Quasitriangular structures of the double of a finite group
aa r X i v : . [ m a t h . QA ] A ug QUASITRIANGULAR STRUCTURES OF THE DOUBLEOF A FINITE GROUP
MARC KEILBERG
Abstract.
We give a classification of all quasitriangular struc-tures and ribbon elements of D ( G ) explicitly in terms of grouphomomorphisms and central subgroups. This can equivalently beinterpreted as an explicit description of all braidings with whichthe tensor category Rep( D ( G )) can be endowed. We also charac-terize their equivalence classes under the action of Aut( D ( G )) anddetermine when they are factorizable. Introduction
Quasitriangular (quasi-)bialgebras were first introduced by Drinfel ′ d[9, 10, 11] as a way of producing solutions to the quantum Yang-Baxterequation. These have applications in statistical mechanics, where theyyield exactly solvable lattice models [14], as well as in quantum com-puting, where they can be characterized as universal quantum gates[17, 36]. A number of knot and link invariants can also be constructedfrom such objects [16, 34]. When we have the additional structure of afactorizable ribbon Hopf algebra, we also obtain projective representa-tions of mapping class groups of compact oriented surfaces of arbitrarygenus with a finite (possible empty) collection of marked boundarycircles [22].From a categorical point of view, if H is a quasitriangular (quasi-)Hopf algebra over the field of complex numbers then Rep( H ), thecategory of finite-dimensional representations of H , is a braided tensorcategory. In particular, the braidings of Rep( H ) are precisely given bythe quasitriangular structures of H [24, Theorem 10.4.2]. The braidedfusion categories Rep( D ( G )), the finite dimensional representations ofthe Drinfel’d double of a group, and more generally Rep( D ω ( G )) [8], Key words and phrases. quasitriangular structures, group doubles, braidings,ribbon elements, factorizability.A portion of this research was conducted during the author’s stay at the Insti-tut de Math´ematiques de Bourgogne, Universit´e de Bourgogne in Dijon, France,and was partially supported through a FABER Grant by the Conseil r´egional deBourgogne. have been of substantial recent interest in their own right. See [12, 23,25, 26, 27, 30] and the references therein.In general a Hopf algebra can have many quasitriangular structures,and there is a characterization of them in terms of certain Hopf algebramorphisms due to Radford [32]. A number of equivalent characteriza-tions of quasitriangular structures, also known as universal R -matrices,have been provided for various types of Hopf algebras [13, 35, 37]. Inthis paper we will provide a complete and explicit description of thequasitriangular structures of D ( G ) over an arbitrary field in terms ofcentral subgroups and group homomorphisms.Investigating the impact of changing the braiding of Rep( H ) for asemisimple ribbon Hopf algebra H , or more generally for any braidedfusion category, is expected to provide additional insights into the cat-egory. For example, one of the most important invariants for a spher-ical category are the Frobenius-Schur indicators [6, 27, 29]. On theone hand the higher indicators can be computed using only that H issemisimple [15], and on the other they can expressed in a number ofcategorical ways, especially when the category is modular [28]. Thus itis natural to ask when a given quasitriangular structure yields a mod-ular category; if inequivalent modular categories can be obtained; andwhat new data can be obtained about the indicators by comparing thecategorical calculations when the braiding is changed. Furthermore,the modular data is connected to many other invariants, such as thefusion rules via Verlinde’s formula [3], and we can similarly questionwhat new insights we obtain about these invariants. Answering some ofthese questions for Rep( D ( G )) is itself a detailed enterprise, however.We will subsequently focus our attention on the group theoretical andclassical Hopf algebra questions for this paper. The author intends toaddress several of these questions for Rep( D ( G )) in a future paper.We note that some of the results in this paper could be stated ingreater generality than given. In particular, Lemma 2.2 is just a spe-cific instance of the general ansatz of splitting a given algebraic objectinto distinct pieces (not necessarily with the same structure) and usingthis decomposition to analyze the original object. The lemma in par-ticular can be used to give various generalizations of Theorem 2.1 and[2, Theorem 3.2] to morphisms between various combinations of bis-mash (co)product Hopf algebras. This in turn becomes a descriptionof weak R -matrices and quasitriangular structures on bismash prod-ucts in much the same fashion. However, doing so in such generalityresults in lengthy lists of equations for which the author is unable tofind meaningful structure or simplifications. As such we have opted to UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 3 restrict focus to the doubles of groups, where the equations take on areasonably straightforward description in group theoretical terms.The paper is structured as follows. In Section 1 we introduce therelevant notation and background. In Section 2 we characterize allHopf algebra morphisms D ( G ) ∗ co → D ( G ). As in [7], we call thesethe weak R -matrices. In Sections 3 and 4 we compute the equivalentconditions for a weak R -matrix to have the appropriate commutationrelationship with the comultiplication. In Section 5 we combine theseresults to describe all quasitriangular structures of D ( G ). We thenshow that a ribbon element exists for each quasitriangular structureand explicitly describe them in Section 6. Section 7 investigates theequivalence of quasitriangular structures under Aut( D ( G )). Finally,Section 8 determines when an arbitrary quasitriangular structure isfactorizable. 1. Preliminaries
Our reference for the general theory of Hopf algebras will be [24].Let H be a Hopf algebra over a field k . Suppose that R ∈ H ⊗ H satisfies the following relations:(∆ ⊗ id)( R ) = R R ;(1.1) (id ⊗ ∆)( R ) = R R ;(1.2) ( ε ⊗ id)( R ) = 1;(1.3) (id ⊗ ε )( R ) = 1 , (1.4)where, writing R = P R (1) ⊗ R (2) we have R = R (1) ⊗ ⊗ R (2) ,and similarly for R and R . Any such element is invertible, with R − = S ⊗ id( R ). Such an element is called a weak R -matrix on H [7].Furthermore, if a weak R -matrix R satisfies h (2) ⊗ h (1) = R ( h (1) ⊗ h (2) ) R − for all h ∈ H then R is said to be a quasitriangular structure, or(universal) R -matrix, of H . If such an element exists, we say that H isquasitriangular, and denote the pair by ( H, R ), or simply H when thestructure is understood from the context. Definition 1.1.
Two quasitriangular Hopf algebras (
H, R ), (
K, R ′ )are said to be isomorphic as quasitriangular Hopf algebras if there isa Hopf algebra isomorphism X : H → K such that X ⊗ X ( R ) = R ′ .Given two quasitriangular structures R, R ′ on H , we say that R and R ′ are equivalent, denoted R ∼ R ′ , if ( H, R ) and (
H, R ′ ) are isomorphicas quasitriangular Hopf algebras. MARC KEILBERG
As noted by Radford [31] there is a k -linear injection F : H ⊗ H → Hom k ( H ∗ , H ) given by F ( a ⊗ b )( p ) = p ( a ) b . In the subsequent, F will always refer to this injection. When H is finite dimensional then R ∈ H ⊗ H satisfies equations (1.1) to (1.4) if and only if F ( R ) is amorphism of Hopf algebras H ∗ co → H . Indeed, there are always finitelymany such R under mild assumptions on H and k . On the other hand,given a morphism of Hopf algebras ψ : H ∗ co → H and any basis B of H then R = X h ∈B h ⊗ ψ ( h ∗ ) , (1.5)where h ∗ is the element dual to h , satisfies F ( R ) = ψ . Thus R is aweak R -matrix, and by injectivity of F it is independent of the choiceof basis of H .Let K, L also be Hopf algebras over k . Given linear maps f : H → K and g : L → K , by f g g we mean that the images of f and g commuteelementwise, and say that f and g commute. Dually, for f : H → K and g : H → L , by f f g we mean that the morphisms cocommute: f ( a (1) ) ⊗ g ( a (2) ) = f ( a (2) ) ⊗ g ( a (1) ) for all a ∈ H . Throughout τ denotesthe map H ⊗ K → K ⊗ H given by τ ( h ⊗ k ) = k ⊗ h .Any linear map f : H → K will be called unitary if f (1 H ) = 1 K ,and counitary if ε K ◦ f = f ◦ ε H . We say that f is biunitary if itis both unitary and counitary. All algebra morphisms are unitary,and all coalgebra maps are counitary, so we will not specify unitaryor counitary in these cases. A counitary algebra morphism is alsocalled a morphism of augmented algebras. All morphisms and spacesof morphisms will be of Hopf algebras or groups as appropriate, unlessotherwise specified.For a finite group G we let k G be its group algebra over k and k G bethe dual Hopf algebra. The group of 1-dimensional k -linear charactersof G is denoted by b G , and is identified with the group-likes of k G . Wedenote the left conjugation actions of G on k G and k G both by ⇀ . For g, h ∈ G we let g h = h − gh and [ g, h ] = g − h − gh . Note that G op ∼ = G via the inversion map. We say that G is purely non-abelian if it hasno non-trivial abelian direct factors. A special case of such groups arethe stem groups, which are those G satisfying Z ( G ) ⊆ G ′ .We now describe D ( G ), the Drinfel’d double of G over k . As acoalgebra this is k G co ⊗ k G . Denoting elements of D ( G ) by f ⊲⊳ g , f ∈ k G co , g ∈ G , the algebra structure is given by the semidirectproduct formula( f ⊲⊳ g ) · ( f ′ ⊲⊳ g ′ ) = f ( g ⇀ f ′ ) ⊲⊳ gg ′ . UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 5
Similarly, D ( G ) ∗ co is k G ⊗ k G op as an algebra. Denoting elements of D ( G ) ∗ co by f g , f ∈ k G , g ∈ G op , the coalgebra structure is given by∆( e x g ) = X s ∈ G e s g ⊗ e xs − s − gs. Note that the conjugate s − gs is computed in G op . In particular wesee that k G co is a Hopf subalgebra of D ( G ) ∗ co , whereas G op is onlyan augmented subalgebra since ∆( ε g ) = P s ∈ G e s g ⊗ ε s − gs . Formore details on these Hopf algebras, we refer the reader to [4, 8, 24]. Example 1.2.
When H = D ( G ) is the Drinfel’d double of a finitegroup, the standard quasitriangular structure is R = X g ∈ G ε g ⊗ e g . (1.6)We also have the following quasitriangular structure, which is some-times used instead of R depending on the choice of notation: R = τ ( R − ) = X g ∈ G e g ⊗ ε g − . (1.7) 2. Classifying weak R -matrices for D ( G )We wish to give a useful description of Hom( D ( G ) ∗ co , D ( G )), whichthen gives us a complete description of the weak R -matrices by (1.5).This will involve a number of computations, so we state the result nowand proceed to prove it in stages. The idea is similar to that used byAgore et al. [2] to classify the morphisms between bismash products ofHopf algebras. D ( G ) is of this form, but D ( G ) ∗ co is a smash coproduct,so we must develop an appropriate version for the present context. Theorem 2.1.
The morphisms ψ ∈ Hom( D ( G ) ∗ co , D ( G )) are in bijec-tive correspondence with the quadruples ( u, r, p, v ) wherei) u : k G co → k G co is a unitary morphism of coalgebras;ii) r : k G op → k G co is a biunitary linear map;iii) p : k G co → k G is a morphism of Hopf algebras;iv) v : k G op → k G is a morphism of augmented algebras; MARC KEILBERG satisfying all of the following, for all a, b ∈ k G co and g, h ∈ G op : p f u ;(2.1) p g v ;(2.2) u ( ab ) = u ( a (1) )( p ( a (2) ) ⇀ u ( b ));(2.3) r ( gh ) = X s ∈ G r ( s − gs )( p ( e s ) v ( g ) ⇀ r ( h ));(2.4) ∆( v ( g )) = X s ∈ G p ( e s ) v ( g ) ⊗ v ( s − gs );(2.5) ∆( r ( g )) = X b,s ∈ G u ( e s ) ( p ( e bs − ) ⇀ r ( g )) ⊗ r ( b − gb );(2.6) u ( a (1) ) (cid:0) p ( a (2) ) ⇀ r ( g ) (cid:1) = X s ∈ G r ( s − gs ) (( p ( e s ) v ( g )) ⇀ u ( a )) . (2.7) For such a quadruple, the morphism is given by e x g X a,b,c ∈ Gabc = x u ( e c )( p ( e b ) ⇀ r ( a − ga )) ⊲⊳ p ( e a ) v ( g ) . (2.8) On the other hand, given any linear map ψ : D ( G ) ∗ co → D ( G ) , wedefine the components u, r, p, v by defining u ( a ) = id ⊗ ε ( ψ ( a r ( g ) = id ⊗ ε ( ψ ( ε g ));(2.10) p ( a ) = ev ⊗ id( ψ ( a v ( g ) = ev ⊗ id( ψ ( ε g )) . (2.12)We use the notation of the theorem throughout the rest of the pa-per without further mention. In particular we implicitly identify amorphism with its quadruple, adding indices or superscripts to thecomponents to identify the particular morphism as necessary. We willdenote trivial morphisms by 0 and identity morphisms by 1.It is easy to see, as in [18, Theorem 2.1], that the component p isuniquely determined by a Hopf isomorphism k A → k B , where A, B are abelian subgroups of G . Subsequently we have isomorphisms b A ∼ = A ∼ = B . Whenever we mention A, B in the subsequent we are referringto these subgroups.We now proceed to prove the theorem. We will show how to obtainthe desired quadruple of maps and compatibility conditions from ψ ∈ Hom( D ( G ) ∗ co , D ( G )). The reverse direction is then a simple check. We UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 7 need the following lemma, a proof of which can be found in [2, Lemma3.1].
Lemma 2.2.
Let
C, D, E be coalgebras and
H, K, L algebras.i) There is a bijection between coalgebra morphisms ψ : C → D ⊗ E and pairs ( γ, δ ) where γ : C → D and δ : D → E are co-commuting morphisms of coalgebras. In particular, ψ ( c ) = γ ( c (1) ) ⊗ δ ( c (2) ) , u = (id ⊗ ε ) ψ , and p = ( ε ⊗ id) ψ .ii) There is a bijection between algebra morphisms φ : H ⊗ K → L and pairs ( α, β ) where α : H → L and β : K → L are commut-ing morphisms of algebras. In particular, φ ( h ⊗ k ) = α ( h ) β ( k ) , α ( h ) = φ ( h ⊗ , and β ( k ) = φ (1 ⊗ k ) . So suppose we are given ψ ∈ Hom( D ( G ) ∗ co , D ( G )). We have that D ( G ) is a tensor product as a coalgebra, and subsequently D ( G ) ∗ co is atensor product as an algebra. Thus both parts of the lemma apply, andwe may write ψ ( f ⊲⊳ g ) = α ( f ) β ( g ) = γ (( f ⊲⊳ g ) (1) ) ⊗ δ (( f ⊲⊳ g ) (2) ).Furthermore, it is easily seen that α, β preserve the counit and that γ, δ preserve the unit.In addition ev ⊗ id : D ( G ) → k G is a morphism of Hopf algebras,whence we conclude that δ is in fact a morphism of Hopf algebras.Applying the lemma again, we may write δ ( f ⊲⊳ g ) = p ( f ) v ( g ) for p : k G co → k G a morphism of Hopf algebras and v : k G op → k G amorphism of augmented algebras satisfying p g v . Since k G co is aHopf subalgebra of D ( G ) ∗ co we also have that α is a morphism ofHopf algebras. Therefore α ( f ) = u ( f (1) ) p ′ ( f (2) ) for u : k G co → k G co a morphism of unitary coalgebras and p ′ : k G co → k G a morphism ofHopf algebras satisfying p ′ f u . Indeedev ⊗ id ( ψ ( f ⊲⊳ p ( f ) = p ′ ( f )for all f ∈ k G co . We define r ( g ) = γ ( ε g ) for all g ∈ G op . This yieldsthe quadruple ( u, r, p, v ) in the theorem. We now need to prove thatthe indicated compatibility relations hold, and that ψ has the indicatedform.We first show that we can write β in terms of r, p, v . Since γ f δ wehave β ( g ) = γ ⊗ δ (∆( ε ⊗ g ))= γ ⊗ δ X s ∈ G ε s − gs ⊗ e s g ! = X s ∈ G r ( s − gs ) p ( e s ) v ( g ) . MARC KEILBERG
Subsequently we have γ ( f g ) = id ⊗ ε ( α ( f ) β ( g )) = u ( f (1) )( p ( f (2) ) ⇀ r ( g )) . By then computing ψ ( f g ) = α ( f ) β ( g ) we find that (2.8) holds.To get (2.3) we first observe that α ( f · h ) = u ( f (1) h (1) ) ⊲⊳ p ( f (2) h (2) )= α ( f ) α ( h )= u ( f (1) ) (cid:0) p ( f (3) ) ⇀ u ( h (1) ) (cid:1) ⊲⊳ p ( f (2) ) p ( h (1) ) . The desired relation then follows by applying id ⊗ ε .Similarly we have β ( gh ) = X s ∈ G r ( s − ghs ) ⊲⊳ p ( e s ) v ( g )= β ( g ) β ( h )= X s,t,x ∈ G r ( s − gs ) (cid:0)(cid:0) p ( e x ) v ( g ) (2) (cid:1) ⇀ r ( t − ht ) (cid:1) ⊲⊳ p ( e sx − ) v ( g ) (1) p ( e t ) v ( h ) . Applying id ⊗ ε we find that (2.4) holds.We can also easily compute that∆ δ ( ε g ) = ∆( p ( ε ))∆( v ( g ))= ∆( v ( g ))= X s ∈ G p ( e s ) v ( g ) ⊗ v ( s − gs ) , which is (2.5). By computing ∆ β ( g ) in two different ways we similarlyfind that (2.6) holds.In order for α g β to hold we see that for all f ∈ k G co , g ∈ G op X s ∈ G u ( f (1) ) (cid:0) p ( f (3) ) ⇀ r ( s − gs ) (cid:1) ⊲⊳ p ( f (2) e s ) v ( g )must be equal to X s,t ∈ G r ( s − gs ) (cid:0) p ( e t ) v ( g ) (1) ⇀ u ( f (1) ) (cid:1) ⊲⊳ p ( e st − ) v ( g ) (2) p ( f (2) ) . Applying id ⊗ ε to both expressions we find that (2.7) holds.This completes the proof.By the bijective correspondence between the weak R -matrices andHom( D ( G ) ∗ co , D ( G )) we have the following description of the weak R -matrices. UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 9
Theorem 2.3.
Given ( u, r, p, v ) ∈ Hom( D ( G ) ∗ co , D ( G )) then R = X a,b,c,s ∈ G e s ⊲⊳ abc ⊗ u ( e c ) ( p ( e b ) ⇀ r ( a ⇀ s )) ⊲⊳ p ( e a ) v ( s )(2.13) is a weak R -matrix with F ( R ) = ( u, r, p, v ) .Remark . Expressing the comultiplication of D ( G ) ∗ co with the moregeneral form of a semidirect coproduct would permit one to write R in terms of arbitrary bases for k G co and k G . This does not provide ameaningful benefit in the subsequent, so we choose to express R in thestandard bases. Example 2.5.
The standard quasitriangular structure R of D ( G ) in(1.6) corresponds to the morphism (1 , , , Example 2.6.
For any morphism of Hopf algebras u : k G co → k G co ,( u, , ,
0) corresponds to the weak R -matrix R u = X g ∈ G ε ⊲⊳ g ⊗ u ( e g ) ⊲⊳ . When u = id we get the standard R -matrix. Note that τ ( R − u ) =(0 , , , Su ∗ ). Example 2.7.
For any group homomorphism r : G op → b G , (0 , r, , R -matrix R r = X g ∈ G e g ⊲⊳ ⊗ r ( g ) ⊲⊳ . Example 2.8.
For any Hopf algebra morphism p : k G co → k G , (0 , , p, R -matrix R p = P t ∈ G ε ⊲⊳ t ⊗ ε ⊲⊳ p ( e t ). Example 2.9.
For any v ∈ End( G ), (0 , , , Sv ) gives the weak R -matrix R v = X s ∈ G e s ⊲⊳ ⊗ ε ⊲⊳ v ( s − ) . Note that τ ( R − v ) = ( v ∗ , , , v = id we have R v = R = τ ( R − ).3. Weak R -matrices commuting with G We wish to determine those weak R -matrices which commute withthe coproduct, which we will call the central weak R -matrices, as wellas the quasitriangular structures. To determine the central weak R -matrices and quasitriangular structures of D ( G ) explicitly we need tocheck the equalities(3.1) R ∆( f ⊲⊳ x ) = ∆( f ⊲⊳ x ) R and(3.2) R ∆( f ⊲⊳ x ) = τ ◦ ∆( f ⊲⊳ x ) R respectively.It suffices to check these identities for elements of the form ε ⊲⊳ x and f ⊲⊳
1. In this section we consider the former, and in the nextsection we consider the latter.To this end we compute R ∆( ε ⊲⊳ x ) = X a,b,c,s ∈ G e s ⊲⊳ abcx (3.3) ⊗ ( p ( e b ) ⇀ r ( a ⇀ s )) u ( e c ) ⊲⊳ p ( e a ) v ( s ) x and (∆( ε ⊲⊳ x )) R = X a,b,c,s ∈ G x ⇀ e s ⊲⊳ xabc (3.4) ⊗ x ⇀ (( p ( e b ) ⇀ r ( a ⇀ s )) u ( e c )) ⊲⊳ xp ( e a ) v ( s ) . Applying ev ⊗ id ⊗ ev ⊗ id to (3.3) and (3.4) and equating we find X a ∈ G xa ⊗ xp ( e a ) = X a ∈ G ax ⊗ p ( e a ) x, from which we conclude that p ( g ⇀ f ) = g ⇀ p ( f ). As a conse-quence A ≤ Z ( G ) ⇔ B ≤ Z ( G ). Note that B ≤ Z ( G ) implies that F ( R )( f g ) = u ( f (1) ) r ( g ) ⊲⊳ p ( f (2) ) v ( g ), thus simplifying (2.8).Applying ev ⊗ id ⊗ id ⊗ ε instead we find X c ∈ G cx ⊗ u ( e c ) = X c ∈ G xc ⊗ x ⇀ u ( e c ) , which is equivalent to g ⇀ u ( f ) = u ( g ⇀ f ) for all g ∈ G , f ∈ k G co .Similarly, applying id ⊗ ε ⊗ id ⊗ ε yields r ( h ) = x ⇀ r ( x ⇀ h ), orequivalently that x ⇀ rS ( h ) = rS ( x ⇀ h ). Lastly id ⊗ ε ⊗ ev ⊗ idyields g ⇀ vS ( h ) = vS ( g ⇀ h ). Note that vS : k G → k G is a mor-phism of augmented algebras.This proves necessity in the following, and the sufficiency is a simplecheck. Lemma 3.1.
A weak R -matrix R = ( u, r, p, v ) ∈ D ( G ) ⊗D ( G ) satisfies R ∆( ε x ) = ∆( ε x ) R for all x ∈ G if and only if u, rS, p, and vS allcommute with the conjugation actions of G . UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 11 Weak R -matrices commuting with ∆ k G We now check the equality of (3.1) for elements of the form f ⊲⊳ R ∆( f ⊲⊳
1) = X a,b,c,s,t ∈ G e s ( abc ⇀ f (2) ) ⊲⊳ abc (4.1) ⊗ (cid:0)(cid:0) p ( e t ) v ( s ) (1) (cid:1) ⇀ f (1) (cid:1) u ( e c )( p ( e b ) ⇀ r ( a ⇀ s )) ⊲⊳ p ( e at − ) v ( s ) (2) and ( f (2) ⊲⊳ ⊗ f (1) ⊲⊳ R = X a,b,c,s ∈ G f (2) e s ⊲⊳ abc ⊗ f (1) (cid:0) p ( e b ) ⇀ r ( a ⇀ s ) (cid:1) u ( e c ) ⊲⊳ p ( e a ) v ( s ) . (4.2)Applying ev ⊗ id ⊗ id ⊗ ε to both expressions and equating we get X c ∈ G c ⊗ f u ( e c ) = X a,c ∈ G ac ⊗ u ( e c )( p ( e a ) ⇀ f ) . For the special case f ∈ Im( u ) an application of (2.3) shows that u is amorphism of Hopf algebras, from which it follows that we may identify u ∗ ∈ End( G ). Let c = u ∗ ( h ) for some h ∈ G . Then applying id ⊗ ev h to the above equality we find ( p ( e ) ⇀ f )( h ) = f ( h ). This equationholds for all h ∈ G if and only if B ≤ Z ( G ).Applying id ⊗ ε ⊗ ev ⊗ id to (4.1) and (4.2) and equating we find X s ∈ G f e s ⊗ v ( s ) = X s,a ∈ G e s ( a ⇀ f ) ⊗ p ( e a ) v ( s ) . Thus for all s ∈ G we have e s f ⊗ X a ∈ G e s ( a ⇀ f ) ⊗ p ( e a ) . This forces A ≤ Z ( G ). Then from (2.5) we conclude that v is a grouphomomorphism. Similarly, (2.4) becomes r ( gh ) = r ( g )( v ( g ) ⇀ r ( h )) . (4.3)Since r is unitary we conclude that r ( g ) is invertible for all g ∈ G .Subsequently, (2.7) simplifies to v ( g ) ⇀ u ( a ) = u ( a ) . (4.4) Now applying id ⊗ ε ⊗ id ⊗ ε and equating we have X s ∈ G f (2) e s ⊗ f (1) r ( s ) = X c,s ∈ G e s ( c ⇀ f (2) )(4.5) ⊗ ( v ( s ) ⇀ f (1) ) u ( e c ) r ( s ) . In particular for all s, h ∈ Gf (2) e s ⊗ f (1) r ( s ) e h = X c ∈ G e s ( u ∗ ( h ) ⇀ f (2) ) ⊗ r ( s ) e h (cid:0) v ( s ) ⇀ f (1) (cid:1) . Therefore for any fixed s, h ∈ G we have r ( s )( h ) f (2) ( s ) f (1) ( h ) = r ( s )( h )( u ∗ ( h ) ⇀ f (2) )( s )( v ( s ) ⇀ f (1) )( h )which is equivalent to r ( s )( h ) f ( hs ) = r ( s )( h ) f ( h v ( s ) s u ∗ ( h ) ) . Since r ( s ) is invertible, r ( s )( h ) = 0. The arbitrary choice of f thenmakes this equation equivalent to(4.6) h v ( s ) s u ∗ ( h ) = hs for all s, h ∈ G .The relation obtained by applying ev ⊗ id ⊗ ev ⊗ id is trivially truein all cases. This proves necessity in the following, with sufficiencybeing a simple check. Lemma 4.1.
A weak R -matrix ( u, r, p, v ) ∈ D ( G ) ⊗ D ( G ) satisfies R ∆( f ⊲⊳
1) = ∆( f ⊲⊳ R for all f ∈ k G co if and only if the followingall hold:i) u is a morphism of Hopf algebras, or equivalently u ∗ ∈ End( G ) ;ii) A, B ≤ Z ( G ) ;iii) v is a morphism of Hopf algebras;iv) v ( g ) ⇀ u ( a ) = u ( a ) for all a ∈ k G co , g ∈ G op ;v) Equation (4.6) is satisfied for all s, h ∈ G . Example 4.2.
Any u ∗ , v with central image clearly satisfy (4.6). Wewill see later that these are the only possibilities for a central weak R -matrix.Now when we consider (3.2), instead, we easily observe that all of thepreceding arguments apply, with the exception that (4.6) is replacedwith s u ∗ ( h ) h v ( s ) = hs. (4.7) UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 13
Lemma 4.3.
A weak R -matrix R = ( u, r, p, v ) ∈ D ( G ) ⊗D ( G ) satisfies R ∆( f ⊲⊳
1) = τ (∆( f ⊲⊳ R for all f ∈ k G co if and only if the following all hold:i) u is a morphism of Hopf algebras, or equivalently u ∗ ∈ End( G ) ;ii) A, B ≤ Z ( G ) ;iii) v is a morphism of Hopf algebras;iv) v ( g ) ⇀ u ( a ) = u ( a ) for all a ∈ k G co , g ∈ G op ;v) Equation (4.7) is satisfied for all s, h ∈ G . Example 4.4. If u ∗ has central image and for all s ∈ G op v ( s ) = zs − for some central element z depending on s , then (4.7) is clearly satisfied.Conversely, if v has central image and for all s ∈ G u ∗ ( s ) = zs for somecentral element z depending on s , then once again (4.7) is satisfied. Wewill see later that these are the only possibilities for a quasitriangularstructure when G is indecomposable, and that u ∗ , v are naturally builtfrom such examples on indecomposable factors otherwise.5. The central weak R -matrices and quasitriangularstructures Having computed the commutation relations we can now easily givea precise description of the central weak R -matrices and the quasitri-angular structures.For any finite group G we may use Krull-Schmidt to write G = G × G ×· · ·× G n where G is abelian and G i is an indecomposable non-abelian group for all 1 ≤ i ≤ n . Let ι i , π i be the corresponding injectionand projection respectively for G i , 0 ≤ i ≤ n . For any endomorphism w : G → G define w ij = π i ◦ w ◦ ι j , and set w i = w ii . The w ij arealso endomorphisms of G and uniquely determine w [5]. We make theanalogous description when w : G op → G as well. Proposition 5.1.
A weak R -matrix ( u, r, p, v ) is a central weak R -matrix if and only if the following all hold:i) vS, u ∗ ∈ Hom(
G, Z ( G )) ;ii) r is a bicharacter, meaning r ∈ Hom( G op , b G ) = Hom( G, b G ) ;iii) A, B ≤ Z ( G ) .In this case the weak R -matrix may be written as X a,c,s ∈ G e s ⊲⊳ ac ⊗ r ( s ) u ( e c ) ⊲⊳ p ( e a ) v ( s ) . (5.1) Proof.
The only remaining case is to suppose R is a central weak R -matrix and to show that v, u ∗ ∈ Hom(
G, Z ( G )) follows from (4.6). By Lemma 3.1
Sv, u ∗ are normal group endomorphisms G → G . Decom-pose G and its endomorphisms as before. It follows that without lossof generality we may consider (4.6) under the assumption that G isindecomposable and non-abelian. We note that u ∗ and ( Su ∗ ) ∗ id aresimultaneously normal group endomorphisms. Therefore by normalityof u ∗ and assumptions on G either u ∗ or ( Su ∗ ) ∗ id is a central automor-phism. In this case the other is necessarily in Hom( G, Z ( G )). Similarly,either vS is a central automorphism or vS ∈ Hom(
G, Z ( G )). It is eas-ily checked that (4.6) then holds if and only if u ∗ , vS ∈ Hom(
G, Z ( G )),as desired. (cid:3) As a consequence we have the following.
Corollary 5.2.
Define Z k ( G ) to be the maximal subgroup of Z ( G ) allof whose subgroups are isomorphic to their character groups over k .Then the central weak R -matrices form an abelian group isomorphic to Hom(
G, Z ( G )) × Hom( G, b G ) × End( Z k ( G )) , where the multiplication of central weak R -matrices is given by compo-nentwise convolution products. Using similar arguments to those in the proof of Proposition 5.1 weobtain the following explicit description of the quasitriangular struc-tures of D ( G ). Theorem 5.3.
A weak R -matrix ( u, r, p, v ) is a quasitriangular struc-ture of D ( G ) if and only if the following all hold:i) A, B ≤ Z ( G ) ;ii) r ∈ Hom( G, b G ) is a bicharacter;iii) u ∗ , Sv are normal endomorphisms of G ;iv) For each ≤ i ≤ n exactly one of the following holds:(a) Sv i ∈ Hom( G i , Z ( G i )) , u ∗ i ∈ Aut c ( G i ) ;(b) Sv i ∈ Aut c ( G i ) , u ∗ i ∈ Hom( G i , Z ( G i )) .In this case we have ( u, r, p, v ) = X a,b,s ∈ G e s ⊲⊳ ab ⊗ r ( s ) u ( e b ) ⊲⊳ p ( e a ) v ( s ) . (5.2) In particular there is a n -to-1 correspondence between quasitriangu-lar structures and the central weak R -matrices. When G is abelian thequasitriangular structures and central weak R -matrices are (trivially)the same.Proof. As noted, the calculations for the central weak R -matrices ap-plies to this case as well, with much the same arguments showing that(4.7) yields the stated description of the components for v and u ∗ . UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 15
For the last claim, suppose we are given a set E ∈ P ( { , ..., n } ). Thenfor a central weak R -matrix ( u, r, p, v ) we construct a quasitriangularstructure ( u ′ , r, p, v ′ ) by defining u ′ ij = ( u ij i = j or i ESu i ∗ id i = j and i ∈ E ; v ′ ij = ( v ij i = j or i ∈ Ev i ∗ S i = j and i E. That reversing this process yields a central weak R -matrix follows from[1]. (cid:3) Note that the correspondence is dependent upon the choice of de-composition of G . Since the quasitriangular structures themselves areindependent of this decomposition, we will simply leave a fixed butotherwise arbitrary choice of decomposition for G implicit. The quasi-triangular structures associated to the sets E = ∅ and E = { , ..., n } ,however, are canonically determined and do not depend on the choiceof decomposition. The quasitriangular structures obtained from thetrivial weak R -matrix are the standard quasitriangular structures R and R = τ ( R − ) = P g ∈ G e g ⊲⊳ ⊗ ε ⊲⊳ g − irrespectively. Remark . The last two conditions from the theorem can be restatedas follows. Let ˆ v : G/G → G/G be given by ˆ v ij = Sv ij for i, j > u . Then the last two conditions are equivalent toˆ v, ˆ u ∗ being normal and ˆ u ∗ ∗ ˆ v ∈ Aut c ( G/G ). In other words, ˆ v andˆ u ∗ give a convolution factorization of a central automorphism of G/G into normal endomorphisms. It is worth pointing out that neither v nor u need be an isomorphism, and indeed that u ∗ ∗ v need not be acentral automorphism of G . Example 5.5.
For any quasitriangular structure R = ( u, r, p, v ) as-sociated to E ∈ P ( { , ..., n } ) we can easily check that τ ( R − ) =( Sv ∗ , Sr ∗ , Sp ∗ , Su ∗ ) is a quasitriangular structure associated to E c . In-deed, R is obtained from the trivial central weak R -matrix if and onlyif τ ( R − ) is obtained from the trivial central weak R -matrix.6. Ribbon elements
We now recall the basic facts about ribbon Hopf algebras, which canbe found in [33]. Given a quasitriangular Hopf algebra (
H, R ), we definethe Drinfel’d element to be u R = m ( τ ( R − )), where m : H ⊗ H → H is the multiplication of H . This element satisfies S ( h ) = uhu − for all h ∈ H . We say that ν ∈ H is a ribbon element of ( H, R ) if ν = uSu , ν is central and invertible in H , ε ( ν ) = 1, Sν = ν , and τ ( R ) R ∆( ν ) = ν ⊗ ν. When such a ν exists, we say that ( H, R, ν ), or just (
H, R ) or H whenthere is no ambiguity, is a ribbon Hopf algebra.In general a ribbon element is not necessarily uniquely defined whenit exists, but by taking the ratio of any two ribbon elements we see thatthey differ by multiplication by a central group-like element of H thathas order dividing 2. In the case where D ( G ) = H , the group-likes areprecisely b G × G , which has center b G × Z ( G ).We will now show that ( D ( G ) , R ) admits a ribbon element for anychoice of quasitriangular structure. Theorem 6.1.
Let R = ( u, r, p, v ) be a quasitriangular structure of D ( G ) . Then for the quasitriangular Hopf algebra ( D ( G ) , R ) the Drin-fel’d element is u R = X a,s ∈ G r ( s ) e s − ⊲⊳ p ( e a ) a − v ( s ) u ∗ ( s ) . Furthermore, u R is also a ribbon element.Proof. Let R = ( u, r, p, v ) be a quasitriangular structure. By definition u R = m ( τ ( R − )). We have R − = id ⊗ S ( R ) = X s,a,b ∈ G e s − ⊲⊳ a − b − ⊗ r ( s ) u ( e b ) ⊲⊳ p ( e a ) v ( s ) , whence u R = m ( τ ( R − )) = X s,a,b ∈ G r ( s ) u ( e b )( v ( s ) ⇀ e s − ) ⊲⊳ p ( e a ) v ( s ) a − b − = X s,a,b ∈ G r ( s ) u ( e b ) e s − ⊲⊳ p ( e a ) a − v ( s ) b − , where we have used that the components of v are either central orcentral automorphisms, and A, B ≤ Z ( G ). We then observe that u ( e b ) e s − = 0 if and only if u ∗ ( s − ) = b . This gives the desired for-mula for u R .Now since any component of v which is not central implies that thesame component of u ∗ is central, and vice versa, we see that u ∗ g v .Since all components of u ∗ are also either central or central automor-phisms, we conclude that Su R = u R . Since D ( G ) is involutory u R mustbe central. UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 17
The only relation u R must then satisfy which is non-trivial and notyet established is that τ ( R ) R ∆( u R ) = u R ⊗ u R . This is equivalent to R ∆( u R ) = τ ( R − ) u R ⊗ u R . We will show this relation holds when u ∗ ∈ Aut c ( G ) is a central isomorphism and Sv ∈ Hom(
G, Z ( G )) is a centralhomomorphism. A similar argument, which we omit, then shows thatthe relation also holds when u ∗ ∈ Hom(
G, Z ( G )) and Sv ∈ Aut c ( G ).This establishes the result for an indecomposable non-abelian group,and the general case then follows by breaking u, v into components.For simplicity of performing calculations, we may suppose that v hasdomain G (rather than G op ) by replacing it with Sv ; meaning that all v ( s ) terms in expressions will be replaced with v ( s − ).For ease of reference, we recall the following identities. R = X s,a,t ∈ G e s ⊲⊳ au ∗ ( t ) ⊗ r ( s ) e t ⊲⊳ p ( e a ) v ( s − );(6.1) R − = X s,a,t ∈ G e s − ⊲⊳ a − u ∗ ( t − ) ⊗ r ( s ) e t p ( e a ) v ( s − );(6.2) τ ( R − ) = X s,a,t ∈ G r ( s ) e t ⊲⊳ p ( e a ) v ( s − ) ⊗ e s − ⊲⊳ a − u ∗ ( t − );(6.3) u R = X y,b ∈ G r ( y − ) e y ⊲⊳ p ( e b ) b − v ( y ) u ∗ ( y − ) . (6.4)We then have∆ u R = X y,h,c,b ∈ G r ( y − ) e h ⊲⊳ p ( e c ) b − v ( y ) u ∗ ( y − ) ⊗ r ( y − ) e yh − ⊲⊳ p ( e bc − ) b − v ( y ) u ∗ ( y − ) . (6.5)We can then compute that R ∆ u R = X y,s,t,h,c,b,a ∈ G r ( y − ) e s e tht − ⊲⊳ p ( e c ) b − av ( y ) u ∗ ( ty − ) ⊗ r ( sy − ) e t e yh − ⊲⊳ p ( e a ) p ( e bc − ) b − v ( s − y ) u ∗ ( y − )= X y,h,c,b − G r ( y − ) e yhy − ⊲⊳ p ( e c ) c − v ( y ) u ∗ ( yh − y − ) ⊗ r ( hy − ) e yh − ⊲⊳ p ( e bc − ) b − v ( h − y ) u ∗ ( y − )= X y,x,c,b ∈ G r ( y − ) e x − ⊲⊳ p ( e c ) c − v ( y ) u ∗ ( x ) ⊗ r ( y − x − ) e xy ⊲⊳ p ( e bc − ) b − v ( xy ) u ∗ ( y − ) . (6.6) Next we have u R ⊗ u R = X g,h,b,c ∈ G r ( g − ) e g ⊲⊳ p ( e c ) c − v ( g ) u ∗ ( g − ) ⊗ r ( h − ) e h ⊲⊳ p ( e b ) b − v ( h ) u ∗ ( h − ) , (6.7)and thus τ ( R − ) u R ⊗ u R = X s,g,t,a,c,h,b ∈ G r ( sg − ) e t e g ⊲⊳ p ( e a ) p ( e c ) c − v ( s − g ) u ∗ ( g − ) ⊗ r ( h − ) e s − e t − ht ⊲⊳ p ( e b ) b − a − v ( h ) u ∗ ( t − h − )= X h,g,b,c ∈ G r ( h − g − ) e g ⊲⊳ p ( e c ) c − v ( hg ) u ∗ ( g − ) ⊗ r ( h − ) e g − hg ⊲⊳ p ( e b ) b − c − v ( h ) u ∗ ( g − h − )= X h,g,b,c ∈ G r ( h − g ) e g − ⊲⊳ p ( e c ) c − v ( hg − ) u ∗ ( g ) ⊗ r ( h − ) e ghg − ⊲⊳ p ( e b ) b − c − v ( hg ) u ∗ ( gh − )= X g,y,b,c ∈ G r ( y − ) e g − p ( e c ) c − v ( y ) u ∗ ( g ) ⊗ r ( y − g − ) e gy ⊲⊳ p ( e b ) b − c − v ( gy ) u ∗ ( y − )= X x,y,b,c ∈ G r ( y − ) e x − ⊲⊳ p ( e c ) c − v ( y ) u ∗ ( x ) ⊗ r ( y − x − ) e xy ⊲⊳ p ( e bc − ) c − v ( xy ) u ∗ ( y − ) . (6.8)This is precisely equation (6.6), and so completes the proof that u R isa ribbon element. (cid:3) Remark . An informal proof that u R is a ribbon element is as fol-lows. By Theorem 5.3 the only meaningful impact the field has ona quasitriangular structure R of D ( G ), and subsequently its Drinfel’delement u R and the desired identity τ ( R ) R ∆( u R ) = u R ⊗ u R , is thatit limits the choices for p and r . Thus if the identity always holds inone field, it must hold in any field compatible with p, r . Furthermore,it is well-known that any semisimple and cosemisimple quasitriangularHopf algebra H has its Drinfel’d element as a ribbon element. Since D ( G ) always has these properties when k is algebraically closed withcharacteristic zero, u R must always be a ribbon element. UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 19
Example 6.3.
For ( D ( G ) , R ) we have u R = P g ∈ G e g ⊲⊳ g is a ribbonelement. For ( D ( G ) , R ) we have u R = P g ∈ G e g ⊲⊳ g − . Both of theseare well-known. 7. Equivalence under
Aut( D ( G ))In this section we investigate the equivalence relation on quasitrian-gular structures given in Definition 1.1. We note that if R ∼ R ′ are twoequivalent quasitriangular structures of D ( G ), then any isomorphismof the quasitriangular Hopf algebras ( D ( G ) , R ) and ( D ( G ) , R ′ ) inducesa braided equivalence of their representation categories. In general,though, it is possible for non-isomorphic quasitriangular Hopf algebrasto have representation categories which are equivalent as braided tensorcategories. As has been mentioned, it is beyond the scope of this paperto settle equivalence at the categorical level, but in light of the con-jectural connection between Aut( D ( G )) and the structure of (braided)autoequivalences of Rep( D ( G )) [20, 21], the classical picture may infact resolve most, if not all, of the categorical one.We first recall the fundamental properties of Aut( D ( G )). Theorem 7.1. [18, 19, 20] Every automorphism X of D ( G ) can bedescribed by a matrix X = (cid:18) α βγ δ (cid:19) where α ∗ , δ are normal group homomorphisms, β ∈ Hom( G, b G ) is abicharacter, and γ : k G → k G is a morphism of Hopf algebras associ-ated to central subgroups A, B . The automorphism is explicitly givenby f ⊲⊳ x β ( x ) α ( f (1) ) ⊲⊳ γ ( f (2) ) δ ( x ) for all f ∈ k G co and x ∈ G .Any X ∈ Aut( D ( G )) also satisfies X ∗ ∈ Aut( D ( G )) , where X ∗ isthe linear dual of X given by X ∗ = (cid:18) δ ∗ β ∗ γ ∗ α ∗ . (cid:19) When G is purely non-abelian then α ∗ , δ ∈ Aut( G ) and δα ∗ ∈ Aut c ( G ) .Moreover, in this case Aut( D ( G )) consists precisely of those matricessatisfying these conditions on α, β, γ, and δ .Remark . If R = ( u, r, p, v ) is a central weak R -matrix or a quasi-triangular structure for D ( G ), then we may similarly write F ( R ) as a matrix (cid:18) u rp v (cid:19) . These matrix descriptions for R and Aut( D ( G )) are compatible, in thesense that we may perform matrix multiplications in the usual fashion,but where multiplication is composition and addition is convolutionproduct. Some examples of this are given in example 7.8 below. Definition 7.3.
We say that ribbon Hopf algebras (
H, R, ν ) , ( K, R ′ , ν ′ )are isomorphic as ribbon Hopf algebras if there exists an isomorphism ofquasitriangular Hopf algebras X : ( H, R ) → ( K, R ′ ) such that X ( ν ) = ν ′ . In this case X is called an isomorphism of ribbon Hopf algebras.By [18, Theorem 3.5] we have that D ( G ) ∼ = D ( H ) for finite groups G, H if and only if G ∼ = H . The following standard results then showthat Definition 7.3 is essentially the same as Definition 1.1 when H = K = D ( G ). Lemma 7.4.
Let ( H, R ) , ( K, R ) be two quasitriangular Hopf algebraswith Drinfel’d elements u H , u K respectively. Suppose X : ( H, R ) → ( K, R ′ ) is an isomorphism of quasitriangular Hopf algebras. Then X ( u H ) = u K .Proof. Let
X, R, R ′ be as in the statement with X ⊗ X ( R ) = R ′ . Then u K = m ( τ ( R ′− ))= m ( τ ( X ⊗ X ( R − )))= m ( X ⊗ Xτ ( R − ))= X ( m ( τ ( R − )))= X ( u H ) . (cid:3) Corollary 7.5. If R, R ′ are two quasitriangular structures of D ( G ) and X ∈ Aut( D ( G )) is such that X ⊗ X ( R ) = R ′ , then X is anisomorphism of ribbon Hopf algebras ( D ( G ) , R, u R ) → ( D ( G ) , R ′ , u R ′ ) .Proof. Apply the preceding lemma and Theorem 6.1. (cid:3)
More generally, we can describe the action of Aut( D ( G )) on thequasitriangular structures as follows. Theorem 7.6.
Let R = ( u, r, p, v ) , R ′ = ( u ′ , r ′ , p ′ , v ′ ) be quasitriangularstructures of D ( G ) , and let X = (cid:18) α βγ δ (cid:19) ∈ Aut( D ( G )) . UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 21
Then the following are equivalent:i) X ⊗ X ( R ) = R ′ ;ii) X ◦ ( F ( R ) ◦ X ∗ ) = F ( R ′ ) ;iii) The following four identities all hold: u ′ = αrγ ∗ + αuδ ∗ + βpδ ∗ + βvγ ∗ ;(7.1) r ′ = αrα ∗ + αuβ ∗ + βpβ ∗ + βvα ∗ ;(7.2) p ′ = γrγ ∗ + γuδ ∗ + δpδ ∗ + δvγ ∗ ;(7.3) v ′ = γrα ∗ + γuβ ∗ + δpβ ∗ + δvα ∗ , (7.4) where addition denotes convolution product.Proof. Let
X, R, R ′ be as in the statement. Then by [19] X ∗ = (cid:18) δ ∗ β ∗ γ ∗ α ∗ (cid:19) ∈ Aut( D ( G ) ∗ co )is given by X ∗ ( f s ) = P t ∈ G β ∗ ( s ) δ ∗ ( f (1) ) γ ∗ ( f (2) ) α ∗ ( s ) for all f g ∈D ( G ) ∗ co .Now X ⊗ X ( R ) = R ′ is equivalent to F ( R ′ )( f h ) = X s,g ∈ G ev h ⊗ f ( X ( e s ⊲⊳ g )) ( X ◦ F ( R ))( e g s )(7.5)for all f h ∈ D ( G ) ∗ co . We then have F ( R ′ )( ε h ) = X s,g ∈ G ev h ( β ( g ) α ( e s )) ( X ◦ F ( R ))( e g s )= X g ∈ G β ( g, h ) ( X ◦ F ( R ))( e g α ∗ ( h ))= ( X ◦ F ( R ))( β ∗ ( h ) α ∗ ( h ))= ( X ◦ ( F ( R ) ◦ X ∗ ))( ε h ) . Furthermore, F ( R ′ )( f X s,g ∈ G f ( γ ( e s ) δ ( g ))( X ◦ F ( R ))( e g s )= X s,g ∈ G f (2) ( γ ( e s )) f (1) ( δ ( g ))( X ◦ F ( R ))( e g s )= X ◦ F ( R )( δ ∗ ( f (1) ) γ ∗ ( f (2) )) . = ( X ◦ ( F ( R ) ◦ X ∗ ))( f . This shows that the first two items are equivalent.That the second item is equivalent to the third follows from Theo-rem 2.1; equations (2.9) to (2.12) in particular. (cid:3)
Corollary 7.7.
Let
R, R ′ be quasitriangular structures of D ( G ) and X ∈ Aut( D ( G )) . Then X ⊗ X ( R ) = R ′ if and only if X ⊗ X ( τ ( R − )) = τ ( R ′− ) .Proof. The result follows from example 5.5 and Theorem 7.6. (cid:3)
Example 7.8.
Consider the quasitriangular structures R = (1 , r, ,
0) and R ′ = (1 , , p, D ( G ). We have X = (cid:18) r ∗ (cid:19) ∈ Aut( D ( G )); Y = (cid:18) p (cid:19) ∈ Aut( D ( G )) . Moreover, we can use the matrix notations to compute X ◦ ( F ( R ) ◦ X ∗ ): X ( (cid:18) (cid:19) (cid:18) r (cid:19) ) = (cid:18) r ∗ (cid:19) (cid:18) r (cid:19) = (cid:18) r (cid:19) , which shows that X ⊗ X ( R ) = R . A similar calculation shows that Y ⊗ Y ( R ) = R ′ .Theorem 7.6 then implies that X is an isomorphism of ribbon Hopfalgebras ( D ( G ) , R , u R ) → ( D ( G ) , R, u R ) , and that Y is an isomorphism of ribbon Hopf algebras( D ( G ) , R , u R ) → ( D ( G ) , R ′ , u R ′ ) . More generally, we have the following description of the orbit of R . Proposition 7.9.
Let G be a purely non-abelian finite group and let R = ( u, r, p, v ) be a quasitriangular structure of D ( G ) . Then R ∼ R if and only if u ∗ ∈ Aut c ( G ) and v = pu − r .Proof. By Theorem 7.6, R ∼ R ′ if and only if u = αδ ∗ ,r = αβ ∗ ,p = γδ ∗ ,v = γβ ∗ , for some X = (cid:18) α βγ δ (cid:19) ∈ Aut( D ( G )) . UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 23
By Theorem 7.1 we have α ∗ , δ ∈ Aut( G ) and u ∗ = δα ∗ ∈ Aut c ( G ).Then given α, δ we can always solve r = αβ ∗ and p = γδ ∗ for r, p or β, γ when given the other two. Indeed, we have v = γβ ∗ = pδ ∗− α − r = p ( αδ ∗ ) − r = pu − r. This completes the proof. (cid:3)
In the case when G has an abelian direct factor, the main obstructionis that it need no longer be the case that α ∗ , δ ∈ Aut( G ) or that δα ∗ ∈ Aut c ( G ). Indeed, when G is abelian then α, δ can both be trivial,since γ, β can then be isomorphisms. Nevertheless, since Aut( D ( G )) isknown even in the purely non-abelian case [19, 20], one can in principlealways determine the equivalence class of R . Corollary 7.10.
Let G be a purely non-abelian group. Then there areprecisely | Aut c ( G ) | · | Hom( G, b G ) | · | End( Z ( G )) | quasitriangular struc-tures of D ( G ) which are equivalent to R .Proof. By [18, Theorem 6.7] we have | Aut( D ( G )) | = | Aut( G ) | · | Aut c ( G ) | · | Hom( G, b G ) | · | End( Z ( G )) | and also by [18, Example 9.6] that the stabilizer of R is (isomorphicto) Aut( G ). Thus the size of the orbit follows. Alternatively, the orderalso follows directly from Proposition 7.9. (cid:3) Indeed, for such a G it follows from Theorem 5.3 that there areprecisely | Aut c ( G ) | · | Hom( G, b G ) | · | End( Z ( G )) | · | Hom(
G, Z ( G )) | quasitriangular structures ( u, r, p, v ) of D ( G ) with u ∗ ∈ Aut c ( G ). Thusthere are quasitriangular structures with u ∗ ∈ Aut c ( G ) which are in-equivalent to R if and only if Hom( G, Z ( G )) is non-trivial, or equiva-lently that gcd([ G : G ′ ] , | Z ( G ) | ) = 1.In general, it need not be the case that every v ∈ Hom(
G, Z ( G ))appears in a quasitriangular structure ( u, r, p, v ) that is equivalent to R . At the extreme end, sometimes v must in fact always be trivial. Corollary 7.11.
Let G be a finite group. Then the following are equiv-alent.i) G is a stem group. ii) Every quasitriangular structure ( u, r, p, v ) of D ( G ) with ( u, r, p, v ) ∼ R has v trivial.Proof. From equation (7.4) if ( u, r, p, v ) ∼ R then v = γβ ∗ . That thiscomposition is trivial for all choices of γ, β is equivalent to Z ( G ) ⊆ G ′ ,which is the definition of a stem group. (cid:3) Example 7.12. G = D , the dihedral group with 8 elements, is a stemgroup with Hom( G, Z ( G )) non-trivial. Thus for R = (1 , , , z ) with z ∈ Hom(
G, Z ( G )) non-trivial the ribbon Hopf algebras ( D ( G ) , R , u R )and ( D ( G ) , R, u R ) are non-isomorphic. Example 7.13.
On the other hand, Proposition 7.9 also says that anyquasitriangular structure of the form (1 , r, p,
0) with pr = 0 is necessar-ily not equivalent to R . The preceding corollary also guarantees thatsuch choices for p, r always exist when G is not a stem group.8. Factorizability
A quasitriangular Hopf algebra (
H, R ) is said to be factorizable if τ ( R ) R is a non-degenerate tensor on H ⊗ H . Equivalently, if the linearmap H ∗ → H given by f ( m ◦ ( f ⊗ id))( τ ( R ) R ) is bijective. Weconclude the paper by considering this property for quasitriangularstructures of D ( G ). We need a number of basic definitions and lemmasto proceed.As usual, we fix a finite group G and a decomposition G = G × G × · · · G n , where G is abelian and G , ..., G n are indecomposable non-abeliangroups. Definition 8.1.
Let E ⊆ { , ..., n } . We make the following definitions.i) π E is the canonical retraction to the subgroup Q i ∈ E G i . This isgiven by the canonical surjection G → Q i ∈ E G i followed by thecanonical injection Q i ∈ E G i → G .ii) π E c is the canonical retraction to the subgroup Q
Lemma 8.2.
For any E ⊆ { , ..., n } the following is a basis of D ( G ) ∗ co : { ( π E ( x ) ⇀ e y ) π E c ( y ) ⇀ x ) | x, y ∈ G } . Proof.
Indeed, we can show that this is precisely the standard basis of D ( G ) ∗ co . So for g, h, x, y ∈ G we have ( π E ( x ) ⇀ e y ) π E c ( y ) ⇀ x ) = e g h if and only if, in the notation of Definition 8.1, the following allhold y = g ; y E c = g E c ; y E = g x E E = g h E E ; x = h ; x E = h E ; x E c = h y Ec E c = h g Ec E c . Thus g, h uniquely determine x, y and conversely, and the desired claimfollows. (cid:3)
Proposition 8.3.
Let R = ( u, r, p, v ) be a quasitriangular structure of D ( G ) associated to the set E ⊆ { , ..., n } as in the proof of Theorem 5.3.Then we have a well-defined morphism Ψ R = (cid:18) v ∗ + u r ∗ + rp ∗ + p u ∗ + v (cid:19) ∈ End( D ( G E )) . (8.1) Proof.
The definition of G E in Definition 8.1 and the properties of thecomponents of R in Theorem 5.3 combined with the results of [2, 18]show that Ψ R is a well-defined endomorphism of D ( G E ). (cid:3) Since G ∼ = G E canonically, we can always identify the underlyingvector spaces of D ( G ) and D ( G E ). We do so whenever convenientwithout further mention. Example 8.4.
For R we have that Ψ R is the identity of D ( G ). Inparticular, Ψ R is an automorphism, and R is well-known (and easilyshown) to be factorizable.Similarly, for R we have that Ψ R is the identity of D ( G op ). ThusΨ R is an automorphism, and R is well-known to be factorizable.Indeed, we can now complete our goal by showing that this relationbetween Ψ R and the factorizability of R holds for arbitrary R . Theorem 8.5.
Let R = ( u, r, p, v ) be a quasitriangular structure of D ( G ) associated to the set E ⊆ { , ..., n } as in the proof of Theorem 5.3.Define Ψ R as in Proposition 8.3.Then F ( τ ( R ) R ) and Ψ R have the same image. Therefore R is fac-torizable if and only if Ψ R ∈ Aut( D ( G E )) . Proof.
By Theorem 5.3 we have that u ∗ , v are normal group endomor-phisms such that u ∗ g v . With this we can easily compute that τ ( R ) R = X s,t,a,b ∈ G ( v ( t ) ⇀ e s ) ⊲⊳ abv ( t ) u ∗ ( t ) ⊗ r ∗ ( s ) r ( s )( u ∗ ( s ) ⇀ e t ) ⊲⊳ p ∗ ( e a ) p ( e b ) u ∗ ( s ) v ( s )= X s,t,a,b ∈ G e s ⊲⊳ abu ∗ ( t s ) v ( t ) ⊗ r ∗ ( s ) r ( s ) e t ⊲⊳ p ∗ ( e a ) p ( e b ) u ∗ ( s ) v ( s t ) . From this we then have F ( τ ( R ) R )( f x ) = X t ∈ G r ∗ ( x ) r ( x ) f (3) ( u ∗ ( t x ) v ( t )) e t ⊲⊳ p ( f (1) ) p ∗ ( f (2) ) u ∗ ( x ) v ( x t ) . (8.2)Note that F ( τ ( R ) R ) is in general not a morphism of Hopf algebras.Let E ⊆ { , ..., n } be the set associated to R for this decompositionas in the proof of Theorem 5.3. Now Definition 8.1, equation (8.2),and Theorem 5.3 show that F ( τ ( R ) R )(( π E ( x ) ⇀ f ) x )= X t ∈ G r ∗ ( x ) r ( x )( u + v ∗ )( f (1) ) e t ⊲⊳ ( p + p ∗ )( f (2) ) u ∗ ( x ) v ( x t ) . (8.3)Next we consider the special case f = e y for some y ∈ G . Thenequation (8.3) becomes F ( τ ( R ) R )(( π E ( x ) ⇀ e y ) x )= X c,t ∈ Gu ∗ ( t ) v ( t ) c = y r ∗ ( x ) r ( x ) e t ⊲⊳ ( p + p ∗ )( e c ) u ∗ ( x ) v ( x t ) . (8.4)Let C, D be the subgroups of Z ( G ) determined by p + p ∗ ; in particular, p + p ∗ gives an isomorphism b C → D , and in the above summation wemust have c ∈ C for the term to be non-zero.As we have noted, u ∗ g v and v is normal. So if u ∗ ( t ) v ( t ) c = y forsome c ∈ C and t, y ∈ G then v ( x t ) = v ( x v ( t ) )= v ( x u ∗ ( t ) v ( t ) c )= v ( x y ) . (8.5) UASITRIANGULAR STRUCTURES FOR GROUP DOUBLES 27
So equations (8.4) and (8.5), Definition 8.1, Proposition 8.3, and The-orem 5.3 combine to give F ( τ ( R ) R )(( π E ( x ) ⇀ e y ) π E c ( y ) ⇀ x ))= X c,t ∈ Gu ∗ ( t ) v ( t ) c = y r ∗ ( x ) r ( x ) e t ⊲⊳ ( p + p ∗ )( e c ) u ∗ ( x ) v ( x )= Ψ R ( e y ⊲⊳ x ) . (8.6)By Lemma 8.2 we therefore conclude that F ( τ ( R ) R ) and Ψ R have thesame image. Thus F ( τ ( R ) R ) is bijective if and only if Ψ R is bijective.This completes the proof. (cid:3) Corollary 8.6.
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