ Y i βS ( ¯ Z i ))ˇ α ¯ X i Applying the inverse antipode and using that S − (ˇ α ) = ˆ u − α by the precedinglemma, we get S − ( u ) = X i S − ( ¯ X i )ˆ u − α ¯ Y i βS ( ¯ Z i ) = ˆ u − X i S ( ¯ X i ) α ¯ Y i βS ( ¯ Z i ) = ˆ u − where the last step follows from the duality axiom. This shows that u = S (ˆ u − ),as asserted. (cid:3) We note that this proposition and the preceding lemma imply immediately thatˇ α = S ( α ) u , which is an identity that appears in [1], Eq. (3.9), p. 88.13 .2 The choice of the R-matrix R as the twisting element T does not onlylead to the elements ˆ α , ˆ β , and ˆ u , but also, as we saw in Paragraph 1.6, to newversions of the elements γ , δ , and F , which we denote by ˆ γ , ˆ δ , and ˆ F . Similarly,the choice of R ′− as the twisting element T leads to new versions of theseelements that we denote by ˇ γ , ˇ δ , and ˇ F . We have seen in Proposition 1.6 howthe new elements can be expressed in terms of the old ones; we record here onlythe form of ˆ F and ˇ F , where this proposition yields thatˆ F = ( S ⊗ S )( R ′− ) F R − ˇ F = ( S ⊗ S )( R ) F R ′ We now use all of this to derive the fundamental properties of the Drinfel’delement u , as promised in the introduction and in Paragraph 1.5. These funda-mental properties are the following: Proposition u is invertible. Moreover, we have1. ε ( u ) = 12. S ( a ) = uau −
3. ∆( u ) = F − (( S ⊗ S )( F ′ ))( u ⊗ u )( R ′ R ) − Proof.
By Proposition 2.1, we have u = ˇ u , and we have noted already inParagraph 2.1 that ˇ u is invertible. By using Lemma 1.2, Lemma 1.5, and thecounit-associator property, it follows directly from the definition that ε ( u ) = 1,or alternatively ε (ˇ u ) = 1 from its definition. The second property of the Drinfel’delement is just one of the properties of ˇ u that follow directly from its construc-tion in Paragraph 2.1. For the third property, recall that we have describedthe structure elements of the coopposite quasi-Hopf algebra in Paragraph 1.4;in particular, we have seen there that the element F , formed in A cop , is just( S − ⊗ S − )( F ). On the other hand, we have explained in Paragraph 2.1 howthe coopposite coproduct arises by twisting the original coproduct with the helpof the R-matrix, or alternatively with the help of its variant R ′− . As the twostructures were related via ˆ u resp. ˇ u , we get from Proposition 1.3 thatˆ F = (ˆ u ⊗ ˆ u )( S − ⊗ S − )( F )∆ cop (ˆ u − ) ˇ F = (ˇ u ⊗ ˇ u )( S − ⊗ S − )( F )∆ cop (ˇ u − )Because the Drinfel’d element is equal to ˇ u , we focus on the second formula,and substitute for ˇ F the expression from the beginning of this paragraph to get( S ⊗ S )( R ) F R ′ = ( u ⊗ u )( S − ⊗ S − )( F )∆ cop ( u − )But we have ( S ⊗ S )( R ) F = F ′ R by Lemma 1.5, and therefore can use thesecond property of the Drinfel’d element to rewrite the preceding equation as F ′ RR ′ = ( S ⊗ S )( F )( u ⊗ u )∆ cop ( u − )Interchanging tensor factors, this becomes F R ′ R = ( S ⊗ S )( F ′ )( u ⊗ u )∆( u − ),which in turn implies R ′ R ∆( u ) = F − ( S ⊗ S )( F ′ )( u ⊗ u ). But by quasi-cocom-mutativity, we have R ′ R ∆( u ) = ∆( u ) R ′ R , and the third assertion follows. (cid:3)
14t must be emphasized that the preceding proposition is not new: The invert-ibility of u , the first property and in particular the second property were provedby D. Altsch¨uler and A. Coste in [1], Sec. 3, p. 87f. The third property is statedthere as well (cf. Eq. (4.21), p. 95), and the authors also propose a general strat-egy for its proof, of which they carry out the first step explicitly (cf. Eq. (4.20),p. 95), which however, as they say clearly, only works under the assumptionthat α is invertible. The first complete, rigorous proof without this assumptionwas given by D. Bulacu and E. Nauwelaerts in [2], p. 668ff. As its Hopf-algebraicpredecessor (cf. [7], Thm. 10.1.13, p. 181f), it is based on a comparatively in-volved computation, but has the advantage to deduce the result almost directlyfrom the axioms. We now use the machinery developed so far to study ribbon quasi-Hopfalgebras. A quasitriangular quasi-Hopf algebra is called a ribbon quasi-Hopfalgebra if it contains a ribbon element. This means the following:
Definition
A nonzero central element v ∈ A is called a ribbon element if itsatisfies ∆( v ) = ( R ′ R )( v ⊗ v ) and S ( v ) = v Let us clarify how this definition relates to the various competing definitions of aribbon quasi-Hopf algebra that we have already mentioned in the introduction.We will prove below that it follows from our definition that a ribbon elementis invertible. The definitions given in [1], [2], and [3] all work instead with theinverse element; our convention is the one used in [10], Sec. XI.3.1, p. 500. Asalready pointed out in [3], Def. 2.3, p. 6106, it follows from the counitalityproperty and Lemma 1.5 that ε ( v ) = 1; to see this, one just needs to apply ε ⊗ id to the first axiom in our definition above. This shows that, modulo theinversion, our definition matches with the definition in [3], loc. cit.A different definition was given by D. Altsch¨uler and A. Coste in [1], Par. 4.1,p. 89. As noted in [2], Thm. 3.1, p. 667, it follows from the formula for thecoproduct of the Drinfel’d element, which we have just reconfirmed in Propo-sition 2.2.3, that the definition given by Altsch¨uler and Coste is equivalent toour definition and the additional requirement that v − = uS ( u ). Furthermore,it was shown in [3], Prop. 5.5, p. 6119 that this property is automatically satis-fied if α is invertible. We will now show that this restriction is unnecessary. Forpreparation, we need the following lemma: Lemma
We have v ˇ α = ˆ α and v ˆ β = ˇ β . Proof.
Because the ribbon element is central and invariant under the antipode,we have v ˆ β = X l v s l βS ( t l ) = X l s l vβS ( t l v )15he above definition also yields R ′− ∆( v ) = R ( v ⊗ v ). Inserting this into thepreceding formula, we get v ˆ β = X l ¯ t l v (1) βS (¯ s l v (2) ) = X l ¯ t l v (1) βS ( v (2) ) S (¯ s l ) = X l ¯ t l βS (¯ s l ) = ˇ β by the right antipode equation and the fact that ε ( v ) = 1, which we alreadyrecorded above. This proves the second assertion. The proof of the first assertionis similar: Since ∆( v ) R − = ( v ⊗ v ) R ′ , we have v ˇ α = X l S ( vt l ) αvs l = X l S ( v (1) ¯ s l ) αv (2) ¯ t l = X l S (¯ s l ) α ¯ t l = ˆ α by the left antipode equation. (cid:3) The proof of our main result is now almost immediate:
Theorem v − = uS ( u ) Proof.
By construction, we have S − ( α ) = ˆ u − ˆ α = ˇ u − ˇ α . Comparing thiswith the first assertion of the lemma, we see that v ˇ α = ˆ α = ˆ u ˇ u − ˇ α . Now theduality axiom for the twisted quasi-Hopf algebra yields X i S ( Z i )ˇ αY i ˇ βS ( X i ) = 1Because both v and ˆ u ˇ u − are central, this implies v = X i S ( Z i ) v ˇ αY i ˇ βS ( X i ) = X i S ( Z i )ˆ u ˇ u − ˇ αY i ˇ βS ( X i ) = ˆ u ˇ u − In view of Proposition 2.1, this means that v = S − ( u − ) u − . Inverting this,we get v − = uS − ( u ). But as u is invariant under the square of the antipodeby Proposition 2.2.2, this implies the assertion. (cid:3) In Paragraph 1.4, we have described how to turn the coproduct into thecoopposite coproduct. But we can also simultaneously turn the product intothe opposite product. In this way, we arrive at the opposite and cooppositequasi-Hopf algebra A op cop , which is again a quasi-Hopf algebra with respect tothe following structure elements: Its counit and antipode are unchanged, but itsassociator is P i Z i ⊗ Y i ⊗ X i , its evaluation element is β , and its coevaluationelement is α (cf. [4], Rem. 4, p. 1424; [6], Exerc. XV.6.2, p. 381). Furthermore,if A was quasitriangular, then A op cop is still quasitriangular with respect to thesame R-matrix. Therefore, its Drinfel’d element is˜ u := X i,l ¯ Z i s l βS ( t l ) S ( S ( ¯ X i ) α ¯ Y i )16ll the elements that we have introduced in Paragraph 2.1 can also be formedin A op cop . But it turns out that we do not get any new elements in this way;rather these elements coincide with other elements formed in A . For example,the element ˆ α , if formed in A op cop , is equal to the original element ˇ β as formedin A . The following table indicates which elements formed in A op cop are equalto which elements formed in A :In A op cop ˆ α ˆ β ˇ α ˇ β ˆ u ˇ u In A ˇ β ˇ α ˆ β ˆ α ˇ u − ˆ u − These correspondences can be applied to prove the following fact:
Proposition u = S (˜ u ) Proof.
By Proposition 2.1, we have u = ˇ u . In A op cop , this means ˜ u = ˆ u − .But we have already seen in Proposition 2.1 that ˇ u = S (ˆ u − ). (cid:3) This result can also be proved by direct computation, which is quite tedious.However, there is another comparatively short proof: The result follows fromLemma 1.6, because A op cop is isomorphic to a twist of A by [4], Prop. 1.2,p. 1426. Let us explain this in greater detail. The element γ introduced inParagraph 1.2 satisfies ( ε ⊗ id)( γ ) = (id ⊗ ε )( γ ) = ε ( α ) α by the counit-associatorproperty. It then follows from the duality axiom that the element F , which wehave also defined there, satisfies ( ε ⊗ id)( F ) = (id ⊗ ε )( F ) = ε ( α )1, so that theelement T := ε ( β ) F satisfies the requirement ( ε ⊗ id)( T ) = (id ⊗ ε )( T ) = 1imposed in Paragraph 1.6; recall that ε ( α ) ε ( β ) = 1 by Lemma 1.2. As explainedin [4], loc. cit., the compatibility conditions stated in Proposition 1.2 now yieldthat the antipode, considered as a map from A op cop to A T , is a quasi-bialgebraisomorphism. However, it is not a quasi-Hopf algebra isomorphism; we ratherhave S ( β ) = ε ( β ) α T S ( α ) = ε ( α ) β T as we see from [2], Eq. (2.14), p. 665 via a small correction. This means thatthe antipode becomes a quasi-Hopf algebra morphism if the evaluation elementand the coevaluation element of A T are adjusted as indicated in Paragraph 1.3,using the element x := ε ( β ) A op cop to the Drinfel’d element of A T , with the adjustments just indicated.By Proposition 1.5, this means in formulas that S (˜ u ) = xS ( x − ) u T . However, wehave S ( x ) = x in our case, and therefore S (˜ u ) = u T . But u T = u by Lemma 1.6,which completes the second derivation of our proposition above.17 eferenceseferences