Some ribbon elements for the quasi-Hopf algebra D ω (H)
aa r X i v : . [ m a t h . QA ] D ec SOME RIBBON ELEMENTS FOR THE QUASI-HOPF ALGEBRA D ω ( H ) DANIEL BULACU AND FLORIN PANAITE
Abstract.
We construct an explicit isomorphism between the quasitriangular quasi-Hopf alge-bra D ω ( H ) defined in [6] and a certain quantum double quasi-Hopf algebra. We give also newcharacterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to constructsome ribbon elements for D ω ( H ). Introduction
Dijkgraaf, Pasquier and Roche constructed in [11], from a finite group G and a normalized 3-cocycle ω on it, the famous quasi-Hopf algebra D ω ( G ) (called the twisted quantum double of G ). Theimportance of this construction stems from the fact that the irreducible representations of D ω ( G )allow to recover the fusion rules, the S matrix and the conformal weights of a certain RationalConformal Field Theory described in [12]. On the other hand, ribbon (quasi-) quantum groups giverise to topological invariant of knots and links: following the constructions of Reshetikhin and Turaev[23, 24], for such an algebra one can define regular isotopy invariants of coloured ribbon graphs, thecolours being finite-dimensional representations. This result was applied to the ribbon quasi-Hopfalgebra D ω ( G ) in [1] by considering surgery on the ribbon graphs coloured by a representation of D ω ( G ), leading thus to a 3-manifold invariant.In [6] we generalized the construction of D ω ( G ) to an arbitrary finite-dimensional cocommutativeHopf algebra H and ω : H ⊗ H ⊗ H → k a normalized 3-cocycle in the cohomology of commutativealgebras over cocommutative Hopf algebras introduced by Sweedler in [26]. We denoted this newquasi-Hopf algebra by D ω ( H ) and showed that D ω ( H ) is always a quasitriangular (QT for short)quasi-Hopf algebra. We expected D ω ( H ) to always be ribbon, as D ω ( G ) is, but we were able toprovide a ribbon element for it only if an extra condition is satisfied; see [6, Proposition 3.3]. Thegoal of this paper is to overcome this problem and to explain the meaning of the extra conditionrequired in [6]. Towards this end, we identify D ω ( H ) with the quantum double (in the sense ofHausser and Nill [14, 15]) of H ∗ ω , the finite-dimensional quasi-Hopf algebra introduced in [21]; seeTheorem 4.2. This gives us for free the QT quasi-Hopf algebra structure of D ω ( H ) as well asthe modular elements of D ω ( H ) in terms of the modular elements of H . The latter occur in thecomputation of the ribbon elements for D ω ( H ), owing to Theorem 3.1 and its Corollary 3.3 below,and the fact that D ω ( H ) is unimodular. It then comes out (see Theorem 5.3) that to any square root ζ in G ( H ∗ ) of the modular element µ H we can associate a ribbon element for D ω ( H ), where G ( H ∗ )stands for the group of grouplike elements of H ∗ , that is algebra maps from H to k . Situationswhen such an element ζ exists are uncovered at the end of Section 5, for instance when H has odddimension or G ( H ∗ ) is of odd order, or when the characteristic of k does not divide the dimensionof H (and consequently when k has characteristic zero). Example 5.6 says that the extra conditionin [6, Proposition 3.3] that guarantees a ribbon element for D ω ( H ) is satisfied if and only if H is unimodular. As the Hopf group algebra k [ G ] is always unimodular, this explains why for thequasi-Hopf algebra D ω ( G ) one can always construct a ribbon element. We should also mention here Mathematics Subject Classification.
Key words and phrases.
Quasi-Hopf algebra; twisted quantum double; ribbon element. that our approach provides new ribbon elements even for some of the quasi-Hopf algebras D ω ( G ),and thus new 3-manifold invariants as in [1].2. Preliminaries
We present, briefly, the definition an the basic properties of a (quasitriangular) quasi-Hopf alge-bra. For more information we refer to [13] or [18], [19]. We work over a field k . All algebras, linearspaces, etc. will be over k ; unadorned ⊗ means ⊗ k .A quasi-bialgebra is a 4-tuple ( H, ∆ , ε, Φ), where H is an associative algebra with unit 1 H , Φ is aninvertible element in H ⊗ H ⊗ H , and ∆ : H → H ⊗ H and ε : H → k are algebra homomorphismssuch that ∆ is coassociative up to conjugation by Φ and ε is counit for ∆; furthermore, Φ is anormalized 3-cocycle.In what follows we denote ∆( h ) = h ⊗ h , for all h ∈ H , the tensor components of Φ by capitalletters and the ones of Φ − by lower case letters. H is called a quasi-Hopf algebra if, moreover, there exists an anti-morphism S of the algebra H and elements α, β ∈ H such that, for all h ∈ H , we have: S ( h ) αh = ε ( h ) α and h βS ( h ) = ε ( h ) β, (2.1) X βS ( X ) αX = 1 H and S ( x ) αx βS ( x ) = 1 H . (2.2)A quasi-Hopf algebra with Φ = 1 H ⊗ H ⊗ H and α = β = 1 H is an ordinary Hopf algebra.For a quasi-Hopf algebra H we introduce the following elements in H ⊗ H :(2.3) p R = p ⊗ p := x ⊗ x βS ( x ) and q R = q ⊗ q := X ⊗ S − ( αX ) X . Note that our definition of a quasi-Hopf algebra is different from the one given by Drinfeld [13]in the sense that we do not require the antipode to be bijective. Anyway, the bijectivity of theantipode S will be implicitly understood in the case when S − , the inverse of S , appears is formulasor computations. According to [3], S is always bijective, provided that H is finite dimensional.The antipode of a Hopf algebra is an anti-morphism of coalgebras. For a quasi-Hopf algebra H there is an invertible element f = f ⊗ f ∈ H ⊗ H , called the Drinfeld twist, such that ε ( f ) f = ε ( f ) f = 1 H and f ∆( S ( h )) f − = ( S ⊗ S )(∆ cop ( h )), for all h ∈ H , where ∆ cop ( h ) = h ⊗ h .For H a quasi-bialgebra, the category of left H -modules H M is monoidal (see [18, Chapter XV]for the definition of a monoidal category). The tensor product is defined by ⊗ endowed with the H -module structure given by ∆ and unit object k , considered as an H -module via ε ; the associativityconstraint is determined by Φ and the left and right unit constraints are given by the canonoicalisomorphisms in the category of k -vector spaces.A quasi-bialgebra H is called quasitriangular (QT for short) if, moreover, the category H M isbraided in the sense of [18, Definition XIII.1.1]. This is equivalent to the existence of an invertibleelement R = R ⊗ R ∈ H ⊗ H (formal notation, summation implicitly understood), called R -matrix,satisfying certain conditions. Record that any R -matrix obeys(2.4) ε ( R ) R = ε ( R ) R = 1 H . When we refer to a QT quasi-bialgebra or quasi-Hopf algebra we always indicate the R -matrix R that produces the QT structure by pointing out the couple ( H, R ).For (
H, R ) a QT quasi-Hopf algebra, u is the element of H defined by(2.5) u = S ( R x βS ( x )) αR x . By [5], u is an invertible element of H and the following equalities hold ( R := R ⊗ R ): S ( h ) = uhu − , ∀ h ∈ H, (2.6) S ( α ) u = S ( R ) αR , (2.7) ∆( u ) = f − ( S ⊗ S )( f )( u ⊗ u )( R R ) − . (2.8) IBBON QUASI-HOPF ALGEBRAS 3 Ribbon quasi-Hopf algebras
The definition of a ribbon quasi-Hopf algebra is designed in such a way that the category H M fd offinite-dimensional left H -modules is a ribbon category as in [18, Definition XIV.3.2]. More exactly,by [7] a QT quasi-Hopf algebra ( H, R ) is ribbon if there exists an invertible central element ν ∈ H such that(3.9) ∆( ν ) = ( ν ⊗ ν )( R R ) − and S ( ν ) = ν. We can provide the following characterization for ribbon quasi-Hopf algebras. Note that (3.10)below was proved for the first time in [7, Proposition 5.5] under the condition α invertible; that itworks in general was proved in [25, Section 2.3]. In what follows we present an easy proof of this,based mostly on the arguments used in [7]. Theorem 3.1.
A QT quasi-Hopf algebra ( H, R ) is ribbon if and only if there exists a central element ν ∈ H (called ribbon element) such that the conditions in (3.9) hold and (3.10) ν = uS ( u ) , where, as before, u is the element defined in (2.5).Proof. Suppose that (
H, R ) is ribbon and let ν ∈ H be an invertible central element such thatthe conditions in (3.9) are satisfied. To prove that (3.10) is satisfied as well, observe first that byapplying ε ⊗ ε to both sides of the first equality in (3.9) we get ε ( ν ) = 1; see (2.4). Denote η := ν − and restate the first equality in (3.9) as ∆( η ) = ( η ⊗ η ) R R . Thus, by this equality and the factthat η is a central element in H we deduce that α = ε ( η ) α = S ( η ) αη = S ( ηR r ) αηR r . ) = η S ( R r ) αR r . ) = η S ( αr ) ur . ) = η S ( S ( r ) αr ) u ( . ) = η S ( S ( α ) u ) u ( . ) = η S ( u ) uα. This fact allows to compute, for all
A, B ∈ H : AαB = Aη S ( u ) uαB = η S ( uS − ( A )) uαB ( . ) = η S ( u ) S ( A ) uαB ( . ) = η S ( u ) uAαB. In particular, by taking A ⊗ B = S ( x ) ⊗ x βS ( x ), by (2.2) we conclude that 1 H = η S ( u ) u , andtherefore ν = η − = S ( u ) u , as needed. Note that S ( u ) u = uS ( u ) is a central element of H .Conversely, let ν be a central element of H such that the relations in (3.9) and (3.10) are fulfilled.Then ν is invertible because so is u , and therefore ν is an invertible central element of H . It followsnow that ( H, R, ν ) is a ribbon quasi-Hopf algebra. (cid:3)
It is a well established result that ribbon categories are pivotal (or, equivalently, sovereign)braided categories satisfying a certain condition related to some canonical twists; see [17, PropositionA.4]. In the case when the category is H M fd , with ( H, R, ν ) a ribbon quasi-Hopf algebra as inTheorem 3.1, this result is encoded in [6, Theorem 3.5]. It asserts that l ul defines a one toone correspondence between { l ∈ G ( H ) | l = u − S ( u ) and S ( h ) = l − hl, ∀ h ∈ H } and centralinvertible elements ν ∈ H satisfying (3.9) and (3.10), i.e. ribbon elements of H . Here G ( H ) = { l ∈ H | l is invertible with l − = S ( l ) and ∆( l ) = ( l ⊗ l )( S ⊗ S )( f − ) f } . Due to the proof of [10, Proposition 3.12], the conditions l invertible and l − = S ( l ) in thedefinition of an element l ∈ G ( H ) are redundant, provided that lS ( h ) = hl , for all h ∈ H .Otherwise stated, we have the following. Corollary 3.2.
Let ( H, R ) be a QT quasi-Hopf algebra and u the element defined in (2.5). Then l ul defines a one to one correspondence between R ( H ) := { l ∈ H | l = u − S ( u ) , ∆( l ) = ( l ⊗ l )( S ⊗ S )( f − ) f and lS ( h ) = hl, ∀ h ∈ H } and ribbon structures ν on H as in Theorem 3.1. DANIEL BULACU AND FLORIN PANAITE
We end this section by pointing out that in the finite-dimensional case the element u − S ( u ) canbe computed in terms of the modular elements g ∈ H and µ ∈ H ∗ , and the R -matrix R = R ⊗ R of H (we refer to [16] for the definitions of g and µ ). Namely, by the formula (6.21) in [9] we have(3.11) u − S ( u ) = µ ( X R p S ( X ) f ) S − ( S ( X ) f ) R p S ( g − ) , where g − is the inverse of g in H and p R = p ⊗ p is the element defined in (2.3). Corollary 3.3. If ( H, R ) is a finite-dimensional unimodular QT quasi-Hopf algebra and u is as in(2.5) then l ul defines a one to one correspondence between { l ∈ H | l = g , ∆( l ) = ( l ⊗ l )( S ⊗ S )( f − ) f and lS ( h ) = hl, ∀ h ∈ H } and ribbon elements of H , where g is the modular element of H .Proof. When H is unimodular, i.e. µ = ε , the formula in (3.11) gives the equality u − S ( u ) = S ( g − ).As uS ( u ) = S ( u ) u and S ( u ) = u this implies S ( g − ) = S ( u − S ( u )) = S ( u ) S ( u − ) = uS ( u ) − = S ( u ) − u = ( u − S ( u )) − = S ( g ) . By using the bijectivity of S , we obtain that S ( g − ) = g , and so u − S ( u ) = g . Now everythingfollows from Corollary 3.2. (cid:3) The quasi-Hopf algebras D ω ( H ) and D ω ( G )Let H be a cocommutative Hopf algebra with antipode S over a base field k . As H is cocommu-tative, we can introduce a simplified version of Sweedler’s sigma notation: for h ∈ H , we denote∆( h ) = h ⊗ h, (Id H ⊗ ∆)(∆( h )) = (∆ ⊗ Id H )(∆( h )) = h ⊗ h ⊗ h, and so on. With this notation, the antipode and counit axioms read: S ( h ) h = hS ( h ) = ε ( h )1 H , ε ( h ) h = hε ( h ) = h. We recall some facts concerning Hopf crossed products and cohomology (see [26]). Let H be acocommutative Hopf algebra and A a commutative left H -module algebra, with H -action denotedby H ⊗ A → A , h ⊗ a h · a . Assume that we are given a linear map σ : H ⊗ H → A , whichis normalized (that is, σ (1 H , h ) = σ ( h, H ) = ε ( h )1 A for all h ∈ H ) and convolution invertible.Suppose that, moreover, σ satisfies the 2-cocycle condition:(4.1) σ ( x, y ) σ ( xy, z ) = [ x · σ ( y, z )] σ ( x, yz ) , ∀ x, y, z ∈ H. Then, if we define a multiplication on A ⊗ H by ( a h )( b g ) = a ( h · b ) σ ( h, g ) hg (for a ∈ A and h ∈ H we write a h in place of a ⊗ h in order to distinguish this structure), this multiplicationis associative and 1 A ⊗ H is a unit. We denote A ⊗ H with this algebra structure by A σ H andcalled it the Hopf crossed product of A and H .From now on, for the rest of this section, we assume that H is a finite dimensional cocommutativeHopf algebra. Thus, H ∗ is a commutative Hopf algebra with unit ε , counit ε ( ϕ ) = ϕ (1 H ), multi-plication ( ϕψ )( h ) = ϕ ( h ) ψ ( h ), comultiplication ∆( ϕ ) = ϕ ⊗ ϕ if and only if ϕ ( hg ) = ϕ ( h ) ϕ ( g ),and antipode S ( ϕ ) = ϕ ◦ S , where ϕ, ψ ∈ H ∗ are arbitrary as well as h, g ∈ H .Assume that we are given a k -linear map ω : H ⊗ H ⊗ H → k that is convolution invertible andsatisfies the conditions: ω ( x, y, zt ) ω ( xy, z, t ) = ω ( y, z, t ) ω ( x, yz, t ) ω ( x, y, z ) , ∀ x, y, z, t ∈ H, (4.2) ω (1 H , x, y ) = ω ( x, H , y ) = ω ( x, y, H ) = ε ( x ) ε ( y ) , ∀ x, y ∈ H. (4.3)Note that such a map ω is noting but a normalized 3-cocycle in the Sweedler cohomology of H withcoefficients in k , as defined in [26].Since H is finite dimensional, we can identify ( H ⊗ H ⊗ H ) ∗ with H ∗ ⊗ H ∗ ⊗ H ∗ , so we can regard ω ∈ H ∗ ⊗ H ∗ ⊗ H ∗ ; we denote ω = ω ⊗ ω ⊗ ω and its convolution inverse ω − = ω ⊗ ω ⊗ ω . IBBON QUASI-HOPF ALGEBRAS 5
We define the element Φ ∈ H ∗ ⊗ H ∗ ⊗ H ∗ by Φ := ω − = ω ⊗ ω ⊗ ω . Since H ∗ is a commutativealgebra, ( H ∗ , ∆ , ε, Φ) is a quasi-bialgebra, where ∆ and ε are the ones that give the usual coalgebrastructure of H ∗ (dual to the algebra structure of H ). Moreover, if we define β ∈ H ∗ by the formula β ( h ) = ω ( h, S ( h ) , h ), then ( H ∗ , ∆ , ε, Φ , S, α = ε, β ) is a quasi-Hopf algebra, which will be denotedby H ∗ ω , see [21].We can consider the diagonal crossed product ( H ∗ ω ) ∗ ⊲⊳ H ∗ ω as in [8, 14]. On the other hand, wewill construct a certain Hopf crossed product H ∗ σ H as follows, see [6].We introduce first the following notation: g ⊳ x = S ( x ) gx , for all g, x ∈ H . Next, we define thelinear map θ : H ⊗ H ⊗ H → k , by θ ( g ; x, y ) = ω ( g, x, y ) ω ( x, y, g ⊳ ( xy )) ω − ( x, g ⊳ x, y ) , (4.4)for all g, x, y ∈ H , where ω − is the convolution inverse of ω .It is easy to see that θ is also normalized and convolution invertible. By [6], we have θ ( g ; x, y ) θ ( g ; xy, z ) = θ ( g ⊳ x ; y, z ) θ ( g ; x, yz ) , ∀ g, x, y, z ∈ H. (4.5)Since H is cocommutative, H ∗ becomes a commutative left H − module algebra, with action H ⊗ H ∗ → H ∗ , h ⊗ ϕ h • ϕ , where h • ϕ = h ⇀ ϕ ↼ S ( h ), where we denoted by ⇀ and ↼ theleft and right regular actions of H on H ∗ given by ( h ⇀ ϕ )( a ) = ϕ ( ah ) and ( ϕ ↼ h )( a ) = ϕ ( ha )for all h, a ∈ H and ϕ ∈ H ∗ . Hence, ( h • ϕ )( a ) = ϕ ( a ⊳ h ) for all h, a ∈ H and ϕ ∈ H ∗ .Define now the linear map σ : H ⊗ H → H ∗ by σ ( x, y )( g ) = θ ( g ; x, y ). Since θ is normalizedand convolution invertible, σ is also normalized and convolution invertible, and the relation (4.5) isequivalent to the fact that σ is a 2-cocycle, that is (4.1) holds if we replace the action · by • . Hence,we can consider the Hopf crossed product H ∗ σ H , denoted by D ω ( H ), which is an associativealgebra with unit ε H . Its multiplication is given, for all ϕ, ϕ ′ ∈ H ∗ , h, h ′ ∈ H , by( ϕ ⊗ h )( ϕ ′ ⊗ h ′ ) = ϕ ( h ⇀ ϕ ′ ↼ S ( h )) σ ( h, h ′ ) ⊗ hh ′ . (4.6) Theorem 4.1.
The linear map w : ( H ∗ ω ) ∗ ⊲⊳ H ∗ ω → H ∗ σ H defined by w ( h ⊲⊳ ϕ ) = ω ( h ) ω ( S ( h )) ω ( h ⇀ ϕ ↼ S ( h )) h, ∀ h ∈ H, ϕ ∈ H ∗ , (4.7) is an algebra isomorphism, with inverse W : H ∗ σ H → ( H ∗ ω ) ∗ ⊲⊳ H ∗ ω given by W ( ϕ h ) = p ( h ) p ( S ( h ))( ϕ ⇀ h ↼ S ( ϕ )) ⊲⊳ p ϕ , ∀ ϕ ∈ H ∗ , h ∈ H, (4.8) where we denoted by p ⊗ p = x ⊗ x βS ( x ) the element for H ∗ ω given by (2.3) and by ⇀ and ↼ the regular actions of H on H ∗ and of H ∗ on H .Proof. We will construct the map w by using the Universal Property of the diagonal crossed product([2, Proposition 8.2]). We define the linear maps γ : H ∗ ω → H ∗ σ H, γ ( ϕ ) = ϕ H ,v : H = ( H ∗ ω ) ∗ → H ∗ σ H, v ( h ) = ε h. One can easily see that γ is an algebra map and the relations (8.2) and (8.4) in [2, Proposition 8.2]are satisfied, that is we have γ ( ϕ ) v ( h ↼ ϕ ) = v ( ϕ ⇀ h ) γ ( ϕ ) , v (1 H ) = ε H , ∀ ϕ ∈ H ∗ , h ∈ H. So the only thing left to prove is the relation (8.3) in [2, Proposition 8.2], namely ε hh ′ = ( ω H )( ε ω ω ⇀ h ↼ ω )( ω H )( ε ω ⇀ h ′ ↼ ω ω )( ω H ) , where we denoted by ω − = ω ⊗ ω ⊗ ω another copy of ω − . We compute:( ω H )( ε ω ω ⇀ h ↼ ω )( ω H )( ε ω ⇀ h ′ ↼ ω ω )( ω H )= ( ω ω ω ⇀ h ↼ ω )( ω ω ⇀ h ′ ↼ ω ω )( ω H )= ω ( h ) ω ( h ) ω ( h ) ω ( h ′ ) ω ( h ′ ) ω ( h ′ )( ω h )( ω ( h ′ ⇀ ω ↼ S ( h ′ )) h ′ ) DANIEL BULACU AND FLORIN PANAITE = ω ( h ) ω ( h ) ω ( h ) ω ( h ′ ) ω ( h ′ ) ω ( h ′ )( ω ( h ⇀ ω ↼ S ( h ))( hh ′ ⇀ ω ↼ S ( hh ′ )) σ ( h, h ′ ) hh ′ ) . When we evaluate this in g ⊗ ϕ ∈ H ⊗ H ∗ we obtain: ω ( h, S ( h ) gh, h ′ ) ω − ( h, h ′ , S ( hh ′ ) ghh ′ ) ω − ( g, h, h ′ ) θ ( g ; h, h ′ ) ϕ ( hh ′ )= ω ( h, S ( h ) gh, h ′ ) ω − ( h, h ′ , S ( hh ′ ) ghh ′ ) ω − ( g, h, h ′ ) ω ( g, h, h ′ ) ω ( h, h ′ , S ( hh ′ ) ghh ′ ) ω − ( h, S ( h ) gh, h ′ ) ϕ ( hh ′ )= ε ( g ) ϕ ( hh ′ ) = ( ε hh ′ )( g ⊗ ϕ ) , q.e.d. Thus, Proposition 8.2 from [2] yields an algebra map w : ( H ∗ ω ) ∗ ⊲⊳ H ∗ ω → H ∗ σ H , defined by w ( h ⊲⊳ ϕ ) = γ ( q ) v ( h ↼ q ) γ ( ϕ ) , ∀ ϕ ∈ H ∗ , h ∈ H, where q ⊗ q = X ⊗ S − ( αX ) X is the element for H ∗ ω given by (2.3). An easy computationshows that this map is identical to the one given by (4.7).To prove that w is bijective with inverse W , since the underlying vector spaces have the same(finite) dimension, it is enough to prove that w ◦ W = Id. We compute: w ( W ( ϕ h )) = w ( p ( h ) p ( S ( h )) ϕ ( h ) ϕ ( S ( h )) h ⊲⊳ p ϕ )= p ( h ) p ( S ( h )) ϕ ( h ) ϕ ( S ( h )) ω ( h ) ω ( S ( h )) ω ( h ⇀ p ϕ ↼ S ( h )) h. When we evaluate this in g ⊗ ψ ∈ H ⊗ H ∗ we obtain: p ( h ) p ( S ( h )) ϕ ( h ) ϕ ( S ( h )) ω ( h ) ω ( S ( h )) p ( S ( h ) gh ) ϕ ( S ( h ) gh ) ω ( g ) ψ ( h )= p ( gh ) p ( S ( h )) ϕ ( g ) ω − ( g, h, S ( h )) ψ ( h )= ω ( gh ) ω ( S ( h )) β ( S ( h )) ω ( h ) ω − ( g, h, S ( h )) ϕ ( g ) ψ ( h )= ω ( gh, S ( h ) , h ) ω ( S ( h ) , h, S ( h )) ω − ( g, h, S ( h )) ϕ ( g ) ψ ( h ) . To finish the proof it will be enough to prove that ω ( gh, S ( h ) , h ) ω ( S ( h ) , h, S ( h )) ω − ( g, h, S ( h )) = ε ( g ) ε ( h ) . The 3-cocycle condition for ω applied to the elements x = h , y = S ( h ), z = h , t = S ( h ) yields ω ( S ( h ) , h, S ( h )) = ω − ( h, S ( h ) , h ) . (4.9)So it is enough to prove that ω ( gh, S ( h ) , h ) ω − ( h, S ( h ) , h ) ω − ( g, h, S ( h )) = ε ( g ) ε ( h ) . But this relation follows immediately by applying the 3-cocycle condition for ω to the elements x = g , y = h , z = S ( h ), t = h . (cid:3) The quantum double D ( H ∗ ω ) of the quasi-Hopf algebra H ∗ ω has as underlying algebra structurethe diagonal crossed product ( H ∗ ω ) ∗ ⊲⊳ H ∗ ω , so Theorem 4.1 implies: Theorem 4.2. D ω ( H ) = H ∗ σ H is a QT quasi-Hopf algebra. It turns out that the QT quasi-Hopf algebra structure otained on D ω ( H ) by transferring thestructure from D ( H ∗ ω ) via the isomorphism (4.7) coincides with the one introduced in [6]. Namely: • the reassociator: Φ = ( ω H ) ⊗ ( ω H ) ⊗ ( ω H ) ∈ D ω ( H ) ⊗ D ω ( H ) ⊗ D ω ( H ). • the comultiplication: define the linear map γ : H ⊗ H ⊗ H → k by γ ( g, h ; x ) = ω ( g, h, x ) ω ( x, g ⊳ x, h ⊳ x ) ω − ( g, x, h ⊳ x ) , IBBON QUASI-HOPF ALGEBRAS 7 for all g, h, x ∈ H . Then define the linear map ν : H → ( H ⊗ H ) ∗ , ν ( h )( x ⊗ y ) = γ ( x, y ; h ).Identifying ( H ⊗ H ) ∗ with H ∗ ⊗ H ∗ , we will write, for any h ∈ H , ν ( h ) = ν ( h ) ⊗ ν ( h ) ∈ H ∗ ⊗ H ∗ .Then the comultiplication of D ω ( H ) is defined, for all ϕ ∈ H ∗ , h ∈ H , by∆ : D ω ( H ) → D ω ( H ) ⊗ D ω ( H ) , ∆( ϕ h ) = ( ν ( h ) ϕ h ) ⊗ ( ν ( h ) ϕ h ) . (4.10) • the counit: ε : D ω ( H ) → k , ε ( ϕ h ) = ϕ (1 H ) ε ( h ), for all ϕ ∈ H ∗ , h ∈ H . • the antipode: α D ω ( H ) = ε H , β D ω ( H ) = β H and s : D ω ( H ) → D ω ( H ) given by s ( ϕ h ) = [ ε S ( h )][ σ − ( h, S ( h )) S ( ϕν − ( h )) ν − ( h ) H ] , for all ϕ ∈ H ∗ , h ∈ H , where we denoted by ν − the convolution inverse of ν , with notation ν − ( h ) = ν − ( h ) ⊗ ν − ( h ) ∈ H ∗ ⊗ H ∗ . • the R -matrix: R = n X i =1 ( e i H ) ⊗ ( ε e i ) ∈ D ω ( H ) ⊗ D ω ( H ) , where { e i , e i } i are dual bases in H and H ∗ .Moreover, by [6], β is convolution invertible with inverse β − = β ◦ S and we have the relation:(4.11) s ( ϕ h ) = ( β − H )( ϕ h )( β H ) , ∀ ϕ ∈ H ∗ , h ∈ H. Let now G be a finite group, with multiplication denoted by juxtaposition and unit denotedby e . Let ω be a normalized 3-cocycle on G , i.e. ω : G × G × G → k ∗ is a map such that ω ( x, y, z ) ω ( tx, y, z ) − ω ( t, xy, z ) ω ( t, x, yz ) − ω ( t, x, y ) = 1 for all t, x, y, z ∈ G , and ω ( x, y, z ) = 1whenever x , y or z is equal to e . We can take H = k [ G ], the group algebra of G , which is a finitedimensional cocommutative Hopf algebra, and extend ω by linearity to a map ω : H ⊗ H ⊗ H → k ,which turns out to be a Sweedler 3-cocycle on H . So, we can consider the QT quasi-Hopf algebra D ω ( H ), which will be denoted by D ω ( G ). This QT quasi-Hopf algebra structure of D ω ( G ) wasintroduced in [11]. 5. Some ribbon elements for D ω ( H ) and D ω ( G )Let H be a finite dimensional cocommutative Hopf algebra, ω a normalized 3-cocycle on H and D ω ( H ) the quasi-Hopf algebra constructed in Section 4. So D ω ( H ) is a QT quasi-Hopf algebraisomorphic to the quantum double D ( H ∗ ω ).In what follows, in order to avoid any confusion, we denote by µ H ∈ H ∗ and g H ∈ H the modularelements of H as a Hopf algebra. Similar notation, µ H ∗ ω ∈ H and g H ∗ ω ∈ H ∗ , is used for the modularelements of the quasi-Hopf algebra H ∗ ω . As H is cocommutative it follows that g H = µ H ∗ ω = 1 H , i.e. H ∗ and H ∗ ω are unimodular as Hopf and respectively quasi-Hopf algebras. Also, it is clear that a left(and at the same time right) integral in the quasi-Hopf algebra H ∗ ω is nothing but a left (and at thesame time right) integral on the Hopf algebra H , i.e. an element λ ∈ H ∗ obeying λ ( h ) h = λ ( h )1 H ,for all h ∈ H . Finally, K will always denote a finite dimensional quasi-Hopf algebra and H a finitedimensional cocommutative Hopf algebra.In this section we show that the element ν = u ( ζ H ) β D ω ( H ) = u ( ζβ H ) is a ribbon elementfor D ω ( H ), provided that ζ : H → k is an algebra map such that ζ = µ H ; here, as before, u isthe element in (2.5) corresponding to D ω ( H ) and the other notation is as in Section 4. Note thatwhen dim k H = 0 in k or H is unimodular we can take ζ = ε , and therefore D ω ( H ) is ribbon withribbon element ν = u ( β H ). This applies for instance to a finite dimensional Hopf group algebra H = k [ G ], and so D ω ( G ) is always a ribbon quasi-Hopf algebra.To this end, we start by describing the space of left cointegrals on H ∗ ω . By the comments madeafter the proof of [4, Theorem 3.7] we have that a non-zero left cointegral on a quasi-Hopf algebra K for which α, β are invertible elements is a non-zero morphism λ ∈ K ∗ satisfying λ ( t ) t = DANIEL BULACU AND FLORIN PANAITE λ ( t ) βS − ( α ), for any left integral t in K (in [4] the assumption β invertible is omitted and in placeof S − ( α ) appears α ; we correct these facts now). Lemma 5.1.
Let t ∈ H be a non-zero left integral in H , i.e. ht = ε ( h ) t , for all h ∈ H . Then t := β ( S ( t )) t ∈ H is a non-zero left cointegral for H ∗ ω .Proof. For the quasi-Hopf algebra H ∗ ω the elements α, β are invertible, so a non-zero left cointegralon H ∗ ω is a non-zero element t ∈ H satisfying λ ( h t ) = λ ( t ) β ( h ), for all h ∈ H .It is well known that for any non-zero left integral t in H the map H ∗ ∋ h ∗ h ∗ ( t ) t ∈ H isbijective (result valid for any finite-dimensional Hopf algebra, not necessarily cocommutative). Thuswe find a unique element h ∗ ∈ H ∗ such that h ∗ ( t ) t = t ; in particular, h ∗ ( t ) = 0. We have, for all h ∈ H , that h ∗ ( t ) λ ( ht ) = h ∗ ( t ) λ ( t ) β ( h ), which is equivalent to h ∗ ( S ( h ) ht ) λ ( ht ) = λ ( t ) h ∗ (1 H ) β ( h ),which in turn is equivalent to λ ( t ) h ∗ ( S ( h )) = λ ( t ) h ∗ (1 H ) β ( h ).As λ ( t ) = 0, it follows that h ∗ = h ∗ (1 H ) β ◦ S , which implies β ( S ( t )) = 0. Therefore, by rescaling,we can assume without loss of generality that t = β ( S ( t )) t , as stated.Conversely, t = β ( S ( t )) t is a left cointegral on H ∗ ω since λ ( h t ) = β ( S ( t )) λ ( ht ) = β ( S ( ht ) h ) λ ( ht ) = β ( h ) λ ( ht ) = β ( h ) λ ( t ) , and this is equal to λ ( t ) β ( h ) because λ ( t ) β ( h ) = β ( S ( t )) λ ( t ) β ( h ) = λ ( t ) β ( S (1 H )) β ( h ) = λ ( t ) β ( h ) . So our proof ends. (cid:3)
For a finite-dimensional quasi-Hopf algebra K the modular element g ∈ K is defined by g = λ ( S − ( q t p )) q t p , where λ ∈ K ∗ is a left cointegral, t ∈ K is a left integral such that λ ( S − ( t )) =1 and p R = p ⊗ p and q R = q ⊗ q are the elements defined in (2.3). Corollary 5.2.
We have that g H ∗ ω = β µ H . Consequently, the modular element g D ( H ∗ ω ) of thequantum double D ( H ∗ ω ) equals H ⋊⋉ β µ H .Proof. Recall that µ H is defined by th = µ H ( h ) t , for all h ∈ H and t ∈ H a non-zero left integral.Also, since H ∗ is unimodular we have λ ◦ S = λ . Thus, by specializing the above definition of g to H ∗ ω we compute, for all h ∈ H , that g H ∗ ω = q ( S ( t )) p ( S ( t )) λ ( S ( t h )) q ( S ( h )) p ( S ( h ))= ω ( S ( h ) , S ( t ) , t ) β ( S ( t )) ω − ( S ( h ) , t , S ( t )) λ ( S ( t h ))= ω ( S ( h ) , hS ( t h ) , t hS ( h )) β ( hS ( t h )) ω − ( S ( h ) , t hS ( h ) , hS ( t h )) λ ( t h )= ω ( S ( h ) , h, S ( h )) β ( h ) ω − ( S ( h ) , h, S ( h )) λ ( t h )= β ( h ) β ( S ( t )) λ ( th )= β ( h ) β ( hS ( th )) λ ( th )= β ( h ) µ H ( h ) λ ( t ) . But the pair ( λ, t ) obeys λ ( t ) = 1 or, equivalently, β ( S ( t )) λ ( t ) = 1. The latter is equivalent to λ ( t ) = 1, and so g H ∗ ω = β µ H , as desired.Finally, we have µ H ∗ ω = 1 H , hence the formula in [4, Proposition 5.9] yields g D ( H ∗ ω ) = 1 H ⋊⋉ g − H ∗ ω ◦ S − = 1 H ⋊⋉ g − H ∗ ω ◦ S = 1 H ⋊⋉ g H ∗ ω , since g − H ∗ ω = β − µ − H with β − = β ◦ S and µ − H = µ H ◦ S , and together with S = Id H this implies g − H ∗ ω ◦ S = g H ∗ ω . (cid:3) IBBON QUASI-HOPF ALGEBRAS 9
The computation performed before [6, Proposition 3.3] ensures that the distinguished element β D ω ( H ) = β H ∈ D ω ( H ) satisfies(5.1) ∆( β D ω ( H ) ) = ( β D ω ( H ) ⊗ β D ω ( H ) )( s ⊗ s )( f − ) f , where f ∈ D ω ( H ) ⊗ D ω ( H ) is the Drinfeld twist. Consequently, the same relation is satisfied byany element of the form ζβ H ∈ D ω ( H ), provided that ζ : H → k is an algebra map.We have now all the necessary ingredients in order to prove the following: Theorem 5.3.
Let H be a finite dimensional cocommutative Hopf algebra H , ω a normalized -cocycle on H and ζ : H → k an algebra map. Then the element ν = u ( ζβ H ) is ribbon if andonly if ζ = µ H .Proof. The quasi-Hopf algebra D ω ( H ) is unimodular; this follows from Theorem 4.1 and [9, Theorem6.5]. Furthermore, by Corollary 5.2 and the definition of the isomorphism w in (4.7) we can seethat the modular element g D ω ( H ) of D ω ( H ) is g D ω ( H ) = w ( g D ( H ∗ ω ) ) = g D ( H ∗ ω ) H = β µ H H . So, according to Corollary 3.3, it suffices to see when l := ζβ H ∈ D ω ( H ) satisfies the relations l = β µ H H , (5.2) ∆( l ) = ( l ⊗ l )( s ⊗ s )( f − ) f , (5.3) l S ( ϕ h ) = ( ϕ h ) l , ∀ ϕ h ∈ D ω ( H ) . (5.4)It can be easily checked that (5.2) is equivalent to ζ β = β µ H , and the latter is equivalent to ζ = µ H , because H ∗ is commutative and β is convolution invertible. Also, the comments madeafter (5.1) guarantees that (5.3) is always satisfied.We look at (5.4). By (4.11) we have that (5.4) is equivalent to ( ζ H )( ϕ h ) = ( ϕ h )( ζ H ),for all ϕ h ∈ D ω ( H ). The last equation becomes ζϕ h = ϕ ( h ⇀ ζ ↼ S ( h )) h , and it holds forany ϕ h ∈ D ω ( H ) since ζ is an algebra map and H ∗ is commutative. (cid:3) We end this section with some concrete examples. In what follows by G ( H ∗ ) := { ζ : H → k | ζ is an algebra map } we denote the set of grouplike elements of H ∗ , assuming, as before, that H isa finite-dimensional cocommutative Hopf algebra. G ( H ∗ ) is a group under convolution, and so thegroup Hopf algebra k [ G ( H ∗ )] is a Hopf subalgebra of H ∗ . By the freeness theorem proved in [20] itfollows that | G ( H ∗ ) | divides dim k ( H ). Example 5.4. If µ H has odd order in G ( H ∗ ) then D ω ( H ) is a ribbon quasi-Hopf algebra. Proof. If µ H has order 2 m + 1 in G ( H ∗ ) then ζ := µ m +1 H : H → k is an algebra map such that ζ = µ H . By Theorem 5.3 we obtain that u ( µ m +1 H β H ) is a ribbon element for D ω ( H ). (cid:3) Example 5.5.
Suppose that either dim k ( H ) or | G ( H ∗ ) | is an odd number. Then D ω ( H ) is aribbon quasi-Hopf algebra. Proof.
Since | G ( H ∗ ) | divides dim k ( H ), in either case we get that µ H has odd order, and soExample 5.4 applies. (cid:3) The next example shows that the condition in [6, Proposition 3.3] is equivalent to the unimodu-larity of H . Example 5.6.
The element ν = u ( β H ) is a ribbon element for D ω ( H ) if and only if H isunimodular. Proof.
Take ζ = ε in Theorem 5.3; we obtain that ν = u ( β H ) is a ribbon element if and only if µ H = ε , i.e. H is unimodular. (cid:3) The next example refers precisely to the ribbon structure on D ω ( G ) defined by (5.18) in [1]. Example 5.7.
The quasi-Hopf algebra D ω ( G ) is ribbon. Proof.
We have D ω ( G ) = D ω ( k [ G ]) and k [ G ] is unimodular. Indeed, t = P g ∈ G g is a left and rightintegral in k [ G ]. (cid:3) Example 5.8.
Suppose that dim k ( H ) = 0 in k (this happens for instance when k is of characteristiczero). Then ν = u ( β H ) is a ribbon element for D ω ( H ). Proof.
Since S = Id H , by the trace formula proved in [22, Proposition 2 (c)] we get that H issemisimple, and so unimodular, too. Hence ν = u ( β H ) is a ribbon element for D ω ( H ). (cid:3) References [1] D. Altschuler, A. Coste:
Quasi-quantum groups, knots, three manifolds and topological field theory , Comm.Math. Phys. (1992), 83–107.[2] H. Albuquerque, F. Panaite,
On quasi-Hopf smash products and twisted tensor products of quasialgebras , Algebr.Represent. Theory (2009), 199–234.[3] D. Bulacu, S. Caenepeel: Integrals for (dual) quasi-Hopf algebras. Applications , J. Algebra (2003), 552–583.[4] D. Bulacu, S. Caenepeel:
On integrals and cointegrals for quasi-Hopf algebras , J. Algebra (2012), 390–425.[5] D. Bulacu, E. Nauwelaerts:
Quasitriangular and ribbon quasi-Hopf algebras , Comm. Algebra (2003), 657–672.[6] D. Bulacu, F. Panaite: A generalization of the quasi-Hopf algebra D ω ( G ), Comm. Algebra (1998), 4125–4141.[7] D. Bulacu, F. Panaite, F. Van Oystaeyen: Quantum traces and quantum dimensions for quasi-Hopf algebras ,Comm. Algebra (1999), 6103–6122.[8] D. Bulacu, F. Panaite, F. Van Oystaeyen: Generalized diagonal crossed products and smash products for quasi-Hopf algebras. Applications , Comm. Math. Phys. , 355–399 (2006).[9] D. Bulacu, B. Torrecillas:
Factorizable quasi-Hopf algebras. Applications , J. Pure Appl. Algebra (2004),39–84.[10] D. Bulacu, B. Torrecillas:
On sovereign, balanced and ribbon quasi-Hopf algebras . Preprint 2018, submitted.[11] R. Dijkgraaf, V. Pasquier, P. Roche:
Quasi-Hopf algebras, group cohomology and orbifold models , Nuclear Phys.B. Proc. Suppl.
18 B (1990), 60–72.[12] R. Dijkgraaf, C. Vafa, E. Verlinde, H. Verlinde:
Operator algebra of orbifold models , Comm. Math. Phys. (1989), 485–526.[13] V. G. Drinfeld:
Quasi-Hopf algebras , Leningrad Math. J. (1990), 1419–1457.[14] F. Hausser, F. Nill: Diagonal crossed products by duals of quasi-quantum groups , Rev. Math. Phys. (1999),553–629.[15] F. Hausser, F. Nill: Doubles of quasi-quantum groups , Comm. Math. Phys. (1999), 547–589.[16] F. Hausser, F. Nill:
Integral theory for quasi-Hopf algebras , Unpublished 1999, arXiv:9904164.[17] A. Henriques, D. Penneys, J. Tener:
Categorified trace for module tensor categories over braided tensor cate-gories , Documenta Math. (2016), 1089-1149.[18] C. Kassel: Quantum Groups, Graduate Texts in Mathematics , Springer Verlag, Berlin, 1995.[19] S. Majid: Foundations of quantum group theory, Cambridge University Press, 1995.[20] W. D. Nichols, M. B. Zoeller: A Hopf algebra freeness theorem , Amer. J. Math. (1989), 381–385.[21] F. Panaite, F. Van Oystaeyen:
Quasi-Hopf algebras and the centre of a tensor category , in ”Hopf algebras andquantum groups” (eds. S. Caenepeel and F. Van Oystaeyen), 221–235, Lecture Notes in Pure and Appl. Math. , Marcel Dekker, New York, 2000.[22] D. E. Radford:
The trace function and Hopf algebras , J. Algebra (1994), 583–622.[23] N. Reshetikhin, V. G. Turaev:
Invariants of -manifolds via link polynomials and quantum groups , Invent.Math. (1991), 547–597.[24] N. Reshetikhin, V. G. Turaev: Ribbon graphs and their invariants derived from quantum groups , Comm. Math.Phys. (1990), 1–26.[25] Y. Sommerh¨auser:
On the notion of a ribbon quasi-Hopf algebra , Rev. Uni´on Mat. Argent. (2010), 177-192.[26] M. E. Sweedler: Cohomology of algebras over Hopf algebras , Trans. Amer. Math. Soc. , 205–239 (1968).
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