Quantum Lines for Dual Quasi-Bialgebras
aa r X i v : . [ m a t h . QA ] D ec QUANTUM LINES FOR DUAL QUASI-BIALGEBRAS
ALESSANDRO ARDIZZONI, MARGARET BEATTIE, AND CLAUDIA MENINI
Abstract.
In this paper, the theory to construct quantum lines for general dual quasi-bialgebrasis developed followed by some specific examples where the dual quasi-bialgebras are pointed withcyclic group of points.
Contents
1. Introduction 12. Preliminaries 22.1. Definitions 23. Quasi-Yetter-Drinfeld data for dual quasi-bialgebras 53.1. Yetter-Drinfeld modules 53.2. Quasi-Yetter-Drinfeld data 74. Quantum lines 85. Quasi-Yetter-Drinfeld data for bosonizations 156. Examples 186.1. Group cohomology 186.2. Quasi-
Y D data for k C n Y D data for R k C n Introduction
For H a bialgebra over a field k and R a bialgebra in the category of Yetter-Drinfeld modules HH YD , the Radford biproduct or bosonization R H is a well-known construction giving a newbialgebra. Similarly, if H is a dual quasi-bialgebra and R is a bialgebra in a suitably definedcategory of Yetter-Drinfeld modules over H , then by [AP], a new dual quasi-bialgebra R H canbe defined, called the bosonization of R and H .Given a bialgebra H , however, finding R is nontrivial. For H = k G , a group algebra, finding R is the key to constructing pointed bialgebras with G as the group of points and finding finitedimensional R is crucial to the classification of pointed Hopf algebras of finite dimension. If R isfinite dimensional and is generated as an algebra by a one-dimensional vector space in HH YD , then R is called a quantum line. For various H semisimple of even dimension, not necessarily groupalgebras, the question of the existence of a quantum line for H was completely settled in [CDMM]as well as the question of liftings of the bosonizations.In this paper, we adapt the methods and language of [CDMM] to dual quasi-bialgebras. We findnecessary and sufficient conditions for quantum lines to exist for a given dual quasi-bialgebra H and we compute several examples for the dual quasi-bialgebra ( H, ω ) where H is the group algebra Mathematics Subject Classification.
Primary 16W30; Secondary 16S40.
Key words and phrases.
Dual quasi-bialgebras; quantum lines; bosonizations.This paper was written while the first and the third authors were members of GNSAGA. The first authorwas partially supported by the research grant “Progetti di Eccellenza 2011/2012” from the “Fondazione Cassa diRisparmio di Padova e Rovigo”. The second author was supported by an NSERC Discovery Grant. Her stay, as avisiting professor at University of Ferrara in 2011, was supported by INdAM. of a cyclic group of any order and ω is a 3-cocycle. Finally we give an example of the existence ofa quantum line for a bosonization R k C n where n is an even integer.The duals of our examples will be quasi-bialgebras as studied in the papers of Angiono [An],Gelaki [Ge], and Etingof and Gelaki [EGe1][EGe2][EGe3].The first section of this paper is used for notation and some preliminary material. In the secondsection, we define quasi-Yetter-Drinfeld data for dual quasi-bialgebras, and then in the next sectionwe construct quantum lines. Section 5 discusses conditions to construct a quantum line for abosonization and then the last section gives examples of these constructions. The examples arebased on knowledge of the dual quasi-bialgebra k C n with Drinfeld reassociator given by a nontrivial3-cocycle. 2. Preliminaries
Throughout we work over k , an algebraically closed field of characteristic zero. The tensorproduct over k will be denoted by ⊗ . Vector spaces, algebras and coalgebras are all understood tobe over k and all maps are understood to be k -linear. The usual twist map from the tensor space V ⊗ W to W ⊗ V will be denoted τ , i.e., τ ( v ⊗ w ) = w ⊗ v . The multiplicative group of nonzeroelements of k is denoted by k × .For any coalgebra C and algebra A , ∗ will denote the convolution product in Hom ( C, A ).Composition of functions may be written as concatenation if the emphasis of the symbol ◦ is notrequired. The tensor product of a map with itself will often be written exponentially, i.e., we willwrite φ ⊗ to denote φ ⊗ φ ⊗ φ . Similarly H ⊗ H is denoted H ⊗ , etc.The group algebra over a group G will be written k G . The set of grouplike elements of acoalgebra C will be denoted G ( C ) and the subcoalgebra generated by G ( C ) will be denoted k G ( C ).We make the convention that an empty product, for example, a product of the form Q ≤ j ≤ a with a <
1, is defined to be 1.2.1.
Definitions. A coalgebra with multiplication and unit is a datum ( H, ∆ , ε, m, u ) where ( H, ∆ , ε )is a coalgebra, m : H ⊗ H → H is a coalgebra homomorphism called multiplication and u : k → H is a coalgebra homomorphism called unit [Ka, dual to page 368]. Denote u (1 k ) by 1 H .For H a coalgebra with multiplication and unit, a convolution invertible map ω : H ⊗ → k iscalled a 3-cocycle if and only if(1) ( ε ⊗ ω ) ∗ ω ( H ⊗ m ⊗ H ) ∗ ( ω ⊗ ε ) = ω ( H ⊗ H ⊗ m ) ∗ ω ( m ⊗ H ⊗ H ) , and we say that a cocycle ω is unitary or normalized if for all h, h ′ ∈ H ,(2) ω ( h ⊗ H ⊗ h ′ ) , or equivalently either ω (1 ⊗ h ⊗ h ′ ) or ω ( h ⊗ h ′ ⊗ , is ε ( h ) ε ( h ′ ) . If H, L are coalgebras with multiplication and unit, a coalgebra map φ : L → H is a morphismof coalgebras with multiplication and unit, if m H ( φ ⊗ φ ) = φm L , φu L = u H . If φ : L → H is a morphism of coalgebras with multiplication and unit and ω is a (normalized)3-cocycle for H , then ω ◦ φ ⊗ is a (normalized) 3-cocycle for L .For H a coalgebra with unit, a convolution invertible map v : H ⊗ → k such that v (1 ⊗ h ) = v ( h ⊗
1) = ε ( h ) for all h ∈ H , i.e., v is unitary or normalized, is called a gauge transformation .Note that for H a cocommutative bialgebra and v : H ⊗ H → k a (normalized) convolutioninvertible map then the map ∂ v : H ⊗ H ⊗ H → k , defined by ∂ v := ( ε ⊗ v ) ∗ v − ( m ⊗ H ) ∗ v ( H ⊗ m ) ∗ ( v − ⊗ ε ) , is a (normalized) cocycle called a (normalized) coboundary. Conversely, if v is convolution invertibleand ∂ v is normalized then v (1 ⊗ h ) = v ( h ⊗
1) = ε ( h ) v (1 ⊗
1) and so v (1 ⊗ − v is normalized.(See also Lemma 2.3.) UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 3
Now we define the objects of interest in this paper.
Definition . A dual quasi-bialgebra is a datum ( H, ∆ , ε, m, u, ω ) where ( H, ∆ , ε, m, u ) is acoalgebra with multiplication and unit and ω : H ⊗ H ⊗ H → k is a normalized 3-cocycle such that[ m ( H ⊗ m )] ∗ ( uω ) = ( uω ) ∗ [ m ( m ⊗ H )];(3) m (1 H ⊗ h ) = h = m ( h ⊗ H ) , for all h ∈ H. (4)Unless it is needed to emphasis the structure of the coalgebra H with multiplication and unit, wewill write ( H, ω ) for a dual quasi-bialgebra. The map ω is called the (Drinfeld) reassociator of thedual quasi-bialgebra. Note that if H is cocommutative, then ( H, ω ) has associative multiplicationfor every reassociator ω .Following [Sc, Section 2], we say that Φ : ( H, ω ) → ( H ′ , ω ′ ) is a morphism of dual quasi-bialgebras if Φ : H → H ′ is a morphism of coalgebras with multiplication and unit and ω ′ ◦ Φ ⊗ = ω. A bijective morphism of dual quasi-bialgebras is an isomorphism.A dual quasi-subbialgebra of a dual quasi-bialgebra ( H ′ , ω ′ ) is a dual quasi-bialgebra ( H, ω )such that H is a subcoalgebra of H ′ and the canonical inclusion σ : H → H ′ is a morphism of dualquasi-bialgebras.We note that by (3) multiplication in the dual quasi-subbialgebra k G ( H ) of H is associative.Let ( H, ω ) be a dual quasi-bialgebra. It is well-known that the category H M of left H -comodulesbecomes a monoidal category as follows. Given a left H -comodule V , we denote the left coactionof V by ρ = ρ lV : V → H ⊗ V, ρ ( v ) = v − ⊗ v . The tensor product of two left H -comodules V and W is a comodule via diagonal coaction i.e. ρ ( v ⊗ w ) = v − w − ⊗ v ⊗ w . The unit is the trivialleft H -comodule k , i.e. ρ ( k ) = 1 H ⊗ k . The associativity and unit constraints are defined, for all U, V, W ∈ H M and u ∈ U, v ∈ V, w ∈ W, k ∈ k , by H a U,V,W ( u ⊗ v ⊗ w ) := ω − ( u − ⊗ v − ⊗ w − ) u ⊗ ( v ⊗ w ) , (5) l U ( k ⊗ u ) := ku and r U ( u ⊗ k ) := uk. The monoidal category we have just described will be denoted by ( H M , ⊗ , k , H a, l, r ) . The monoidal categories ( M H , ⊗ , k , a H , l, r ) and ( H M H , ⊗ , k , H a H , l, r ) are defined similarly.We just point out that a HU,V,W ( u ⊗ v ⊗ w ) := u ⊗ ( v ⊗ w ) ω ( u ⊗ v ⊗ w ) , H a HU,V,W ( u ⊗ v ⊗ w ) := ω − ( u − ⊗ v − ⊗ w − ) u ⊗ ( v ⊗ w ) ω ( u ⊗ v ⊗ w ) . For (
H, ω ) a dual quasi-bialgebra and v : H ⊗ → k a convolution invertible map, define maps m v : H ⊗ → H and ω v : H ⊗ → k by setting m v : = v ∗ m ∗ v − (6) ω v : = ( ε ⊗ v ) ∗ v ( H ⊗ m ) ∗ ω ∗ v − ( m ⊗ H ) ∗ (cid:0) v − ⊗ ε (cid:1) . (7)If v is a gauge transformation, then the datum( H, ω ) v = ( H v , ω v ) = ( H, m v , u, ∆ , ε, ω v )is a dual quasi-bialgebra with reassociator ω v called the twisted dual quasi-bialgebra of H by v . Remark . (i) If a ∈ k × and v is convolution invertible, then so is av , the composition of v withmultiplication by a , and av has inverse a − v − . Note that m v = m av and ω v = ω av .(ii) Note that ( m v ) v − = m . It is straightforward to verify that ( ω v ) v − = ω , remembering thatthe multiplication in ( H v , ω v ) is m v . Thus, since v − is a gauge transformation for ( H v , ω v ), wehave that ( H v , ω v ) v − ∼ = ( H, ω ).(iii) Note that if H is cocommutative then ω v = ∂ v ∗ ω so that ( H v , ε v ) = ( H, ∂ v ). ALESSANDRO ARDIZZONI, MARGARET BEATTIE, AND CLAUDIA MENINI
Lemma . For ( H, ω ) a dual quasi-bialgebra, suppose v : H ⊗ H → k is a convolution invert-ible map such that ω v as defined in (7) is normalized, i.e., satisfies (2). Then av is a gaugetransformation for a = v (1 ⊗ − ∈ k × .Proof. For all h, h ′ ∈ H , ε ( h ) ε ( h ′ ) (2) = ω v ( h ⊗ ⊗ h ′ ) (7) = v (1 ⊗ h ′ ) v − ( h ⊗ . Setting h and h ′ equal to 1 in turn, we obtain for all h ∈ H , ε ( h ) = v (1 ⊗ v − ( h ⊗
1) = a − v − ( h ⊗
1) and ε ( h ) = v (1 ⊗ h ) v − (1 ⊗
1) = av (1 ⊗ h ) . Since a − v − ( h ⊗
1) = ε ( h ) and av is the inverse of a − v − then av ( h ⊗
1) = ε ( h ) also. (cid:3) Corollary . Let H be a cocommutative bialgebra. If v : H ⊗ H → k is a convolution invertiblemap such that ∂ v is a normalized cocycle, then av is normalized for a = v (1 ⊗ − ∈ k × .Proof. Take ω = ε in Lemma 2.3. Since H is cocommutative, ω v = ∂ v. (cid:3) Proposition . For σ : ( H, ω H ) → ( A, ω A ) a morphism of dual quasi-bialgebras and v : A ⊗ → k a gauge transformation for A , then v ◦ ( σ ⊗ σ ) is a gauge transformation for H . Also ω vA ◦ σ ⊗ = ω v ◦ σ ⊗ H and m vA ◦ σ ⊗ = σm v ◦ σ ⊗ H . Thus σ : ( H v ◦ σ ⊗ , ω v ◦ σ ⊗ H ) → ( A v , ω vA ) is also a morphism of dual quasi-bialgebras between thetwisted dual quasi-bialgebras obtained from ( H, ω H ) and ( A, ω A ) .Proof. We have ω vA ◦ σ ⊗ = (cid:2) ( ε A ⊗ v ) ∗ v ( A ⊗ m A ) ∗ ω A ∗ v − ( m A ⊗ A ) ∗ (cid:0) v − ⊗ ε A (cid:1)(cid:3) ◦ σ ⊗ = (cid:0) ε H ⊗ vσ ⊗ (cid:1) ∗ vσ ⊗ ( H ⊗ m H ) ∗ ω H ∗ v − σ ⊗ ( m H ⊗ H ) ∗ (cid:0) v − σ ⊗ ⊗ ε H (cid:1) (7) = ω vσ ⊗ H . Also m vA σ ⊗ = (cid:2) v ∗ m A ∗ v − (cid:3) (cid:0) σ ⊗ (cid:1) = vσ ⊗ ∗ m A σ ⊗ ∗ v − σ ⊗ = vσ ⊗ ∗ σm H ∗ [ vσ ⊗ ] − = σm vσ ⊗ H . (cid:3) Definition . Dual quasi-bialgebras A and B are called quasi-isomorphic (or equivalent) when-ever ( A, ω A ) ∼ = ( B v , ω vB ) as dual quasi-bialgebras for some gauge transformation v ∈ ( B ⊗ B ) ∗ .By Remark 2.2-ii., if ( A, ω A ) ∼ = ( B v , ω vB ), then ( B, ω B ) ∼ = ( A, ω A ) v − . Corollary . If σ : ( H, ω H ) → ( A, ω A ) is a morphism of dual quasi-bialgebras and ( A, ω A ) isquasi-isomorphic to an ordinary bialgebra so is ( H, ω H ) .Proof. Suppose γ A : A ⊗ A → k is a gauge transformation such that A γ A has trivial reassociator.Then γ H := γ A ( σ ⊗ σ ) : H ⊗ H → k is a gauge transformation, and, by Proposition 2.5, the map σ : ( H γ H , ω γ H H ) → ( A γ A , ω γ A A = ε A ⊗ ) is a morphism of dual quasi-bialgebras. Hence ω H γH = ω γ H H = ω γ A A ◦ σ ⊗ = ε A ⊗ ◦ σ ⊗ = ε H ⊗ so that H γ H has trivial reassociator. (cid:3) UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 5 Quasi-Yetter-Drinfeld data for dual quasi-bialgebras
Yetter-Drinfeld modules.
In this subsection, we first recall some facts from [AP] aboutthe category of Yetter-Drinfeld modules for a dual quasi-bialgebra.
Definition . For (
H, ω ) a dual quasi-bialgebra, the category HH YD ofYetter-Drinfeld modules over H is defined as follows. An object is a tern ( V, ρ V , ⊲ ) , where( V, ρ ) is an object in H M and µ : H ⊗ V → V is a k -linear map written h ⊗ v h ⊲ v suchthat, for all h, l ∈ H and v ∈ V ( hl ) ⊲ v = (cid:20) ω − ( h ⊗ l ⊗ v − ) ω (cid:0) h ⊗ ( l ⊲ v ) − ⊗ l (cid:1) ω − (( h ⊲ ( l ⊲ v ) ) − ⊗ h ⊗ l ) ( h ⊲ ( l ⊲ v ) ) (cid:21) , (8) 1 H ⊲ v = v and(9) ( h ⊲ v ) − h ⊗ ( h ⊲ v ) = h v − ⊗ ( h ⊲ v ) . (10)A morphism f : ( V, ρ, ⊲ ) → ( V ′ , ρ ′ , ⊲ ′ ) in HH YD is a morphism f : ( V, ρ ) → ( V ′ , ρ ′ ) in H M suchthat f ( h ⊲ v ) = h ⊲ ′ f ( v ) . Remark . The category HH YD is isomorphic to the weak right center of H M , see [AP, TheoremA.2.]. As a consequence HH YD has a prebraided monoidal structure given as follows. The unit is k regarded as an object in HH YD via the trivial structures ρ k ( k ) = 1 H ⊗ k and h ⊲ k = ε H ( h ) k. The tensor product is defined by(
V, ρ V , ⊲ ) ⊗ ( W, ρ W , ⊲ ) = ( V ⊗ W, ρ V ⊗ W , ⊲ )where ρ V ⊗ W ( v ⊗ w ) = v − w − ⊗ v ⊗ w and(11) h ⊲ ( v ⊗ w ) = (cid:20) ω ( h ⊗ v − ⊗ w − ) ω − (cid:0) ( h ⊲ v ) − ⊗ h ⊗ w − (cid:1) ω (cid:0) ( h ⊲ v ) − ⊗ ( h ⊲ w ) − ⊗ h (cid:1) ( h ⊲ v ) ⊗ ( h ⊲ w ) (cid:21) . The constraints are the same as H M i.e. H a U,V,W ( u ⊗ v ⊗ w ) : = ω − ( u − ⊗ v − ⊗ w − ) u ⊗ ( v ⊗ w ) ,l U ( k ⊗ u ) : = ku and r U ( u ⊗ k ) := uk. viewed as morphisms in HH YD . The prebraiding c V,W : V ⊗ W → W ⊗ V is given by(12) c V,W ( v ⊗ w ) = ( v − ⊲ w ) ⊗ v . Remark . The coproduct of a family (cid:0) V i (cid:1) i ∈ I of objects in HH YD is the vector space ⊕ i ∈ I V i regarded as an object in HH YD via the action and the coaction defined by h ⊲ (cid:0) v i (cid:1) i ∈ I = (cid:0) h ⊲ v i (cid:1) i ∈ I and ρ (cid:16)(cid:0) v i (cid:1) i ∈ I (cid:17) = X i ∈ I v i − ⊗ u i (cid:0) v i (cid:1) respectively, where u i : V i → ⊕ i ∈ I V i is the canonical injection. Let W ∈ HH YD . By the universalproperty of the coproduct, the canonical morphisms W ⊗ u i : W ⊗ V i → W ⊗ (cid:0) ⊕ i ∈ I V i (cid:1) yield amorphism in HH YD ⊕ i ∈ I (cid:0) W ⊗ V i (cid:1) → W ⊗ (cid:0) ⊕ i ∈ I V i (cid:1) . This is bijective because it is bijective at the level of vector spaces. This proves that the functor W ⊗ ( − ) : HH YD → HH YD commutes with coproducts. Similarly ( − ) ⊗ W : HH YD → HH YD commuteswith coproducts.Therefore we can apply [Mac, Theorem 2, page 172] to construct a left adjoint T : HH YD →
Mon (cid:0) HH YD (cid:1) of the forgetful functor. For every V ∈ HH YD , the algebra T ( V ) will be called the tensor algebra of V in HH YD . By standard arguments one can endow T := T ( V ) with a bialgebrastructure in HH YD where the comultiplication ∆ T and the counit ε T are uniquely defined by setting∆ T ( v ) = v ⊗ T + 1 T ⊗ v and ε T ( v ) = 0 . ALESSANDRO ARDIZZONI, MARGARET BEATTIE, AND CLAUDIA MENINI
The next theorem from [AP] gives the structure of a bosonization in this setting.
Theorem . Let ( H, ω H ) be a dual quasi-bialgebra.Let ( R, µ R , ρ R , ∆ R , ε R , m R , u R ) be a bialgebra in HH YD and use the following notations h ⊲ r : = µ R ( h ⊗ r ) , r − ⊗ r := ρ R ( r ) ,r · R s : = m R ( r ⊗ s ) , R := u R (1 k ) ,r ⊗ r : = ∆ R ( r ) . Consider on B := R ⊗ H the following structures: m B [( r ⊗ h ) ⊗ ( s ⊗ k )] = ω − H ( r − ⊗ h ⊗ s − k ) ω H ( h ⊗ s − ⊗ k ) ω − H [( h ⊲ s ) − ⊗ h ⊗ k ] ω H ( r − ⊗ ( h ⊲ s ) − ⊗ h k ) r · R ( h ⊲ s ) ⊗ h k u B ( k ) = k R ⊗ H ∆ B ( r ⊗ h ) = ω − H ( r − ⊗ r − ⊗ h ) r ⊗ r − h ⊗ r ⊗ h ε B ( r ⊗ h ) = ε R ( r ) ε H ( h ) ω B (( r ⊗ h ) ⊗ ( s ⊗ k ) ⊗ ( t ⊗ l )) = ε R ( r ) ε R ( s ) ε R ( t ) ω H ( h ⊗ k ⊗ l ) . Then ( B, ∆ B , ε B , m B , u B , ω B ) is a dual quasi-bialgebra. Definition . For
H, R, B as in Theorem 3.4, the dual quasi-bialgebra B will be called the bosonization of R by H and denoted by R H . Elements of B may be written r h instead of r ⊗ h to emphasize that we are working in the bosonization. Remark . Let A := R H, B := S L where H, L are cosemisimple dual quasi-bialgebras and
R, S are bialgebras in the categories of Yetter-Drinfeld modules over H and L respectively suchthat A = k H and B = k L . Then if A and B are quasi-isomorphic, so are H and L .For suppose that there is an isomorphism ϕ : A → B v of dual quasi-bialgebras. Since ϕ is acoalgebra isomorphism, ϕ ( A ) = ( B v ) = B .Write ϕ (1 ⊗ h ) as 1 ⊗ ϕ ′ ( h ) for some ϕ ′ ( h ) ∈ L . In this way we get the following commutativediagram. H ϕ ′ / / σ H (cid:15) (cid:15) L v ( σ L ⊗ σ L ) σ L (cid:15) (cid:15) R H ϕ / / ( S L ) v By the same argument using ϕ − we get an inverse for ϕ ′ . By Proposition 2.5, the right-hand sidevertical map is an injective morphism of dual quasi-bialgebras. Since σ H and ϕ are also morphismsof dual quasi-bialgebras we get that ϕ ′ also is. Thus ϕ ′ is an isomorphism of dual quasi-bialgebrasas required.The proof of the next lemma is straightforward and so is left to the reader. Lemma . Take the hypothesis and notations of Theorem 3.4. Let π : R H → H be defined by π ( r h ) = ε R ( r ) h. Then π is a morphism of dual quasi-bialgebras and (13) π (( r h ) ) ⊗ ( r h ) π (( r h ) ) = r − h ⊗ ( r h ) ⊗ h . Remark . Note that for R H as above, the map σ : H ֒ → R H defined by σ ( h ) = 1 R h isalso a morphism of dual quasi-bialgebras and, for π as defined in Lemma 3.7, πσ = Id H . Corollary2.7 then implies that H is quasi-isomorphic to an ordinary bialgebra if and only if R H is. UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 7
Quasi-Yetter-Drinfeld data.
Let (
H, ω ) be a dual quasi-bialgebra. In this subsection westudy one-dimensional vector spaces in HH YD and the bialgebras in HH YD generated by these. Proposition . Let ( H, ω ) be a dual quasi-bialgebra and let V be a one-dimensional vectorspace. Then V is an object in HH YD if and only if for all v ∈ V , h, l ∈ H , (i) V ∈ H M and ρ ( v ) = g ⊗ v for some g ∈ G ( H ) ; (ii) There is a unitary map χ ∈ H ∗ such that for g the grouplike in ( i ) , (14) χ ( hl ) = ω − ( h ⊗ l ⊗ g ) χ ( l ) ω ( h ⊗ g ⊗ l ) χ ( h ) ω − ( g ⊗ h ⊗ l ) , (iii) For g the grouplike from part ( i ) , (15) gχ ( h ) h = h χ ( h ) g. Proof.
First let V ∈ HH YD . Since V is one-dimensional, there is a grouplike element g ∈ H suchthat ρ ( v ) = g ⊗ v for all v ∈ V , and also there is a map χ : H → k such that h ⊲ v = χ ( h ) v .Equation (8) of the definition of Yetter-Drinfeld modules now translates to (14) and equation (9)implies that χ is unitary. Finally here equation (10) of Definition 3.1 is equivalent to (15) above.Now suppose that V is a one-dimensional vector space satisfying ( i ) to ( iii ) above. Then it iseasy to see that V with coaction given by ρ ( v ) = g ⊗ v and action given by h ⊲ v = χ ( h ) v for all v ∈ V is an object in HH YD . (cid:3) Definitions . Let (
H, ω ) be a dual quasi-bialgebra. For g ∈ G ( H ) and χ ∈ H ∗ , χ unitary,the triple (( H, ω ) , g, χ ) is called a quasi-Yetter-Drinfeld datum , abbreviated to quasi- Y D datum,whenever equations (14) and (15) above hold. If q := χ ( g ) , we also say that (( H, ω ) , g, χ ) is a quasi-Yetter-Drinfeld datum for q. Remark . (i) When H is a Hopf algebra, ω is trivial and q = 1, then the previous definitionreduces to [CDMM, Definition 2.1].(ii) Note that Proposition 3.9, roughly speaking, says that a one-dimensional vector space V with action and coaction defined by χ and g , is an object in HH YD if and only if (( H, ω ) , g, χ ) is aquasi- Y D datum.(iii) Equation (15) implies that if ((
H, ω ) , g, χ ) is a quasi- Y D datum and ℓ ∈ G ( H ) with χ H ( ℓ ) =0, then gℓ = ℓg . Lemma . Let G be a group and let ω be a normalized -cocycle on G. Let g ∈ G and χ : k G → k . The following are equivalent. ( i ) (( k G, ω ) , g, χ ) is a quasi- Y D datum. ( ii ) g ∈ Z ( G ) , χ is unitary and (14) holds for all h, ℓ ∈ G .Proof. It suffices to prove that ( i ) implies that g ∈ Z ( G ). Let h ∈ G. Then1 = χ (1) = χ (cid:0) h − h (cid:1) (14) = ω − (cid:0) h − ⊗ h ⊗ g (cid:1) χ ( h ) ω (cid:0) h − ⊗ g ⊗ h (cid:1) χ (cid:0) h − (cid:1) ω − (cid:0) g ⊗ h − ⊗ h (cid:1) , so that χ ( h ) is invertible, and gh = hg by Remark 3.11. (cid:3) Definition . For ((
H, ω ) , g, χ ) and (( L, α ) , l, ξ ) quasi- Y D data, a dual quasi-bialgebra ho-momorphism ϕ : ( H, ω ) → ( L, α ) such that ϕ ( g ) = l and ξϕ = χ is called a morphism of quasi- Y D data . Lemma . Let π : ( A, ω A ) → ( H, ω H ) be a morphism of dual quasi-bialgebras and (( H, ω ) , g, χ ) a quasi- Y D datum. If there exists a ∈ G ( A ) such that π ( a ) = g and aχπ ( b ) b = b χπ ( b ) a, for every b ∈ A , then (( A, ω A ) , a, χ A := χπ ) is also a quasi- Y D datum and π is a morphism ofquasi- Y D data.Proof.
We need only verify (14) for ((
A, ω ) , a, χ A ). For h, l ∈ A , since (14) holds for χ , we have: ω − A ( h ⊗ l ⊗ a ) χ A ( l ) ω A ( h ⊗ a ⊗ l ) χ A ( h ) ω − A ( a ⊗ h ⊗ l )= ω − H ( π ( h ) ⊗ π ( l ) ⊗ g ) χ [ π ( l ) ] ω H ( π ( h ) ⊗ g ⊗ π ( l ) ) χ [ π ( h ) ] ω − H ( g ⊗ π ( h ) ⊗ π ( l ) ) ALESSANDRO ARDIZZONI, MARGARET BEATTIE, AND CLAUDIA MENINI(14) = χ ( π ( h ) π ( l )) = χπ ( hl ) = χ A ( hl ) . (cid:3) Lemma . Suppose ( H, ω ) is a dual quasi-bialgebra and (( H, ω ) , g, χ ) is a quasi- Y D datum.Then for c ∈ G ( H ) and ≤ t , χ (cid:0) c t (cid:1) = χ ( c ) t Y ≤ i ≤ t − (cid:2) ω − (cid:0) c i ⊗ c ⊗ g (cid:1) ω (cid:0) c i ⊗ g ⊗ c (cid:1) ω − (cid:0) g ⊗ c i ⊗ c (cid:1)(cid:3) , (16) and, in particular, χ (cid:0) g t (cid:1) = χ ( g ) t Y ≤ i ≤ t − ω − (cid:0) g ⊗ g i ⊗ g (cid:1) . (17) Proof.
Let s > χ (cid:0) c s − c (cid:1) = ω − (cid:0) c s − ⊗ c ⊗ g (cid:1) χ ( c ) ω (cid:0) c s − ⊗ g ⊗ c (cid:1) χ (cid:0) c s − (cid:1) ω − (cid:0) g ⊗ c s − ⊗ c (cid:1) . Equation (16) now follows by induction on t ≥
1. For t = 1, there is nothing to prove. Let t > t − . Then by (18), χ (cid:0) c t (cid:1) = χ (cid:0) c t − (cid:1) (cid:2) χ ( c ) ω − (cid:0) c t − ⊗ c ⊗ g (cid:1) ω (cid:0) c t − ⊗ g ⊗ c (cid:1) ω − (cid:0) g ⊗ c t − ⊗ c (cid:1)(cid:3) and if we then expand χ ( c t − ) using (16), the result is immediate. (cid:3) Remark . If ω = ω ( H ⊗ τ ), then equation (16) simplifies to:(19) χ (cid:0) c t (cid:1) = χ ( c ) t Y ≤ i ≤ t − ω − (cid:0) g ⊗ c i ⊗ c (cid:1) . Quantum lines
Our first lemma will be useful in the computations to follow.
Lemma . Let ( H, ω ) be a dual quasi bialgebra and let g ∈ G ( H ) . For all ≤ a, b, c, (20) ω − (cid:0) g a ⊗ g b ⊗ g c (cid:1) = Y ≤ j ≤ a − ω − (cid:0) g ⊗ g j + b ⊗ g c (cid:1) ω − (cid:0) g ⊗ g j ⊗ g b (cid:1) ω (cid:0) g ⊗ g j ⊗ g b + c (cid:1) . Proof.
Let Φ( j ) := ω − (cid:0) g ⊗ g j + b ⊗ g c (cid:1) ω − (cid:0) g ⊗ g j ⊗ g b (cid:1) ω (cid:0) g ⊗ g j ⊗ g b + c (cid:1) . The proof is by in-duction on a ≥ . For a = 0 , a > a − . By (1) evaluated on g ⊗ g a − ⊗ g b ⊗ g c , we have ω (cid:0) g a − ⊗ g b ⊗ g c (cid:1) ω (cid:0) g ⊗ g a + b − ⊗ g c (cid:1) ω (cid:0) g ⊗ g a − ⊗ g b (cid:1) = ω (cid:0) g ⊗ g a − ⊗ g b + c (cid:1) ω (cid:0) g a ⊗ g b ⊗ g c (cid:1) so that ω (cid:0) g a − ⊗ g b ⊗ g c (cid:1) ω − (cid:0) g a ⊗ g b ⊗ g c (cid:1) = Φ( a − , and the statement then follows from the induction assumption. (cid:3) Next we introduce some useful notation.
Notation . Let ( H, ω ) be a dual quasi-bialgebra. For U, V, W, Z in HH YD , we define Ω U,V,W,Z :( U ⊗ V ) ⊗ ( W ⊗ Z ) → ( U ⊗ W ) ⊗ ( V ⊗ Z ) by (21) Ω U,V,W,Z := a − U,W,V ⊗ Z ( U ⊗ a W,V,Z )( U ⊗ c V,W ⊗ Z )( U ⊗ a − V,W,Z ) a U,V,W ⊗ Z , where a := H a is the associativity constraint (5) in H M . If U = V = W = Z , we write Ω U inplace of Ω U,U,U,U . As observed in Remark 3.3, we can consider the tensor algebra T ( V ) of V in HH YD for anyobject V in HH YD . Explicitly T ( V ) := ⊕ n ∈ N T n ( V ) , where T ( V ) = k , T ( V ) = V and, for n > , T n ( V ) := V ⊗ T n − ( V ) . Thus, for instance, T ( V ) = V ⊗ V and T ( V ) = V ⊗ ( V ⊗ V ) . Note that the order of the brackets is important here.
UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 9
Let ((
H, ω ) , g, χ ) be a quasi- Y D datum for q . Let V = k v be a one-dimensional vector spaceand then ( V, ρ, µ ) with ρ ( v ) = g ⊗ v and µ ( h ⊗ v ) = χ ( h ) v is an object in HH YD by Remark 3.11.Set v [0] := 1 , v [1] := v and, for n > , v [ n ] := v ⊗ v [ n − . As a vector space T ( V ) may be identifiedwith the polynomial ring k [ X ] via the correspondence v [ n ] ↔ X n . However the multiplication isdifferent. Proposition . With hypothesis and notations as above, the tensor algebra T ( V ) has basis (cid:0) v [ n ] (cid:1) n ∈ N and has the following bialgebra structure in HH YD .(i) The left coaction of H on T ( V ) is given by (22) ρ (cid:16) v [ n ] (cid:17) = v [ n ] − ⊗ v [ n ]0 = g n ⊗ v [ n ] . The left action of H on T ( V ) is given by (23) h ⊲ v [ n ] = χ [ n ] ( h ) v [ n ] , where χ [ n ] ∈ H ∗ is defined iteratively by setting χ [0] := ε , and for n ≥ , (24) χ [ n ] := ω (cid:0) − ⊗ g ⊗ g n − (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − (cid:1) ∗ χ [ n − ∗ ω (cid:0) g ⊗ g n − ⊗ − (cid:1) . Furthermore, for n ≥ χ [ n ] = ∗ Y ≤ i ≤ n − ω (cid:0) − ⊗ g ⊗ g n − − i (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − − i (cid:1) ∗ ∗ Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ − (cid:1) and in particular (26) χ [ n ] ( g ) = Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ g (cid:1) χ ( g ) n = q n Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ g (cid:1) . (ii) The algebra structure on T ( V ) is given by T ( V ) = 1 k ∈ T ( V ) and (27) v [ a ] v [ b ] = Y ≤ i ≤ a − ω − (cid:0) g ⊗ g i ⊗ g b (cid:1) v [ a + b ] , for a ≥ , b ∈ N , (iii) The coalgebra structure is given by ε T (cid:0) v [ n ] (cid:1) = δ n, , and (28) ∆ T (cid:16) v [ n ] (cid:17) = X ≤ i ≤ n β ( i, n ) v [ i ] ⊗ v [ n − i ] , where we define (29) β ( i, n ) = (cid:18) ni (cid:19) q Y ≤ j ≤ i − ω (cid:0) g ⊗ g j ⊗ g n − i (cid:1) for all ≤ i ≤ n. Note that β (0 , n ) = 1 = β ( n, n ) for all n ≥ .Proof. Equation (22) follows from the definition of the comodule structure on the tensor productin HH YD in Remark 3.2 and the fact that ρ ( v ) = g ⊗ v .Next we compute h ⊲ v [ n ] for h ∈ H . If n = 0, then χ [0] = ε satisfies (23). We prove, byinduction on n ≥ , that (23) holds for χ [ n ] ∈ H ∗ defined inductively by equation (24). Equation(24) gives χ [1] = χ , which satisfies (23). Let n > n − h ⊲ v [ n ] = h ⊲ (cid:16) v ⊗ v [ n − (cid:17) (11) = ω (cid:16) h ⊗ v − ⊗ v [ n − − (cid:17) ω − (cid:16) ( h ⊲ v ) − ⊗ h ⊗ v [ n − − (cid:17) ω (cid:18) ( h ⊲ v ) − ⊗ (cid:16) h ⊲ v [ n − (cid:17) − ⊗ h (cid:19) ( h ⊲ v ) ⊗ (cid:16) h ⊲ v [ n − (cid:17) = " ω (cid:0) h ⊗ g ⊗ g n − (cid:1) ω − (cid:0) g ⊗ h ⊗ g n − (cid:1) ω (cid:16) g ⊗ (cid:0) h ⊲ v [ n − (cid:1) − ⊗ h (cid:17) χ ( h ) v ⊗ (cid:0) h ⊲ v [ n − (cid:1) = (cid:20) ω (cid:0) h ⊗ g ⊗ g n − (cid:1) ω − (cid:0) g ⊗ h ⊗ g n − (cid:1) ω (cid:0) g ⊗ g n − ⊗ h (cid:1) χ ( h ) v ⊗ χ [ n − ( h ) v [ n − (cid:21) = (cid:2) ω (cid:0) − ⊗ g ⊗ g n − (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − (cid:1) ∗ χ [ n − ∗ ω (cid:0) g ⊗ g n − ⊗ − (cid:1)(cid:3) ( h ) v [ n ] . Now, using this formula, we prove by induction on n ≥ n = 1 , since ω isa normalized cocycle and g = 1, then the right hand side of (25) is just χ = χ [1] .Let n > n −
1. Then χ [ n ] = ω (cid:0) − ⊗ g ⊗ g n − (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − (cid:1) ∗ χ [ n − ∗ ω (cid:0) g ⊗ g n − ⊗ − (cid:1) = [ ω (cid:0) − ⊗ g ⊗ g n − (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − (cid:1) ] ∗ " ∗ Q ≤ i ≤ n − ω (cid:0) − ⊗ g ⊗ g n − − i (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − − i (cid:1) ∗ " ∗ Q ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ − (cid:1) ∗ ω (cid:0) g ⊗ g n − ⊗ − (cid:1) = ∗ Y ≤ i ≤ n − ω (cid:0) − ⊗ g ⊗ g n − − i (cid:1) ∗ χ ∗ ω − (cid:0) g ⊗ − ⊗ g n − − i (cid:1) ∗ ∗ Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ − (cid:1) , and so (25) holds for all n ≥
1. We note that if (25) is applied to a cocommutative element, thensince the product for 0 ≤ i ≤ n − ≤ n − − i ≤ n − χ [ n ] = χ n ∗ ∗ Y ≤ i ≤ n − ω (cid:0) − ⊗ g ⊗ g i (cid:1) ∗ ω − (cid:0) g ⊗ − ⊗ g i (cid:1) ∗ ω (cid:0) g ⊗ g i ⊗ − (cid:1) . Equation (26) follows immediately.(ii) Let b ∈ N and we prove by induction on a ≥ a = 1, we have bydefinition v [ a ] v [ b ] = vv [ b ] = v ⊗ v [ b ] = v [1+ b ] = v [ a + b ] . Let a > a −
1. Then v [ a ] v [ b ] = (cid:16) v ⊗ v [ a − (cid:17) v [ b ] = (cid:16) vv [ a − (cid:17) v [ b ] (5) = ω − (cid:16) v − ⊗ v [ a − − ⊗ v [ b ] − (cid:17) v (cid:16) v [ a − v [ b ]0 (cid:17) (22) = ω − (cid:0) g ⊗ g a − ⊗ g b (cid:1) v (cid:16) v [ a − v [ b ] (cid:17) = ω − (cid:0) g ⊗ g a − ⊗ g b (cid:1) v Y ≤ i ≤ a − ω − (cid:0) g ⊗ g i ⊗ g b (cid:1) v [ a − b ] = Y ≤ i ≤ a − ω − (cid:0) g ⊗ g i ⊗ g b (cid:1) vv [ a − b ] = Y ≤ i ≤ a − ω − (cid:0) g ⊗ g i ⊗ g b (cid:1) v [ a + b ] , and so we have proved that (27) holds for a ≥ T (cid:16) v [ n ] (cid:17) = X ≤ i ≤ n (cid:18) ni (cid:19) q Y ≤ j ≤ i − ω (cid:0) g ⊗ g j ⊗ g n − i (cid:1) v [ i ] ⊗ v [ n − i ] , where if i = 0, the empty product is defined to be 1. If n = 0 the formula holds since ∆ T ( v [0] ) =1 ⊗
1. If n = 1, the formula holds since ∆ T (cid:0) v [1] (cid:1) = v ⊗ ⊗ v , (cid:0) (cid:1) q = (cid:0) (cid:1) q = 1, and ω ( − ⊗ ⊗ − ) = 1. Let n > n −
1. Then∆ T (cid:16) v [ n ] (cid:17) = ∆ T m T (cid:16) v ⊗ v [ n − (cid:17) = ( m T ⊗ m T ) Ω T (∆ T ⊗ ∆ T ) (cid:16) v ⊗ v [ n − (cid:17) = ( m T ⊗ m T ) Ω T h ( v ⊗ ⊗ v ) ⊗ ( X ≤ i ≤ n − β ( i, n − v [ i ] ⊗ v [ n − − i ] ) i . UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 11
From the definition of Ω T , it is easily seen that:Ω T (( v ⊗ ⊗ ( v [ i ] ⊗ v [ n − − i ] )) = ω ( g ⊗ g i ⊗ g n − − i )(( v ⊗ v [ i ] ) ⊗ (1 ⊗ v [ n − − i ] ))andΩ T ((1 ⊗ v ) ⊗ ( v [ i ] ⊗ v [ n − − i ] ) = χ [ i ] ( g ) ω ( g ⊗ g i ⊗ g n − − i ) ω − ( g i ⊗ g ⊗ g n − − i )(1 ⊗ v [ i ] ) ⊗ ( v ⊗ v [ n − − i ] ) . Then∆ T (cid:16) v [ n ] (cid:17) = X ≤ i ≤ n − β ( i, n − (cid:20) ω ( g ⊗ g i ⊗ g n − − i ) v [ i +1] ⊗ v [ n − − i ] + χ [ i ] ( g ) ω ( g ⊗ g i ⊗ g n − − i ) ω − ( g i ⊗ g ⊗ g n − − i ) v [ i ] ⊗ v [ n − i ] ] (cid:21) . The coefficient of v [0] ⊗ v [ n ] in the expression above is χ [0] ( g ) ω ( g ⊗ ⊗ g n − ) ω − (1 ⊗ g ⊗ g n − ) =1 = β (0 , n ) and similarly the coefficient of v [ n ] ⊗ v [0] is β ( n − , n −
1) = 1. For 1 ≤ j ≤ n −
1, wecompute the coefficient of v [ j ] ⊗ v [ n − j ] in this expression to be: β ( j − , n − ω ( g ⊗ g j − ⊗ g n − j ) + β ( j, n − ω ( g ⊗ g j ⊗ g n − − j ) χ [ j ] ( g ) ω − ( g j ⊗ g ⊗ g n − − j )= "(cid:18) n − j − (cid:19) q Y ≤ k ≤ j − ω ( g ⊗ g k ⊗ g n − j ) ω ( g ⊗ g j − ⊗ g n − j )+ "(cid:18) n − j (cid:19) q Y ≤ i ≤ j − ω ( g ⊗ g i ⊗ g n − − j ) ω ( g ⊗ g j ⊗ g n − − j ) χ [ j ] ( g ) ω − ( g j ⊗ g ⊗ g n − − j )= (cid:18) n − j − (cid:19) q Y ≤ k ≤ j − ω ( g ⊗ g k ⊗ g n − j )+ (cid:18) n − j (cid:19) q hY ≤ i ≤ j ω ( g ⊗ g i ⊗ g n − − j ) i χ [ j ] ( g ) ω − ( g j ⊗ g ⊗ g n − − j )= (cid:18) n − j − (cid:19) q Y ≤ k ≤ j − ω ( g ⊗ g k ⊗ g n − j )+ (cid:18) n − j (cid:19) q hY ≤ s ≤ j − ω ( g ⊗ g s +1 ⊗ g n − − j ) i χ [ j ] ( g ) ω − ( g j ⊗ g ⊗ g n − − j ) . By Lemma 4.1 ω − (cid:0) g j ⊗ g ⊗ g n − − j (cid:1) = Y ≤ s ≤ j − ω − (cid:0) g ⊗ g s +1 ⊗ g n − − j (cid:1) ω − ( g ⊗ g s ⊗ g ) ω (cid:0) g ⊗ g s ⊗ g n − j (cid:1) so that the last summand in the expression above becomes: (cid:18) n − j (cid:19) q χ [ j ] ( g ) Y ≤ s ≤ j − ω − ( g ⊗ g s ⊗ g ) Y ≤ t ≤ j − ω ( g ⊗ g t ⊗ g n − j ) (26) = (cid:18) n − j (cid:19) q q j Y ≤ s ≤ j − (cid:0) ω ( g ⊗ g s ⊗ g ) ω − ( g ⊗ g s ⊗ g ) (cid:1) Y ≤ t ≤ j − ω ( g ⊗ g t ⊗ g n − j )= (cid:18) n − j (cid:19) q q j Y ≤ t ≤ j − ω ( g ⊗ g t ⊗ g n − j ) . Thus the coefficient of v [ j ] ⊗ v [ n − j ] in ∆ T (cid:0) v [ n ] (cid:1) is "(cid:18) n − j − (cid:19) q + (cid:18) n − j (cid:19) q q j ≤ t ≤ j − ω ( g ⊗ g t ⊗ g n − j ) = (cid:18) nj (cid:19) q Y ≤ t ≤ j − ω ( g ⊗ g t ⊗ g n − j ) , and this is indeed β ( j, n ) as required. It is then straightforward to see that ε T ( v [ n ] ) = δ n, . (cid:3) The next technical results allow us to construct a bialgebra quotient of the tensor algebra.
Proposition . Let ( A, m A , u A , ∆ A , ε A ) be a bialgebra in an abelian prebraided monoidal cate-gory ( M , ⊗ , , a, l, r, c ) where the tensor functors are additive and right exact. Let ( I, i I : I → A ) be a subobject of A in M such that ( p R ⊗ p R ) ◦ ∆ A ◦ i I = 0 , (31) ε A ◦ i I = 0 , (32) p R ◦ m A ◦ i K = 0 . (33) where R := A/I , p R : A → R denotes the canonical projection and ( K, i K : K → A ⊗ A ) :=Ker ( p R ⊗ p R ) . Then there are maps m R , u R , ∆ R , ε R such that ( R, m R , u R , ∆ R , ε R ) is a bialgebrain ( M , ⊗ , , a, l, r, c ) and p R is a bialgebra morphism.Proof. In this proof we omit the constraints as in view of the coherence theorem they take care ofthemselves. By (31) and (32), there are morphisms∆ R : R → R ⊗ R and ε R : R → k defined by ∆ R p R = ( p R ⊗ p R ) ∆ A and ε R ( p R ) = ε A . The first equality yields(∆ R ⊗ R ) ∆ R p R = (( p R ⊗ p R ) ⊗ p R ) (∆ A ⊗ A ) ∆ A = ( p R ⊗ ( p R ⊗ p R )) ( A ⊗ ∆ A ) ∆ A = ( R ⊗ ∆ R ) ∆ R p R so that (∆ R ⊗ R ) ∆ R = ( R ⊗ ∆ R ) ∆ R . The other equality leads to counitarity of ∆ R . Since the ten-sor functors are right exact, we have that p R ⊗ p R is an epimorphism and hence ( R ⊗ R, p R ⊗ p R ) =Coker ( i K ) . Thus, by (33), we have that there is a unique map m R : R ⊗ R → R such that m R ( p R ) = ( p R ⊗ p R ) m A . Set u R := p R ( u A ) . The first equality yields m R ( m R ⊗ R ) p ⊗ R = m R ( R ⊗ m R ) p ⊗ R so that, by right exactness of tensor functors, we get m R ( m R ⊗ R ) = m R ( R ⊗ m R ) . Similarly onegets m R ( u R ⊗ R ) = l R and m R ( R ⊗ u R ) = r R . Finally, we have( m R ⊗ m R ) ( R ⊗ c R,R ⊗ R ) (∆ R ⊗ ∆ R ) ( p R ⊗ p R )= ( p R ⊗ p R ) ( m A ⊗ m A ) ( A ⊗ c A,A ⊗ A ) (∆ A ⊗ ∆ A )= ( p R ⊗ p R ) ∆ A m A = ∆ R m R ( p R ⊗ p R ) . Since p R ⊗ p R is an epimorphism, we get ( m R ⊗ m R ) ( R ⊗ c R,R ⊗ R ) (∆ R ⊗ ∆ R ) = ∆ R m R . Thus(
R, m R , u R , ∆ R , ε R ) is a bialgebra in ( M , ⊗ , , a, l, r, c ) . Clearly p R is a bialgebra morphism. (cid:3) Lemma . Let ( H, ω ) be a dual quasi-bialgebra and let I be an ideal of a bialgebra A in HH YD .Let z, u ∈ A and assume ∆ A ( u ) ∈ A ⊗ I + I ⊗ A. Then ∆ A ( zu ) ∈ A ⊗ I + I ⊗ A. Proof.
Since ∆ A ( u ) ∈ A ⊗ I + I ⊗ A , then∆ A ( zu ) = ∆ A m A ( z ⊗ u ) = ( m A ⊗ m A ) Ω A (∆ A ⊗ ∆ A ) ( z ⊗ u ) ∈ ( m A ⊗ m A ) Ω A [( A ⊗ A ) ⊗ ( A ⊗ I ) + ( A ⊗ A ) ⊗ ( I ⊗ A )] ⊆ ( m A ⊗ m A ) [( A ⊗ A ) ⊗ ( A ⊗ I ) + ( A ⊗ I ) ⊗ ( A ⊗ A )] ⊆ A ⊗ I + I ⊗ A. (cid:3) Let (
H, g, χ ) be a quasi-
Y D datum for q with q a primitive N -th root of unity with N >
0. Let V = k v ∈ HH YD with coaction and action defined by g and χ as in Remark 3.11. Let (cid:0) v [ n ] (cid:1) n ∈ N bethe basis of T := T ( V ) considered at the beginning of this section.Let I be the two-sided ideal of T generated by v [ N ] , i.e, I =: T ( IT ). Since vv [ n ] = v [ n +1] ,by (27), I is the vector space with basis (cid:0) v [ n ] (cid:1) n ≥ N . Thus,
T /I identifies with K [ X ] / (cid:0) X N (cid:1) . Byformulas (23) and (22), we deduce that I is a subobject of T in HH YD . Hence I is a two-sided idealof T in HH YD . Moreover R := T /I with the induced structures is in HH YD so that the canonicalprojection p R : T → R is in HH YD .To check (31) for I we must show that∆ T (cid:16) v [ n ] (cid:17) ∈ T ⊗ I + I ⊗ T, for every n ≥ N. UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 13
For n = N , this follows from (28) and the fact that, since q has order N , then (cid:0) Ni (cid:1) q = 0 for i = 0 , N . For n ≥ N + 1 , in view of (27) we have v [ n ] = Y ≤ i ≤ n − N − ω (cid:0) g ⊗ g i ⊗ g N (cid:1) v [ n − N ] v [ N ] . Hence, Lemma 4.5 implies that ∆ T (cid:0) v [ n ] (cid:1) ∈ T ⊗ I + I ⊗ T .Since by Proposition 4.3, ε T ( v [ n ] ) = δ n, and N = 0, it is clear that ε T ( I ) = 0 . Since I is a two-sided ideal of T = T ( V ), we have that m T ( T ⊗ I + I ⊗ T ) ⊆ I. Since Ker ( p R ⊗ p R ) = T ⊗ I + I ⊗ T, we deduce that m T (Ker ( p R ⊗ p R )) ⊆ I. By Proposition 4.4, there are maps m R , u R , ∆ R , ε R suchthat ( R, m R , u R , ∆ R , ε R ) is a bialgebra in (cid:0) HH YD , ⊗ , k , H a, l, r, c (cid:1) and p R is a bialgebra morphism.Recall that the Iverson bracket [[ P ]] is a notation that denotes a number that is 1 if the condition P in double square brackets is satisfied, and 0 otherwise.By the above we have the following result. Theorem . Let (( H, ω ) , g, χ ) be a quasi- Y D datum for q > , a primitive N th root of unity.(i) There is a bialgebra R = R (( H, ω ) , g, χ ) in HH YD with basis (cid:0) x [ n ] (cid:1) ≤ n ≤ N − and structuregiven as follows: ρ (cid:16) x [ n ] (cid:17) : = g n ⊗ x [ n ] ,h ⊲ x [ n ] : = χ [ n ] ( h ) x [ n ] , where χ [ n ] ∈ H ∗ is defined in (25) , R : = x [0] ,m R (cid:16) x [ a ] ⊗ x [ b ] (cid:17) = [[ a + b ≤ N − Y ≤ i ≤ a − ω − (cid:0) g ⊗ g i ⊗ g b (cid:1) x [ a + b ] when a, b ≥ , ∆ R (cid:16) x [ n ] (cid:17) = X ≤ i ≤ n β ( i, n ) x [ i ] ⊗ x [ n − i ] , where β ( i, n ) is defined in (29) ,ε R (cid:16) x [ n ] (cid:17) : = δ n, . (ii) For R the bialgebra in HH YD from (i), let B := R H , the bosonization of R by H . Then B ⊆ k R ⊗ H. Proof. (i) Take R := T ( V ) /I as above and set x [ n ] := v [ n ] + I .(ii) For 0 ≤ n < N , let R [ n ] := ⊕ ≤ a ≤ n k x [ a ] . Then, by the structure maps for R in (i), R [ n ] is asubobject of R in HH YD such that∆ R (cid:0) R [ n ] (cid:1) ⊆ X ≤ i ≤ n R [ i ] ⊗ R [ n − i ] . Set B [ n ] := R [ n ] ⊗ H. By the structure maps for B in Theorem 3.4, we see∆ B (cid:0) B [ n ] (cid:1) ⊆ X ≤ i ≤ n B [ i ] ⊗ B [ n − i ] . Since B = ∪ n ∈ N B [ n ] , we have proved that B is a filtered coalgebra so that, by [Sw2, Proposition11.1.1, page 226], B ⊆ B [0] = R [0] ⊗ H = k x [0] ⊗ H = k R ⊗ H. (cid:3) Note that the result in the previous theorem still holds formally if q = 1 but is not so interesting,since, in this case R collapses to the base field k . Definition . Let ((
H, ω ) , g, χ ) be a quasi- Y D datum for q = 1, a primitive N -th root of unity.The bialgebra R = R (( H, ω ) , g, χ ) of the previous theorem will be called a quantum line for thegiven datum. Proposition . The bialgebra R from Theorem 4.6 is a Hopf algebra in HH YD with bijectiveantipode S R : R → R defined by S R (cid:16) x [ n ] (cid:17) = ( − n χ ( g ) n ( n − x [ n ] for ≤ n ≤ N − . Proof.
Consider the basis (cid:0) x [ n ] (cid:1) ≤ n ≤ N − of the bialgebra R = R (( H, ω ) , g, χ ) in HH YD . We want todefine a linear map S R : R → R on the basis which a posteriori is expected to be antimultiplicativein HH YD . Set S R (cid:0) x [1] (cid:1) = − x [1] . Then, for 1 < n ≤ N − , we have S R (cid:16) x [ n ] (cid:17) = S R m R (cid:16) x [1] ⊗ x [ n − (cid:17) = m R ( S R ⊗ S R ) c R,R (cid:16) x [1] ⊗ x [ n − (cid:17) = m R ( S R ⊗ S R ) (cid:16) x [1] − ⊲ x [ n − ⊗ x [1]0 (cid:17) = m R ( S R ⊗ S R ) (cid:16) g ⊲ x [ n − ⊗ x [1] (cid:17) = χ [ n − ( g ) m R ( S R ⊗ S R ) (cid:16) x [ n − ⊗ x [1] (cid:17) = χ [ n − ( g ) S R (cid:16) x [ n − (cid:17) S R (cid:16) x [1] (cid:17) = − χ [ n − ( g ) S R (cid:16) x [ n − (cid:17) x [1] , where c R,R : R ⊗ R → R ⊗ R denotes the braiding of HH YD evaluated in R . Let us check that thisforces S R (cid:16) x [ n ] (cid:17) = ( − n χ ( g ) n ( n − x [ n ] for 0 ≤ n ≤ N − . For n = 0 , n with 1 < n ≤ N − n − . Then S R (cid:16) x [ n ] (cid:17) = − χ [ n − ( g ) S R (cid:16) x [ n − (cid:17) x [1] = − Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ g (cid:1) χ ( g ) n − ( − n − χ ( g ) ( n − n − x [ n − x [1] = Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ g (cid:1) ( − n χ ( g ) n ( n − x [ n − x [1] = Y ≤ i ≤ n − ω (cid:0) g ⊗ g i ⊗ g (cid:1) ( − n χ ( g ) n ( n − Y ≤ i ≤ n − ω − (cid:0) g ⊗ g i ⊗ g (cid:1) x [ n ] = ( − n χ ( g ) n ( n − x [ n ] . We have S R (cid:18)(cid:16) x [ n ] (cid:17) (cid:19) (cid:16) x [ n ] (cid:17) = X ≤ i ≤ n β ( i, n ) S R (cid:16) x [ i ] (cid:17) x [ n − i ] = X ≤ i ≤ n β ( i, n ) ( − i χ ( g ) i ( i − x [ i ] x [ n − i ] = X ≤ i ≤ n β ( i, n ) ( − i χ ( g ) i ( i − Y ≤ j ≤ i − ω − (cid:0) g ⊗ g j ⊗ g n − i (cid:1) x [ n ] . But(34) β ( i, n ) Y ≤ j ≤ i − ω − (cid:0) g ⊗ g j ⊗ g n − i (cid:1) = (cid:18) ni (cid:19) q so that S R (cid:18)(cid:16) x [ n ] (cid:17) (cid:19) (cid:16) x [ n ] (cid:17) = "X ≤ i ≤ n (cid:18) ni (cid:19) q ( − i q i ( i − x [ n ] . UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 15
By [Ka, Proposition IV.2.7] we have that X ≤ i ≤ n (cid:18) ni (cid:19) q ( − i q i ( i − a n − i X i = Y ≤ i ≤ n − (cid:0) a − q i X (cid:1) for any scalar a and variable X. If we take a = 1 and evaluate this polynomial in X = 1 , we get P ≤ i ≤ n (cid:0) ni (cid:1) q ( − i q i ( i − = δ n, . Hence S R (cid:18)(cid:16) x [ n ] (cid:17) (cid:19) (cid:16) x [ n ] (cid:17) = δ n, x [ n ] = δ n, x [0] = ε R (cid:16) x [ n ] (cid:17) R . On the other hand we have (cid:16) x [ n ] (cid:17) S R (cid:18)(cid:16) x [ n ] (cid:17) (cid:19) = X ≤ i ≤ n β ( i, n ) x [ i ] S R (cid:16) x [ n − i ] (cid:17) = X ≤ w ≤ n β ( n − w, n ) x [ n − w ] S R (cid:16) x [ w ] (cid:17) = X ≤ w ≤ n β ( n − w, n ) ( − w χ ( g ) w ( w − x [ n − w ] x [ w ](34) = X ≤ w ≤ n (cid:18) nw (cid:19) q ( − w q w ( w − x [ n ] = δ n, x [ n ] = δ n, x [0] = ε R (cid:16) x [ n ] (cid:17) R . We note that S R : R → R is trivially bijective. (cid:3) Recall the definition of a morphism of quasi-
Y D data from Definition 3.13. Note that if ϕ :(( H, ω ) , g, χ ) → (( L, α ) , ℓ, ξ ) is a morphism of quasi- Y D data, with ((
H, ω ) , g, χ ) a quasi- Y D datum for q then (( L, α ) , ℓ, ξ ) is also a quasi- Y D datum for q since ξ ( ℓ ) = ξϕ ( g ) = χ ( g ). It followseasily from equation (25) that ξ [ n ] ϕ = χ [ n ] for all n ≥
1. The proof of the next proposition isstraightforward and so the details are left to the reader.
Proposition . Let ϕ : (( H, ω ) , g, χ ) → (( L, α ) , l, ξ ) be a morphism of quasi- Y D data with q := χ ( g ) a primitive N -th root of unity , N > . Let (cid:0) x [ n ] (cid:1) ≤ n ≤ N − be the canonical basis for R H := R (( H, ω ) , g, χ ) and (cid:0) y [ n ] (cid:1) ≤ n ≤ N − the canonical basis for R L := R (( L, α ) , l, ξ ) . Considerthe k -linear isomorphism f : R H → R L mapping x [ n ] to y [ n ] for all n ∈ { , . . . , N − } . Then ρ R L f = ( ϕ ⊗ f ) ρ R H , µ R L ( ϕ ⊗ f ) = f µ R H , R L = f (1 R H ) , m R L ( f ⊗ f ) = f m R H , ∆ R L f = ( f ⊗ f ) ∆ R , ε R L f = ε R H . Moreover f ⊗ ϕ : R H H → R L L is a dual quasi-bialgebra homomorphism. Quasi-Yetter-Drinfeld data for bosonizations
In this section we consider quasi-
Y D data for bosonizations R H . In the next lemma, weassume that we have a bosonization B = R H with a quasi- Y D datum and we find that thisyields a quasi-
Y D datum for H . Lemma . For ( H, ω ) a dual quasi-bialgebra and R a bialgebra in HH YD , consider the dual quasi-bialgebra B := R H , the bosonization of R by H , defined in Theorem 3.4. Assume that B ⊆ k R ⊗ H and let (( B, ω B ) , g, χ B ) be a quasi- Y D datum. Then there exists c ∈ G ( H ) such that g = 1 R c , and (( H, ω ) , c, χ B σ ) is a quasi- Y D datum where σ : H ֒ → B is the inclusion. Moreover,for every r ∈ R, h ∈ H we have χ B ( r h ) = ω − H ( r − ⊗ h ⊗ c ) χ B (1 h ) ω H ( r − ⊗ c ⊗ h ) χ B ( r , (35) χ B ( r · R s ω − H ( r − ⊗ s − ⊗ c ) χ B ( s H ) χ B ( r , (36) χ B ( r h ) ch = ( r − h ) χ B ( r h ) c, (37) χ B (cid:0) r r − (cid:1) c ⊲ r = ω − H ( r − ⊗ r − ⊗ c ) r χ B (cid:0) r H (cid:1) . (38) Proof.
Since g ∈ G ( B ) ⊆ B ⊆ k ⊗ H , then g = 1 R c = σ ( c ) for some c ∈ H . Thus c = ( πσ )( c ) = π ( g ) and since the maps π, σ from Section 3.1 are coalgebra maps and g is grouplike, then c isgrouplike.In order to apply Lemma 3.14 to conclude that (( H, ω ) , c, χ B σ ) is a quasi- Y D datum, we mustshow that cχ B σ ( h ) h = h χ B σ ( h ) c for all h ∈ H . Since (( B, ω B ) , g, χ B ) is a quasi- Y D datum,and so satisfies (15), then for every r ∈ R, h ∈ H , gχ B (( r h ) ) ( r h ) = ( r h ) χ B (( r h ) ) g. If we let r = 1 R in the equation above, and apply π to both sides, we obtain cχ B ( σ ( h ) ) π ( σ ( h ) ) = π ( σ ( h ) ) χ B ( σ ( h ) ) c, and since σ, π are coalgebra maps with πσ the identity, then (15) holds for (( H, ω ) , c, χ B σ ) and byLemma 3.14, (( H, ω ) , c, χ B σ ) is a quasi- Y D datum.Since ω B = ω H ◦ π ⊗ then for all x, y ∈ B , by (14) for the quasi- Y D datum for B , we have that χ B ( xy ) is: ω − B ( x ⊗ y ⊗ g ) χ B ( y ) ω B ( x ⊗ g ⊗ y ) χ B ( x ) ω − B ( g ⊗ x ⊗ y )(39)= ω − H ( π ( x ) ⊗ π ( y ) ⊗ c ) χ B ( y ) ω H ( π ( x ) ⊗ c ⊗ π ( y )) χ B ( x ) ω − H ( c ⊗ π ( x ) ⊗ π ( y ))By (13), ( π ⊗ π ⊗ B ⊗ π )∆ B ( r r − ⊗ r − ⊗ ( r ⊗ H ;(40) ( π ⊗ B ⊗ π ⊗ π )∆ B (1 h ) = h ⊗ (1 h ) ⊗ h ⊗ h , (41)and so, letting x = r y = 1 h , we have that χ B ( r h ) is:(42) ω − H ( r − ⊗ h ⊗ c ) χ B (1 h ) ω H ( r − ⊗ c ⊗ h ) χ B ( r ω − H ( c ⊗ ⊗ h ) , and since ω H is normalized, (35) holds.Similarly χ B ( r · R s χ B (( r s π ⊗ B ⊗ π ⊗ π )∆ B ( s H ) = s − ⊗ ( s ⊗ H ⊗ H , it is straightforward to verify (36).Now we prove (37). Since (( B, ω B ) , g, χ B ) satisfies (15), we have, gχ B (( r h ) ) ( r h ) = ( r h ) χ B (( r h ) ) g, for every r ∈ R, h ∈ H. Recall from Theorem 3.4 that∆ B ( r h ) = ω − H ( r − ⊗ r − ⊗ h ) r r − h ⊗ r h , so that applying π to the left hand side of (15) for B we obtain: cχ B ( r r − h ) ω − H ( r − ⊗ r − ⊗ h ) ε ( r ) h = χ B ( r h ) ch . Applying π to the right hand side yields ω − H ( r − ⊗ r − ⊗ h ) ε ( r ) r − h χ B ( r h ) c = χ B ( r h ) r − h c, and thus (37) holds.Equation (38) is verified in a similar fashion. Let h = 1 in the left hand side of equation (15)for B and then apply R ⊗ ε H to obtain χ B (( r )( R ⊗ ε H )[(1 c )( r ] = χ B ( r r − )( R ⊗ ε H )[ c ⊲ r c ] = χ B ( r ⊗ r − ) c ⊲ r . Now let h = 1 in the right hand side of (15) for B and apply R ⊗ ε H to obtain χ B (( r )( R ⊗ ε H )[( r (1 c )] = χ B ( r R ⊗ ε H )[( r r − )(1 c )]= χ B ( r ω − H (( r ) − ⊗ ( r − ) ⊗ c )( r ) ε H (( r − ) c )= χ B ( r ω − H ( r − ⊗ r − ⊗ c ) r , and this finishes the proof of (38). (cid:3) In the next proposition we show how an arbitrary quasi-
Y D datum on a bosonization R H where R := R (( H, ω H ) , g H , χ H ) is related to g H and χ H . UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 17
Proposition . Let (( H, ω H ) , g H , χ H ) be a quasi- Y D datum for a primitive N -th root of unity q and let R = R (( H, ω H ) , g H , χ H ) be the bialgebra in HH YD introduced in Theorem 4.6. Let B = R H , the bosonization of R by H and suppose that (( B, ω B ) , g B , χ B ) is a quasi- Y D datum.Then there exists d ∈ G ( H ) such that g B = 1 R d . If d = g H d , then (i) χ B ( r h ) = ε R ( r ) χ B (1 R h ) , for every r ∈ R, h ∈ H , (ii) χ B (1 R g H ) χ H ( d ) = 1 , (iii) dg H = g H d. Proof.
Theorem 4.6 implies that B ⊆ k R ⊗ H . Then Lemma 5.1 implies that there exists d ∈ G ( H ) such that g B = 1 R d. By (37) with r = x [1] and h = 1 H , χ B (cid:16) x [1] H (cid:17) d = χ B (cid:16) x [1] H (cid:17) g H d. If χ B (cid:0) x [1] H (cid:1) = 0, then d = g H d , contrary to our assumption and so χ B (cid:0) x [1] H (cid:1) = 0.Now, let 2 ≤ n ≤ N − χ B (cid:0) x [ n − H (cid:1) = 0 . Then χ B (cid:16) x [ n ] H (cid:17) = χ B (cid:16) x [1] · R x [ n − H (cid:17) (36) = ω − H (cid:0) g H ⊗ g n − H ⊗ d (cid:1) χ B (cid:16) x [ n − H (cid:17) χ B (cid:16) x [1] H (cid:17) = 0 , so that χ B (cid:16) x [ n ] H (cid:17) = δ n, , for 0 ≤ n ≤ N − . Now χ B (cid:16) x [ n ] h (cid:17) (35) = ω − H ( g nH ⊗ h ⊗ d ) χ B (1 R h ) ω H ( g nH ⊗ d ⊗ h ) χ B (cid:16) x [ n ] H (cid:17) = δ n, χ B (1 R h ) , and so χ B ( r h ) = ε R ( r ) χ B (1 R h ) , for every r ∈ R, h ∈ H. Next we consider equation (38) with r = x [1] .The left hand side is χ B (cid:18)(cid:16) x [1] (cid:17) (cid:16) x [1] (cid:17) − (cid:19) d ⊲ (cid:16) x [1] (cid:17) = χ B (cid:16) x [1] H (cid:17) d ⊲ R + χ B (1 R g H ) d ⊲ x [1] = χ B (1 R g H ) d ⊲ x [1] = χ B (1 R g H ) χ H ( d ) x [1] and the right hand side is ω − H ( (cid:16) x [1] (cid:17) − ⊗ (cid:16) x [1] (cid:17) − ⊗ d ) (cid:16) x [1] (cid:17) χ B (cid:18)(cid:16) x [1] (cid:17) H (cid:19) = ω − H (1 H ⊗ g H ⊗ d ) χ B ( x [1] H ) + ω − H ( g H ⊗ H ⊗ d ) x [1] χ B (1 R H )= x [1] + 1 R χ B (cid:16) x [1] H (cid:17) = x [1] . and we can conclude that χ B (1 R g H ) χ H ( d ) = 1 . Now we apply (15) for the quasi-
Y D datum (
H, d, χ B (1 R ⊗ − )) from Lemma 5.1 with h = g H to obtain χ B (1 R g H ) dg H = g H dχ B (1 R g H ) , and since χ B (1 R g H ) is invertible, we obtain dg H = g H d. (cid:3) Examples
In this section, we present examples illustrating the theory in the previous sections. The problemof course is to find the reassociator explicitly. Our examples are based on the coalgebra k C n wherethe cocycles are well-known.6.1. Group cohomology.
First we set some notation. Our examples will involve cyclic groupsof order n and n , n >
1. We will denote C n = h c i and C n = h c i . We will always denote by q aprimitive n -rd root of unity and set ζ := q n . Let φ : C n → C n be the canonical projection with φ ( c ) = c and denote by the same symbol the corresponding map k C n → k C n . For every a ∈ Z ,let a ′ ∈ { , . . . , n − } be congruent to a modulo n .Since k is an algebraically closed field of characteristic zero, by [Sw1, Theorem 3.1], the Sweedlercohomology can be computed through an isomorphism H tsw ( k C n , k ) ∼ = H t (cid:0) C n , k × (cid:1) , where the latter is the group cohomology computed as in [We, page 167].For 0 ≤ i ≤ n − ≤ a, b, d , define ω ζ i : ( k C n ) ⊗ → k by(43) ω ζ i (cid:0) c a ⊗ c b ⊗ c d (cid:1) = ω ζ i (cid:0) c a ⊗ c d ⊗ c b (cid:1) = ζ ia [[ b ′ + d ′ >n − . Since ζ = q n , it is easy to check that(44) ω ζ i (cid:0) c a ⊗ c b ⊗ c d (cid:1) = ζ ia [[ b ′ + d ′ >n − = q ina [[ b ′ + d ′ >n − = q ia ( b ′ + d ′ − ( b + d ) ′ ) . One can prove that the set of Sweedler 3-cocycles is given by(45) Z sw ( k C n , k ) = (cid:8)(cid:0) ω ζ i (cid:1) v = ω ζ i ∗ ∂ v | ≤ i ≤ n − , v : k C ⊗ n → k is convolution invertible (cid:9) . This follows from the fact (see e.g. [MS, formulas (E.13) and (E.14)] over C ) that the map(46) (cid:8) k ∈ k × | k n = 1 (cid:9) → H sw ( k C n , k ) : k [ ω k ]is a group isomorphism. Proposition . Let k C n be the group algebra with its standard bialgebra structure and ω anormalized -cocycle. Then ( k C n , ω ) is a dual quasi-bialgebra and there is a gauge transformation α : ( k C n ) ⊗ → k , and ≤ i ≤ n − such that ( k C n , ω ) = ( k C n , ω ζ i ) α = ( k C n , ω ζ i ∗ ∂ α ) . Proof.
The first statement follows from the fact that k C n is cocommutative. Since ω is a normalizedSweedler 3-cocycle, by (45) there exists a convolution invertible map v : k C ⊗ n → k and i ∈{ , . . . , n − } such that ω = (cid:0) ω ζ i (cid:1) v = ∂ v ∗ ω ζ i and ( k C n , ω ) = ( k C n , ω vζ i ) = ( k C n , ω ζ i ) v . Since ω and ω ζ i are normalized, so is ∂ v . Thus, by Corollary 2.4, av is a gauge transformation for a = v (1 ⊗ − . Since ( k C n , ω ζ i ) v = ( k C n , ω ζ i ) av , the statement is proved. (cid:3) Remark . In fact, ( k C n , ω ζ i ) is a dual quasi-Hopf algebra, meaning that there exists an antipode S and maps α, β from k C n to k such that for all h ∈ k C n : S ( h ) α ( h ) h = α ( h )1 and h β ( h ) S ( h ) = β ( h )1;(47) ω ζ i ( h β ( h ) ⊗ S ( h ) ⊗ α ( h ) h ) = ω − ζ i ( S ( h ) ⊗ α ( h ) h ⊗ β ( h ) S ( h )) = ε ( h ) . (48)In this case, S is the usual antipode for k C n , α and β are both equal to the counit ε and thensince ω ζ i ( c j ⊗ c − j ⊗ c j ) = 1, the statement follows.Since by the above discussion the maps ω ζ i are not coboundaries, we have the following: Corollary . The dual quasi-bialgebra ( k C n , ω ζ i ) is not quasi-isomorphic to an ordinary bial-gebra, i.e., one with reassociator ε k C ⊗ n . On the other hand k C n with dual quasi-bialgebra structure via the bialgebra epimorphism φ : k C n → k C n , φ ( c ) = c , is quasi-isomorphic to an ordinary bialgebra since if ω is a normalized3-cocycle for k C n , then ωφ ⊗ is a coboundary. In fact, one can see by direct computation that ω ζ i φ ⊗ = ∂ v i where v i : ( k C n ) ⊗ → k is defined by v i ( c a ⊗ c b ) = q ia ( b − b ′ ) . UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 19
Quasi-
Y D data for k C n . To find quasi-
Y D data for ( k C n , ω ζ w ), we apply the results ofSection 3.2. For 0 ≤ z ≤ n −
1, from (19) we will be able to show that if (( k C n , ω ζ w ) , g := c z , χ ) isa quasi- Y D datum then(49) χ ( c t ) = χ ( c ) t for 0 ≤ t ≤ n − χ ( c ) n = ζ wz , and thus, unless χ ( c ) n = 1, i.e., ζ wz = 1, then χ is not a character.We show (49) as follows. From (19) and the definition of ω ζ w , we deduce that χ ( c t ) = χ ( c ) t for1 ≤ t ≤ n −
1. By unitarity of χ , this equality also holds for t = 0. By unitarity of χ and the factthat c n = 1, we get 1 = χ (1) = χ ( c n ). On the other hand a direct computation of χ ( c n ) using (19)and the definition of ω ζ w yields χ ( c n ) = χ ( c ) n ζ − wz and so (49) is proved.Take t ∈ N . Then, since c n = 1, we have χ ( c t ) = χ ( c t ′ ) (49) = χ ( c ) t ′ . Thus (49) is equivalent to(50) χ ( c t ) = χ ( c ) t ′ for t ∈ N and χ ( c ) n = ζ wz . Proposition . Consider the dual quasi-bialgebra ( k C n , ω ζ w ) . Let c z ∈ C n , ≤ z ≤ n − and χ ∈ k C ∗ n . If (( k C n , ω ζ w ) , c z , χ ) is a quasi- Y D datum, then (50), or equivalently (49), holds.Conversely if χ is a unitary map satisfying (50) , or equivalently (49), for some ≤ z ≤ n − then (( k C n , ω ζ w ) , c z , χ ) is a quasi- Y D datum.Proof.
The first assertion follows immediately from Lemma 3.15 and Remark 3.16.Since c z ∈ G ( k C n ), since χ ∈ k C ∗ n is unitary by assumption, and since (15) holds because k C n is both commutative and cocommutative, it remains only to check (14) . We check (14) ongenerators. The equality holds trivially for h = 1 k C n or k = 1 k C n . Hence we can assume that h = c a and k = c b for 1 ≤ a, b ≤ n − . Then the left side of (14) is χ (cid:0) c a c b (cid:1) = χ (cid:0) c a + b (cid:1) (50) = χ ( c ) ( a + b ) ′ = χ ( c ) a + b − [[ a + b ≥ n ]] n (50) = χ ( c ) a + b ζ − [[ a + b ≥ n ]] wz . Since ω ζ w = ω ζ w ( k C n ⊗ τ ) and k C n is cocommutative, the right hand side of (14) is: ω − ζ w (cid:0) c z ⊗ c a ⊗ c b (cid:1) χ ( c a ) χ (cid:0) c b (cid:1) (50) = χ ( c ) a + b ω − ζ w (cid:0) c z ⊗ c a ⊗ c b (cid:1) (43) = χ ( c ) a + b ζ − [[ a + b ≥ n ]] wz . Thus (14) holds and the proof is complete. (cid:3)
Example . Consider the dual quasi-bialgebra ( k C n , ω ζ i ) with i >
0. Let c z ∈ C n with 1 ≤ z ≤ n −
1, and then for χ ∈ ( k C n ) ∗ to satisfy (49), we must have that χ ( c ) n = ζ iz = q niz . Thus if wedefine χ ( c t ) = q izt for 0 ≤ t ≤ n −
1, then (( k C n , ω ζ i ) , c z , χ ) is a quasi Y D -datum for χ ( c z ) = q iz .Note that q iz is a primitive r th root of unity where r = n ( n ,iz ) .More generally, for 0 ≤ j ≤ n −
1, let χ j : k C n → k be defined by χ j ( c t ) = ζ jt q izt if 0 ≤ t ≤ n − χ j ( c ) n = ( ζ j ) n q izn = ζ iz , so that (49) holds, (( k C n , ω ζ i ) , c z , χ j ) is a quasi- Y D datum for χ j ( c z ) = ζ jz q iz and the order of ζ jz q iz is n ( njz + iz ,n ) . Example . Let n = p , a prime. For the quasi- Y D datum (( k C p , ω ζ i ) , c z , χ j ) in Example 6.5with χ j ( c t ) = ζ jz q izt , there are p − i and also for z that give quasi- Y D data for aprimitive p rd root of unity and since j = 0 , . . . , p −
1, there are p choices for j . Thus one mayform p ( p − bosonizations R k C p where R has dimension p . Below we discuss which of thesecan be isomorphic or quasi-isomorphic.Suppose that H := ( k C p , ω ζ i ) and L := ( k C p , ω ζ i ′ ). Then by the discussion in Section 6.1, H and L are quasi-isomorphic if and only if i = i ′ . If R H is quasi-isomorphic to S L for some R, S as in Theorem 4.6, then, by Remark 3.6, H is quasi-isomorphic to L and thus i = i ′ . Thus if twobosonizations as constructed above are quasi-isomorphic, then i = i ′ , i.e., H = L .Now fix H := ( k C p , ω ζ i ), and consider the quasi- Y D data D := ( H, c z , χ j ) and E := ( H, c w , χ k ).Let R (respectively S ) be the Hopf algebra in HH YD constructed from D ( E respectively) withbasis x [ n ] (respectively y [ n ] ). Set x = x [1] , y = y [1] . Suppose that there is a dual quasi-bialgebraisomorphism Φ : R H → S H . By the formula for ∆ R in Theorem 4.6, the comultiplicationformula from Theorem 3.4 and the fact that the coefficients β ( i, n ) are nonzero, we have that P , c i ( R H ) = k (cid:0) − c i (cid:1) + δ i,z k ( x P , c i ( S H ) = k (cid:0) − c i (cid:1) + δ i,w k ( y c z ) = 1 ′ ( c z ).Write Φ ′ ( c z ) = c a with 0 ≤ a ≤ p −
1. Since x ∈ P , c z ( R H ), we get that Φ( x ∈ P , c a ( S H ) = k − c a ) + δ a,w k ( y a = w , then Φ( x ∈ k − c a ) and hence x ∈ k Φ − (1 − c a )) ⊆ k H , a contradiction. Thus a = w and hence Φ(1 c z ) = 1 c w , andΦ( x αy β − c w ). Since Φ − (1 H ) = 1 H then α = 0.Then we have Φ[(1 c z )( x χ j ( c z ) x c z ]= Φ[ χ j ( c z )( x c z )]= χ j ( c z )[ αy β − c w )][1 c w ]= χ j ( c z )[ αy c w + β − c w ) c w ] . However, Φ(1 c z )Φ( x c w )( αy β − c w ))= χ k ( c w ) αy c w + β − c w ) c w . Thus β = 0 and χ j ( c z ) = χ k ( c w ), i.e., ζ jz q iz = ζ kw q iw . Thus p divides p ( jz − kw ) + i ( z − w )so that p divides i ( z − w )( z + w ). Then either z = w or z + w = p .Suppose that z = w . Then p divides z ( j − k ). This is impossible unless j = k and then the twoquasi- Y D data are the same.Suppose that z + w = p . Then p divides jz − kw + i ( z − w ) = jz − k ( p − z ) + i (2 z − p ) so that p divides z ( j + k + 2 i ), i.e., p | ( j + k + 2 i ).In any case, there are at least p ( p −
1) nonisomorphic bosonizations. Fix z = 1. Then there are p choices for j and p − i giving nonisomorphic bosonizations.In the next example, for a change, we consider the group algebra of a nonabelian group andfind a quasi- Y D datum.
Example . Let G := Dic p , the dicyclic group of order 4 p for p an odd prime. Then Dic p = C p ⋊ C = h x, y | x = 1 = y p , xyx − = y − i and Z ( G ) = { , x } . Since C p is a normal subgroupof G then there is a bialgebra projection π from k G to k C = k h c i by π ( y i x j ) = c j . Let ω := ω ζ be the cocycle defined in Subsection 6.1 for k C with q a primitive 16th root of unity and ζ = q .Let ω G : k G ⊗ → k be defined by ω G := ωπ ⊗ and thus ( k G, ω G ) is a dual quasi-bialgebra and π is a dual quasi-bialgebra morphism. By Corollary 2.7, since ( k C , ω ζ ) is nontrivial and since thereis an inclusion σ : k C ֒ → k G such that πσ is the identity, then ( k G, ω G ) is also nontrivial.By Example 6.5 with n = 4, (( k C , ω ζ ) , c , χ ) with χ ( c t ) = q t is a quasi- Y D datum for χ ( c ) = q = ζ , a primitive 4th root of unity. By Lemma 3.14, since π ( x ) = c and x ∈ Z ( G ),then (( G, ω G ) , x , χ G := χπ ) is a quasi- Y D datum for ( k G, ω G ).Note that for a nonabelian group with trivial centre, the construction in the example above canonly yield a trivial Y D datum for q = 1. On the other hand, the same construction is possiblefor any nonabelian group G with a projection onto a cyclic group such that the kernel does notcontain the centre of G .The next example shows that (49) need not hold for a quasi- Y D datum for k C N if ω = ω ζ w , inparticular it can happen that χ ( c t ) = χ ( c ) t for some 0 < t < N . Example . Let φ : k C n = k h c i → k C n = k h c i be the surjection of bialgebras from Section 6.1given by φ ( c ) = c . Then φ induces a morphism of dual quasi-bialgebras from ( k C n , ω ζ φ ⊗ = ∂ v )to ( k C n , ω ζ ) where, by Section 6.1, v ( c a ⊗ c b ) = q a ( b − b ′ ) .By Example 6.5, (( k C n , ω ζ ) , c, χ ) is a quasi Y D -datum, with χ ( c t ) = q t for 0 ≤ t ≤ n −
1, andso by Lemma 3.14, (( k C n , ∂ v ) , c , χφ ) is a quasi- Y D datum also. However, taking t = n < n − χφ ( c n ) = χ ( c n ) = χ (1) = 1 while ( χφ ( c )) n = χ ( c ) n = q n = ζ .Thus in this case (49) is not satisfied. Note that ω ζ φ ⊗ = ω ζ φ ⊗ ( k C n ⊗ τ ) so that (19) still holds. UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 21
Example . Let ((
H, ω ) , g, χ ) be a quasi- Y D datum for some primitive N -th root of unity q , N >
0. Let L = k h g i and let ϕ : ( L, ω L ) ֒ → ( H, ω H ) be the canonical inclusion where ω L = ω | L ⊗ . Note that ϕ : (cid:0) ( L, ω L ) , g, χ | L (cid:1) → (( H, ω ) , g, χ ) is a morphism of quasi- Y D data. ByProposition 4.9, we have a dual quasi-bialgebra homomorphism f ⊗ ϕ : R L L → R H H where R L := R (cid:0) ( L, ω L ) , g, χ | L (cid:1) , R H := R (( H, ω ) , g, χ ) and f : R L → R H is a k -linear isomorphism.Note that, since ϕ is injective, so is f ⊗ ϕ so that R L L identifies with a dual quasi-subbialgebraof R H H. We point out that the following example is dual to one given by Gelaki in [Ge, subsection 3.1].There a quasi-Hopf algebra is given which is quasi-isomorphic to an ordinary Hopf algebra butcontains a sub-quasi-Hopf algebra which is not.
Example . Recall the setting of Example 6.8 where we have a morphism of quasi-
Y D datafrom (( k C n , ω n := ∂ v ) , c , χφ ) to (( k C n , ω ζ ) , c, χ ) with χ ( c t ) = q t , 0 ≤ t ≤ n −
1, induced by thebialgebra surjection φ : k C n → k C n with φ ( c ) = c . Note that both are quasi- Y D -data for q where q is a primitive n -rd root of unity.The isomorphism f : R n → R n from Proposition 4.9 yields a dual quasi-bialgebra surjection f ⊗ φ : R n k C n → R n k C n where R n := R (( k C n , ω n ) , c , χφ ), R n := R (( k C n , ω ζ ) , c, χ ). Set A := R n k C n and B := R n k C n , so that B is a quotient of the dual quasi-bialgebra A. Since ( k C n , ∂ v ) can be twisted by v − to ( k C n , ε ( k C n ) ⊗ ), then by Remark 3.8, A is also quasi-isomorphic to an ordinary bialgebra. Infact, it is easy to check that A should be deformed by the gauge transformation µ := v − ( π ⊗ π ) toobtain an ordinary bialgebra. On the other hand, since ω ζ is not trivial in H ( k C n , k ), ( k C n , ω ζ )cannot be quasi-isomorphic to an ordinary bialgebra and thus by Remark 3.8, neither can B .We can say more about A µ . Since A µ is finite dimensional with coradical k C n , which is a Hopfalgebra, then A µ is also a Hopf algebra [Ta, Remark 36]. Now, let (cid:0) x [ n ] (cid:1) ≤ n ≤ N − be the canonicalbasis for R n . Then X := x [1] k C n is a nontrivial skew-primitive element since∆ A ( x [1] k C n ) = (cid:16) x [1] (cid:17) (cid:16) x [1] (cid:17) − ⊗ (cid:16) x [1] (cid:17) k C n = (cid:0) R n c (cid:1) ⊗ (cid:16) x [1] k C n (cid:17) + (cid:16) x [1] k C n (cid:17) ⊗ (cid:0) R n k C n (cid:1) = (cid:0) R n c (cid:1) ⊗ (cid:16) x [1] k C n (cid:17) + (cid:16) x [1] k C n (cid:17) ⊗ A so that, if we set Γ := 1 R n c , we get∆ A ( X ) = X ⊗ A + Γ ⊗ X. Since A and A µ have the same coalgebra structure, X is a (1 A , Γ)-primitive element also in A µ .Consider the sub-Hopf algebra of A µ generated by X and Γ . This is a Taft algebra of dimension o (Γ) = n . Hence A µ = T q .6.3. Quasi-
Y D data for R k C n . Now we apply Proposition 5.2 to a quasi-
Y D datum used inour examples.
Proposition . Let H := (( k C n , ω ζ ) , c, χ ) with χ ( c t ) = q t for ≤ t ≤ n − , be the quasi- Y D datum from Example 6.5, let R := R (( k C n , ω ζ ) , c, χ ) , and let B := R H . Suppose that there isa quasi- Y D datum for B , (( B, ω B ) , g B , χ B ) as in Proposition 5.2. Then g B = 1 R c w for some ≤ w ≤ n − , and for ≤ t ≤ n − and r ∈ R , (51) χ B (cid:0) r c t (cid:1) = ε R ( r ) q − wt Y ≤ i ≤ t − ω − ζ (cid:0) c w ⊗ c i ⊗ c (cid:1) = ε R ( r ) q − wt . In particular g B and χ B are uniquely determined by w and (( k C n , ω ζ ) , c, χ ) . Proof.
By Proposition 5.2 there exists d = c w such that g B = 1 R d. Since c w = cc w , Proposition5.2(ii) may be applied to get that, χ B (1 R c ) = χ ( c w ) − = q − w . Since by Lemma 5.1 (( k C n , ω ζ ) , c w , χ B (1 R − )) is a quasi- Y D datum, then χ B (1 R − ) must satisfy(19), i.e., χ B (1 R c t ) = χ B (1 R c ) t Y ≤ i ≤ t − ω − ζ ( c w ⊗ c i ⊗ c ) = q − wt Y ≤ i ≤ t − ω − ζ ( c w ⊗ c i ⊗ c ) . The statement now follows from (i) of Proposition 5.2. (cid:3)
Corollary . Let
H, R, B be as in the proposition with n = 2 m . If (( B, ω B ) , g B = 1 c w , χ B ) is a quasi- Y D datum for B with c w = 1 , then w = m .Proof. By Lemma 5.1, if ((
B, ω B ) , g B = 1 c w , χ B ) is a quasi- Y D datum, then (( k C m , ω ζ ) , c w , χ B σ )is also a quasi- Y D datum where σ is the inclusion map. Then by (49), χ B σ ( c ) n = ζ w . By (51), χ B (1 c ) n = q − wn = ζ − w , so that ζ w = 1 and we must have that w = m . (cid:3) We are now able to construct a quasi-
Y D datum on a dual quasi-bialgebra which is a bosoniza-tion of a group algebra. We begin with a useful lemma.
Lemma . Let n, a ∈ N with ≤ a ≤ n − . Then |{ i | ≤ i ≤ a − , i ′ = n − }| = a − a ′ n Proof.
Note that the left hand side of the equation above is the number of nonnegative integerscongruent to n − n and strictly less than a . For t ≥
1, define an interval I t of n integersby I t = { j ∈ N | ( t − n ≤ j ≤ tn − } . Then the left hand side is the number of intervals I t whoseentries are less than a . If a = a ′ + sn , then a ∈ I s +1 and this number is clearly s . (cid:3) In the next example, we find a quasi-
Y D datum for B := R k C n where n is even. As always, q denotes a primitive n -rd root of unity and ζ := q n . Example . Let n = 2 m and let ( B := R k C n , ω B = ω ζ π ) be the dual quasi-bialgebraof dimension n constructed via the quasi- Y D datum (( k C n , ω ζ ) , c, χ ) for q with χ ( c t ) = q t ,0 ≤ t ≤ n −
1, as in Example 6.5. We will construct a quasi-
Y D datum for B for ι := q − m , aprimitive 4-th root of unity.First note that (( k C n , ω ζ ) , c m , e χ ) with e χ ( c t ) = q − mt for 0 ≤ t ≤ n − Y D datum byProposition 6.4 since e χ ( c ) n = q − mn = ζ − m = ζ m since ζ has order 2 m . Also (( k C n , ω ζ ) , c m , e χ ) isa quasi- Y D datum for ι since e χ ( c m ) = q − m .We will now show that (( B, ω B := ω ζ π ⊗ ) , g B := σ ( c m ) , χ B := e χπ ), where π, σ are the usualprojection and inclusion maps from Remark 3.8, is a quasi- Y D datum. Since π is a surjection ofdual quasi-bialgebras from ( B, ω B ) to ( k C n , ω ζ ) with πσ ( c m ) = c m , it remains to show that for all b ∈ B , σ ( c m ) e χπ ( b ) b = b e χπ ( b ) σ ( c m )in order to apply Lemma 3.14 and conclude that (( B, ω B := ω ζ π ) , g B := σ ( c m ) , χ B := e χπ ) is aquasi- Y D datum for ι .Let b = x [ a ] c ℓ for 0 ≤ a ≤ n − ≤ ℓ ≤ n −
1. Since by Theorem 4.6, ∆ R ( x [ a ] ) = P ≤ i ≤ a β ( i, a ) x [ i ] ⊗ x [ a − i ] and since β (0 , a ) = β ( a, a ) = 1, by applying ε H on the left and on theright of (13),(52) π ( b ) ⊗ b = c a c ℓ ⊗ x [ a ] c ℓ = c a c ℓ ⊗ b and b ⊗ π ( b ) = x [ a ] c ℓ ⊗ c ℓ = b ⊗ c ℓ . By the formula for multiplication in B = R k C n in Theorem 3.4, we have that(1 c m )( x [ a ] c ℓ ) = ω ζ ( c m ⊗ c a ⊗ c ℓ ) ω − ζ ( c a ⊗ c m ⊗ c ℓ ) c m ⊲ x [ a ] c m c ℓ , and ( x [ a ] c ℓ )(1 c m ) = ω − ζ ( c a ⊗ c ℓ ⊗ c m ) x [ a ] c ℓ c m = ω − ζ ( c a ⊗ c m ⊗ c ℓ ) x [ a ] c ℓ c m . UANTUM LINES FOR DUAL QUASI-BIALGEBRAS 23
Since c m ⊲ x [ a ] = χ [ a ] ( c m ) x [ a ] where χ [ a ] is defined in Proposition 4.3, it remains to show that ω ζ ( c m ⊗ c a ⊗ c ℓ ) χ [ a ] ( c m ) e χ ( c a c ℓ ) = e χ ( c ℓ ) . By (14) for the quasi-
Y D datum (( k C n , ω ζ ) , c m , e χ ), e χ (cid:0) c a c ℓ (cid:1) = ω − ζ ( c m ⊗ c a ⊗ c ℓ ) e χ ( c a ) e χ ( c ℓ ) , and thus it suffices to prove that e χ ( c a ) χ [ a ] ( c m ) = 1. Since e χ ( c a ) = e χ (cid:16) c a ′ (cid:17) = q − ma ′ , this isequivalent to showing that χ [ a ] ( c m ) = q ma ′ . Since c m is a cocommutative element, by equation (30) χ [ a ] ( c m ) = χ ( c m ) a Y ≤ i ≤ a − ω ζ ( c m ⊗ c ⊗ c i ) = q ma Y ≤ i ≤ a − ω ζ ( c m ⊗ c ⊗ c i ′ )= q ma Y ≤ i ≤ a − ζ m [[1+ i ′ ≥ n ]] = q ma Y ≤ i ≤ a − ζ mδ i ′ ,n − . Thus we have to prove that q ma Y ≤ i ≤ a − ζ mδ i ′ ,n − = q ma ′ . But q ms = q − ms for every s ∈ n Z since, writing s = n b s , q mn b s = q n b s = 1. Thus it suffices toprove that Y ≤ i ≤ a − ζ mδ i ′ ,n − = q − m ( a − a ′ ) = q m ( a − a ′ ) . By Lemma 6.13, we have |{ i | ≤ i ≤ a − , i ′ = n − }| = a − a ′ n , so that Y ≤ i ≤ a − ζ mδ i ′ ,n − = Y ≤ i ≤ a − q nmδ i ′ ,n − = q mn P ≤ i ≤ a − δ i ′ ,n − = q mn a − a ′ n = q m ( a − a ′ ) . This shows that ((
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