The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
ZZMP-HH/17-21Hamburger Beitr¨age zur Mathematik 668
THE SYMPLECTIC FERMION RIBBON QUASI-HOPF ALGEBRAAND THE SL (2 , Z ) -ACTION ON ITS CENTRE V. FARSAD, A.M. GAINUTDINOV, I. RUNKEL
Abstract.
We introduce a family of factorisable ribbon quasi-Hopf algebras Q (N) for Na positive integer: as an algebra, Q (N) is the semidirect product of CZ with the directsum of a Graßmann and a Clifford algebra in 2N generators. We show that Rep Q (N) isribbon equivalent to the symplectic fermion category SF (N) that was computed in [Ru]from conformal blocks of the corresponding logarithmic conformal field theory. The lattercategory in turn is conjecturally ribbon equivalent to representations of V ev , the even part ofthe symplectic fermion vertex operator super algebra.Using the formalism developed in [FGR] we compute the projective SL (2 , Z )-action on thecentre of Q (N) as obtained from Lyubashenko’s general theory of mapping class group actionsfor factorisable finite ribbon categories. This allows us to test a conjectural non-semisimpleversion of the modular Verlinde formula: we verify that the SL (2 , Z )-action computed from Q (N) agrees projectively with that on pseudo trace functions of V ev . keywords: quasi-Hopf algebras, ribbon categories, Kac-Moody (super)algebras, vertex op-erator algebras, Verlinde formula. Contents
1. Introduction 22. The ribbon category SF
63. The ribbon quasi-Hopf algebra Q Rep Q SL (2 , Z )-action on the centre of Q SL (2 , Z )-actions 36Appendix A. Equivalence between SF and Rep S SF and Rep Q a r X i v : . [ m a t h . QA ] A p r V. FARSAD, A.M. GAINUTDINOV, I. RUNKEL Introduction
This paper constitutes the first test of the conjectural modular Verlinde formula for non-semisimple theories proposed in [GR2]. The test is carried out for a family of examplescalled symplectic fermions. These are the simplest logarithmic conformal field theories [Kau],consisting of the untwisted and Z -twisted sector of a free theory (namely purely odd freesuperbosons, see [Ru]). They are parametrised by the natural number N (that counts thenumber of pairs of the fermionic fields) and have central charge c = − C -cofinite, simple and self-dual vertex operator alge-bras V which are non-negatively graded. To prove the modular Verlinde formula one needsto establish the following two statements:(1) Rep V is a factorisable finite ribbon category, see e.g. [FGR, Sec. 4] for conventionsand references.(In Rep V one can then use the categorical Verlinde formula, a purely algebraic state-ment. See [Tu, Thm. 4.5.2] for the semisimple case and [GR2, Thm. 3.9] in general.)(2) The projective SL (2 , Z )-action on the endomorphisms of the identity functor of Rep V computed from [Ly1] agrees with that on the torus one-point functions C ( V ) of V .(The precise formulation is given below; for the Verlinde formula, agreement for the S -generator is enough.)We refer to the introduction of [GR2] for more details and references. In the case that Rep V is in addition semisimple, parts 1 and 2 were proven in [Hu1, Hu2]. The above formulation ofparts 1 and 2 – which is applicable also in the non-semisimple case – was given as conjecturesin [GR2, Sec. 5]. See also [CG] for related results. The precise formulation of part 2 is thatthe two SL (2 , Z )-actions (one projective, one not) on End( Id Rep V ) constructed in (a) and(b) below agree projectively:(a) VOA side:
Pick a projective generator G of Rep V and let E = End V ( G ). Write C ( E ) = { ϕ ∈ E ∗ | ϕ ( f g ) = ϕ ( gf ) for all g, f ∈ E } for the central forms on E . Anelement ϕ ∈ C ( E ) defines a pseudo-trace function ξ ϕG [Mi, AN], providing a linearmap(1.1) ξ G : C ( E ) → C ( V ) , see [GR2, Sec. 4] and Section 6.1 for more details and references. Conjecturally, ξ G is an isomorphism [GR2, Conj. 5.8]. There is a linear (i.e. non-projective) action of SL (2 , Z ) on C ( V ) [Zh] which by pullback along ξ G gives an SL (2 , Z )-action on C ( E ). YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 3 Denote the modular S -transformation on torus one-point functions by S V : C ( V ) → C ( V ) and write δ ∈ C ( E ) for the central form that gets mapped to the modular S -transformation of the vacuum character, ξ δG = S V ( χ V ). Conjecturally, δ providesa non-degenerate central form on E , i.e. it turns E into a symmetric Frobenius al-gebra [GR2, Conj. 5.10]. One obtains an isomorphism between central forms andthe centre of E via C ( E ) → Z ( E ), z (cid:55)→ δ ( z · − ), see e.g. [Br, Lem. 2.5]. Since Z ( E ) ∼ = End( Id Rep V ), altogether we get an isomorphism ˆ δ : End( Id Rep V ) ∼ −→ C ( E ).Via pullback, we finally obtain a linear SL (2 , Z )-action on End( Id Rep V ).(b) Categorical side:
Abbreviate C := Rep V , a factorisable finite ribbon category bypart 1 of the above conjecture. Let L be the universal Hopf algebra in C , which canbe defined as the coend L = (cid:82) U ∈C U ∗ ⊗ U [Ma1, Ly1], see also [FGR, Sec. 2 & 3] formore details. One can equip L with a normalised integral Λ L , which is unique up toa sign. In [Ly1] a projective SL (2 , Z )-action is constructed on the Hom-space C ( , L )using the non-degenerate Hopf pairing on L and the ribbon twist to define the actionof the S - and T -generator, respectively. There is an isomorphism C ( , L ) ∼ = End( Id C ),and one thus obtains a projective SL (2 , Z )-action on End( Id C ), see e.g. [FGR, Sec. 5].This projective action only depends on C and the choice of sign for the integral Λ L [FGR, Prop. 5.3], and different sign choices result in actions that agree projectively.Let us summarise what has been proven in the symplectic fermions example with regard topoints (a) and (b), followed by what is still missing before one has established the Verlindeformula is this case.(a) VOA side for symplectic fermions:
Denote the even part of the vertex operator superalgebra of N pairs of symplectic fermions by V ev [Ab]. Its central charge is c = − SF (N) was obtained via a conformalblock calculation, and it is known that there is a faithful C -linear functor SF (N) → Rep V ev (Proposition 6.1, due to [Ab]), which, conjecturally, is a ribbon equivalence,see below. We review the category SF (N) in Section 2. One can choose a projectivegenerator G in SF (N) and use the above faithful functor to compute pseudo-tracefunctions, their behaviour under modular transformations, and the pullback to C ( E )and to End( Id SF ) ([GR2, Cor. 6.9] and Theorem 6.7).(b) Categorical side for symplectic fermions:
General expressions for the universal Hopfalgebra L and its structure maps are known in the semisimple case, for representationsof Hopf algebras [Ly2, Ke2], and of quasi-Hopf algebras [FGR, Sec. 7]. In this paper,we first generalise the construction of [GR1] to N > Q (N) such that SF (N) (cid:39) Rep Q (N) as ribbon categories. Then we apply theresults in [FGR, Sec. 8] to compute the projective SL (2 , Z )-action on End( Id Rep Q ),and hence on End( Id SF ).The main result of this paper is (Theorem 6.11): V. FARSAD, A.M. GAINUTDINOV, I. RUNKEL
Theorem 1.1.
Under the assumption that ξ G from (1.1) is an isomorphism, the SL (2 , Z ) -actions computed in part (a) and (b) for symplectic fermions agree projectively. This does not yet prove the modular Verlinde formula for symplectic fermions. One stillneeds to show that SF (N) (cid:39) Rep V ev as ribbon categories [DR3, Conj. 7.4] and that thepseudo-trace functions of [AN] indeed provide a bijection between C ( E ) and C ( V ev ) [GR2,Conj. 5.8].After stating the main result, we now describe the contents of this paper in more detail.The present paper is the sequel to [FGR], where the construction of the projective actionof the mapping class group of the torus for factorisable finite ribbon categories from [Ly1]is made explicit for quasi-Hopf algebras. This generalises the treatment for Hopf algebrasin [LM]. Below, we specialise the general results in [FGR] to the symplectic fermion example.To do so, we first need to realise the category SF (N) as the representation category of aquasi-Hopf algebra.From the construction of SF it is natural to make it depend on another discrete parameter β which satisfies β = ( − N , that is, we have factorisable finite ribbon categories SF (N , β )which for the choice β = e − i π N / are conjecturally ribbon-equivalent to Rep V ev , see Sections 2and 6.1. The categories SF (N , β ) all have integral Perron-Frobenius dimensions. By [EGNO,Prop. 6.1.14], such categories can be realised as representations of a quasi-Hopf algebra. Themain technical contribution of this paper is to introduce a new family of factorisable ribbonquasi-Hopf algebras Q (N , β ) such that the following holds (Theorem 3.6): Theorem 1.2. Rep Q (N , β ) (cid:39) SF (N , β ) as C -linear ribbon categories. The proof of this theorem goes via two intermediate quasi-Hopf algebras, one in
Svect andone in vect , and takes up the bulk of this paper.The detailed definition of Q (N , β ) is given in Section 3.1. We refer to [FGR, Sec. 6.1] forour conventions on quasi-Hopf algebras. As an algebra over C , Q (N , β ) is generated by K and f ± i , i = 1 , . . . , N, subject to the relations(1.2) { f ± i , K } = 0 , { f + i , f − j } = δ i,j ( − K ) , { f ± i , f ± j } = 0 , K = , where { x, y } = xy + yx is the anticommutator. Define the central idempotents e := ( + K )and e := ( − K ). Then the coproduct is given by∆( K ) = K ⊗ K − (1 + ( − N ) e ⊗ e · K ⊗ K , (1.3) ∆( f ± i ) = f ± i ⊗ + ( e ± i e ) K ⊗ f ± i . The counit is defined by ε ( K ) = 1 and ε ( f ± i ) = 0. From the coproduct one sees that K isgroup-like only for odd N. The coassociator lives in the Hopf-subalgebra generated by powersof K and takes the simple form(1.4) Φ = ⊗ ⊗ + e ⊗ e ⊗ (cid:8) ( K N − ) e + ( β (i K ) N − ) e (cid:9) . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 5 This should be contrasted with the much more complicated coassociator given in [GR1], where– for a different quasi-Hopf algebra ˆ Q (N , β ) and for N = 1 only – the ribbon equivalence Rep ˆ Q (1 , β ) (cid:39) SF (1 , β ) was first established. In Appendix B.6 we show that ˆ Q (N , β ) istwist-equivalent to Q (N , β ). Remark 1.3. (1) We have not made explicit what the qualifier ‘symplectic’ refers to. But let us at leastpoint out that in [DR1], SF (N , β ) is constructed from a 2N-dimensional symplecticvector space and from the constant β . It is furthermore shown in [DR1, Rem. 5.4]that SF (N , β ) carries a faithful action of SP (2N , C ) by ribbon equivalences.(2) Within a bosonic free-field theory, it was recently shown in [FL] that V ev is the W -algebra constructed as the kernel of (short) screenings in a lattice vertex operatoralgebra of type B N , and with the lattice rescaled by √ p for p = 2. In this context,the symplectic symmetry sp (2N) comes from the action of long screenings. A similarinterplay between B N and C N root lattices exists on the quantum group side due tothe Frobenius homomorphism, studied for this case in [Le].(3) For N = 1, V ev agrees with W ( p ), the vertex operator algebra of the W p -triplet models,in the case p = 2, see [Kau]. For p ≥ W ( p ). The quasi-Hopf algebra in question is a modification of the restrictedquantum group for sl (2) which was used in the study of W p -models in [FGST]. For p = 2, this quasi-Hopf algebra indeed agrees with our Q for N = 1.(4) The categories Rep Q (N , β ) (cid:39) SF (N , β ) provide four of the 16 possible cases in theclassification of factorisable finite tensor categories containing a Lagrangian subcate-gory of a certain form [GS].In Section 2 we review the ribbon category SF (N , β ). In Sections 3 and 4 we studyproperties of Q (N , β ) and the coend L in Rep Q (N , β ). For N even and β = 1, Q (N , β ) isa Hopf algebra, not just a quasi-Hopf algebra. If in addition β = 1, Q (N , β ) is the Drinfelddouble of a generalisation of the Sweedler’s Hopf algebra, see Proposition 3.10. The structuremaps on the coend L = Q (N , β ) ∗ are given in Proposition 4.1.In Section 5 we compute the SL (2 , Z )-action on the centre Z ( Q (N , β )), with the resultstated in Theorem 5.3, and in Section 6 we carry out the comparison to the modular propertiesof pseudo-trace functions as already described above. The three appendices take up almosthalf of the paper and contain the more technical calculations and proofs. Acknowledgements:
We thank Christian Blanchet, Nathan Geer, Simon Lentner, EhudMeir, and Hubert Saleur for helpful discussions. The work of AMG was supported by CNRS.AMG also thanks Mathematics Department of Hamburg University for kind hospitality.
V. FARSAD, A.M. GAINUTDINOV, I. RUNKEL
Convention:
We will make frequent references to the prequel [FGR] of the present paper,and we will abbreviate such references by “I:. . . ”. E.g. we write “Section I:2” instead of[FGR, Sec. 2]. 2.
The ribbon category SF In this section we review the definition of the categories SF (N , β ) introduced in [DR1, Ru],see also [DR2] for a summary. These are finite ribbon categories (see Sections I:2.1 and I:4.1for notation and conventions) which depend on two parameters:(2.1) N ∈ N = { , , . . . } and β ∈ C such that β = ( − N . We will refer to SF (N , β ) as the category of N pairs of symplectic fermions , and we willabbreviate SF := SF (N , β ).2.1. SF as an abelian category. Let Λ be the 2 -dimensional Graßmann algebra over C generated by a i , b i with defining relations { a i , a j } = { b i , b j } = { a i , b j } = 0 , i, j = 1 , . . . , N . (2.2)By giving c ∈ { a , b , . . . , a N , b N } odd parity and defining ∆( c ) = c ⊗ + ⊗ c , ε ( c ) = 0 and S ( c ) = − c we get a Hopf algebra in Svect . Let
Rep s . v . Λ be the category of finite dimensionalsuper-vector spaces with a Λ -action. The category SF = SF (N , β ) is defined as(2.3) SF := SF ⊕ SF where SF := Rep s . v . Λ , SF := Svect . Note that SF is not semi-simple. A choice of representatives for the isomorphism classes ofsimple objects in SF is := C | ∈ SF , T := C | ∈ SF , (2.4) Π := C | ∈ SF , Π T := C | ∈ SF , where Π denotes the parity exchange endofunctor on Svect , and the Λ -action on and Π is trivial. The projective covers of the simple objects are P = Λ , P T = T , (2.5) P Π = Π Λ , P Π T = Π T .
We will now endow SF step by step with the structure of a ribbon category. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 7 Tensor product.
Given two objects
X, Y ∈ SF we define a tensor product functor ∗ : SF × SF → SF as follows:(2.6) X ∗ Y = X Y X ∗ Y SF SF X ⊗ Rep s . v . Λ Y ∈ SF SF SF F ( X ) ⊗ Svect Y ∈ SF SF SF X ⊗ Svect F ( Y ) ∈ SF SF SF Λ ⊗ Svect X ⊗ Svect Y ∈ SF Here, F : Rep s . v . Λ → Svect stands for the forgetful functor. By X ⊗ Rep s . v . Λ Y we mean thetensor product in Svect with Λ -action via the coproduct. In more detail, denote by(2.7) τ s . v .X,Y : X ⊗ Svect Y −→ Y ⊗ Svect
X , τ s . v .X,Y ( x ⊗ y ) = ( − | x || y | y ⊗ x , the symmetric braiding in Svect , where x, y are homogeneous elements. Then the action ofan element g ∈ Λ on x ⊗ y ∈ X ⊗ Svect Y is then given by g. ( x ⊗ y ) = ( ρ X ⊗ ρ Y ) ◦ (id ⊗ τ s . v . Λ ,X ⊗ id) ◦ (∆( g ) ⊗ x ⊗ y )(2.8) = (cid:88) ( g ) ( − | g (cid:48)(cid:48) || x | ( g (cid:48) .x ) ⊗ ( g (cid:48)(cid:48) .y ) where ∆( g ) = (cid:88) ( g ) g (cid:48) ⊗ g (cid:48)(cid:48) , where ρ X and ρ Y give the action of Λ on X and Y .On morphism, we define the tensor product in all cases beside the last one to be f ∗ g = f ⊗ g .If f, g ∈ SF we set f ∗ g = id Λ ⊗ f ⊗ g .For the remainder of this section we will drop the subscripts from the tensor products ⊗ Svect and ⊗ Rep s . v . Λ for brevity.2.3. Associator.
Both, the associator and the braiding depend on β from (2.1) and on acopairing C on Λ given by [DR1, Eqn. (5.5)](2.9) C := N (cid:88) i =1 b i ⊗ a i − a i ⊗ b i ∈ Λ ⊗ Λ . The associator is a natural family of isomorphisms α SF X,Y,Z : X ∗ ( Y ∗ Z ) → ( X ∗ Y ) ∗ Z , which is defined sector by sector by the following eight expressions (see [Ru, Thm. 6.2] and[DR1, Sect. 5.2 & Thm. 2.5]):
X Y Z X ∗ ( Y ∗ Z ) ( X ∗ Y ) ∗ Z α SF X,Y,Z : X ∗ ( Y ∗ Z ) → ( X ∗ Y ) ∗ Z X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z id X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z id X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z exp (cid:0) C (13) (cid:1) V. FARSAD, A.M. GAINUTDINOV, I. RUNKEL X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z id X ⊗ Y ⊗ Z X ⊗ Λ ⊗ Y ⊗ Z Λ ⊗ X ⊗ Y ⊗ Z (cid:104)(cid:8) id Λ ⊗ ( ρ X ◦ ( S ⊗ id X )) (cid:9) ◦ (cid:8) ∆ ⊗ id X (cid:9) ◦ τ s . v .X, Λ (cid:105) ⊗ id Y ⊗ Z Λ ⊗ X ⊗ Y ⊗ Z Λ ⊗ X ⊗ Y ⊗ Z exp (cid:0) C (13) (cid:1) Λ ⊗ X ⊗ Y ⊗ Z Λ ⊗ X ⊗ Y ⊗ Z (cid:8) id Λ ⊗ X ⊗ Y ⊗ ρ Z (cid:9) ◦ (cid:8) id Λ ⊗ τ s . v . Λ ,X ⊗ Y ⊗ id Z (cid:9) ◦ (cid:8) ∆ ⊗ id X ⊗ Y ⊗ Z (cid:9) X ⊗ Λ ⊗ Y ⊗ Z Λ ⊗ X ⊗ Y ⊗ Z (cid:8) φ ⊗ id X ⊗ Y ⊗ Z (cid:9) ◦ (cid:8) τ s . v .X, Λ ⊗ id Y ⊗ Z (cid:9) The underlines mark on which tensor factors Λ acts (hence they appear only when the tripletensor product lies in SF , i.e. when an even number of sectors ‘1’ appear). With C (13) denote C ⊗ ⊗ C where C = (cid:80) ( C ) C ⊗ C . Hence, the action of C (13) is given by(2.10) C (13) ( x ⊗ y ⊗ z ) = ( − ( | x | + | y | ) N (cid:88) i =1 b i .x ⊗ y ⊗ a i .z − a i .x ⊗ y ⊗ b i .z , for homogeneous x, y, z . The linear map φ : Λ → Λ is given by(2.11) φ = (id ⊗ (Λ co Λ ◦ µ Λ )) ◦ (exp( − C ) ⊗ id) , where µ Λ is the multiplication in Λ and Λ co Λ ∈ Λ ∗ is a specific cointegral for Λ [DR1, Eqn. (5.16)].Namely, Λ co Λ is non-vanishing only in the top degree of Λ , and there it takes the value(2.12) Λ co Λ ( a b · · · a N b N ) = β − . Braiding.
For X ∈ Svect denote by(2.13) ω X : X ∼ −−→ X , ω X ( x ) = ( − | x | x , the parity involution on X . The family X (cid:55)→ ω X is a natural monoidal isomorphism of theidentity functor on Svect .The braiding on SF is given – again sector by sector – by the following family of naturalisomorphisms c X,Y (see [Ru, Thm. 6.4] and [DR1, Sect. 5.2 & Thm. 2.8]):(2.14)
X Y c
X,Y : X ∗ Y → Y ∗ X τ s . v .X,Y ◦ exp( − C )0 1 τ s . v .X,Y ◦ (cid:8) κ ⊗ id Y (cid:9) τ s . v .X,Y ◦ (cid:8) id X ⊗ κ (cid:9) ◦ { id X ⊗ ω Y } β · (cid:0) id Λ ⊗ τ s . v .X,Y (cid:1) ◦ (cid:8) R κ − ⊗ id X ⊗ ω Y (cid:9) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 9 Here, κ := exp (cid:0) ˆ C (cid:1) where ˆ C = µ Λ ( C ) = (cid:80) N i =1 − a i b i and R a is the right multiplication with a ∈ Λ : R a : Λ → Λ , R a = µ Λ ◦ ( id Λ ⊗ a ) . (2.15)If a is parity-even, this is indeed a morphism in Svect , and, since it commutes with the Λ -action, it is also a morphism in SF .Recall the definition of a factorisable braided tensor category from Definition I:4.7. A keyproperty of SF is: Proposition 2.1.
The finite braided tensor category SF is factorisable.Proof. In [DR1, Proposition 5.3] it is shown that the full subcategory of transparent objectsin SF is vect . Thus SF fulfils one of equivalent factorisability conditions in [Sh2] (seeSection I:4.3 for a summary in our notation). (cid:3) Left duality.
Our conventions for duality morphisms are given in Section I:2.1. Byconvention, for X ∈ Svect we choose the left-dual X ∗ and the associated duality mapsev Svect X , coev Svect X to be the same as for the underlying vector space.For X ∈ SF i , i = 0 ,
1, we define the left dual object X ∗ ∈ SF i to be the dual in Svect assuper-vector space. For X ∈ SF , X ∗ is furthermore equipped with the Λ -action(2.16) ρ X ∗ : Λ ⊗ X ∗ → X ∗ , g ⊗ ϕ (cid:55)→ ( x (cid:55)→ ( − | g | ( | ϕ | +1) ϕ ( g.x )) . The evaluation and coevaluation maps in SF are induced from those in Svect as follows[DR1, Sec. 3.6] (note that for X ∈ SF we have X ∗ ∗ X = Λ ⊗ X ∗ ⊗ X ):(2.17) X ev SF X : X ∗ ∗ X → coev SF X : → X ∗ X ∗ Svect X coev Svect X ε Λ ⊗ ev Svect X Λ Λ ⊗ coev Svect X Here, ε Λ is the counit for Λ , and Λ Λ = β a b · · · a N b N is the integral for Λ normalised withrespect to the cointegral in (2.12) such that Λ co Λ (Λ Λ ) = 1.2.6. Ribbon twist.
The ribbon twist isomorphisms θ X are given in [DR1, Prop. 4.17]:(2.18) X θ X : X → X − ˆ C )1 β − · ω X The twist isomorphisms satisfy(2.19) θ X ∗ Y = ( θ X ⊗ θ Y ) ◦ c Y,X ◦ c X,Y , θ X ∗ = θ ∗ X . For later reference we note that on the four simple objects in (2.4), the twist is given by(2.20) θ = id , θ Π = id Π , θ T = β − id T , θ Π T = − β − id Π T . We summarise Proposition 2.1 and the ribbon structure on SF reviewed in this section bythe following theorem. Theorem 2.2. SF (N , β ) is a factorisable finite ribbon category for all choices of N and β asin (2.1) . The ribbon quasi-Hopf algebra Q In this section we define the central object of this paper, the family of ribbon quasi-Hopfalgebras Q (N , β ), where the parameters N and β are constrained as in (2.1). The main resultof this section is a ribbon equivalence between Rep Q (N , β ) and SF (N , β ).Our conventions on quasi-Hopf algebras, universal R-matrices, etc., are given in Section I:6.In this section we will abbreviate Q := Q (N , β ).3.1. Definition of Q . In defining the ribbon quasi-Hopf algebra Q , we will first list all of itsdata – starting with the product and ending with the ribbon element – and only afterwardswe will prove that it satisfies the necessary properties.We start by giving Q as an associative unital algebra over C via generators and relations.The generators are(3.1) K and f ± i , i = 1 , . . . , N . Define the elements e := ( + K ) , e := ( − K ) . (3.2)Using these, the defining relations of Q can be written as, for i, j = 1 , . . . , N,(3.3) { f ± i , K } = 0 , { f + i , f − j } = δ i,j e , { f ± i , f ± j } = 0 , K = , where { x, y } = xy + yx is the anticommutator. With these relations, e and e becomecentral idempotents in Q . The corresponding decomposition of Q into ideals is(3.4) Q = Q ⊕ Q where Q i := e i Q . It is easy to check that restricted to each ideal, the algebra structure of Q becomes(3.5) Q = Λ (cid:111) CZ , Q = Cl (cid:111) CZ . Here, Λ is the Graßmann algebra of the 2N generators f ± i e while CZ stands for the groupalgebra of Z generated by K e . Similarly, Cl is the Clifford algebra generated by f ± i e ,while the generator of Z is i K e . In particular, we see that(3.6) dim C Q = 2 and that a basis of Q is(3.7) Q = span C (cid:8) f ε i · · · f ε m i m K n | ≤ m ≤ N , ≤ i < i < . . . < i m ≤ N , ε j = ± , n ∈ Z (cid:9) . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 11 Next we define the quasi-bialgebra structure of Q . The coproduct on generators is∆( K ) = K ⊗ K − (1 + ( − N ) e ⊗ e · K ⊗ K , (3.8) ∆( f ± i ) = f ± i ⊗ + ω ± ⊗ f ± i , ω ± := ( e ± i e ) K . We will show in Lemma 3.3 below that ∆ is well-defined and an algebra map. The counit is(3.9) ε ( K ) = 1 , ε ( f ± i ) = 0 . We remark that K itself is group-like only for odd N. However, one quickly verifies that(3.10) K , K N are group-like for all N . The co-associator and its inverse are(3.11) Φ ± = ⊗ ⊗ + e ⊗ e ⊗ (cid:8) ( K N − ) e + ( β ( ± i K ) N − ) e (cid:9) . For the quasi-Hopf algebra structure on Q we still need to specify the antipode S and theevaluation and coevaluation elements α and β . They are: S ( K ) = K ( − N = ( e + ( − N e ) K , α = , (3.12) S ( f ± k ) = f ± k ( e ± ( − N i e ) K , β = e + β (i K ) N e . Remark 3.1.
For even N and β = 1 we have α = β = and Φ = ⊗ ⊗ , so that(pending the verification that the above data verifies the axioms) in these cases Q is a Hopfalgebra. For odd N on the other hand, the coproduct fails to be coassociative. For example,(∆ ⊗ id) ◦ ∆( f − i ) = f − i ⊗ ⊗ + ω − ⊗ f − i ⊗ (3.13) + K ⊗ K · (cid:0) e ⊗ e − i e ⊗ e − i e ⊗ e − ( − N e ⊗ e (cid:1) ⊗ f − i , (id ⊗ ∆) ◦ ∆( f − i ) = f − i ⊗ ⊗ + ω − ⊗ f − i ⊗ + K ⊗ K · ( e ⊗ e − i e ⊗ e − i e ⊗ e − e ⊗ e ) ⊗ f − i . Next we introduce a quasi-triangular structure on Q . Define the Cartan factor ρ n,m as(3.14) ρ n,m = (cid:88) i,j =0 ( − ij i − in + jm K i ⊗ K j , n, m ∈ { , } . The universal R -matrix and its inverse are defined as R = (cid:16) (cid:88) n,m ∈{ , } β nm ρ n,m e n ⊗ e m (cid:17) · N (cid:89) k =1 ( ⊗ − f − k ω − ⊗ f + k ) , (3.15) R − = N (cid:89) k =1 ( ⊗ + 2 f − k ω − ⊗ f + k ) · (cid:16) (cid:88) n,m ∈{ , } β − nm ρ n,m e n ⊗ e m (cid:17) . Finally, the ribbon element of Q and its inverse are v = ( e − β i K e ) · N (cid:89) k =1 ( − f + k f − k ) , (3.16) v − = ( e − β − i K e ) · N (cid:89) k =1 ( + 2 f + k f − k K ) . (3.17) Proposition 3.2.
The data ( Q , · , , ∆ , ε, Φ , S, α , β , R, v ) defines a ribbon quasi-Hopf algebra. The proof of this proposition will be given after some preparation. The following lemma isstraightforward check on the generators of Q . Lemma 3.3.
The map ∆ defined in (3.8) is an algebra map. Since ∆ is an algebra map, we can use it to define a tensor product functor(3.18) ⊗ : Rep Q × Rep Q → Rep Q . The central idempotents e and e behave under the coproduct as(3.19) ∆( e ) = e ⊗ e + e ⊗ e , ∆( e ) = e ⊗ e + e ⊗ e , so that the tensor product (3.18) respects the Z -grading Rep Q = Rep Q ⊕ Rep Q .Denote by(3.20) τ X,Y : X ⊗ Y −→ Y ⊗ X , τ
X,Y ( x ⊗ y ) = y ⊗ x , the symmetric braiding in vect . By acting with Φ, R nd v − we get families of isomorphismsin Rep Q α Rep Q M,N,K : M ⊗ ( N ⊗ K ) → ( M ⊗ N ) ⊗ K , m ⊗ n ⊗ k (cid:55)→ Φ . ( m ⊗ n ⊗ k ) , (3.21) c Rep Q M,N : M ⊗ N → N ⊗ M , m ⊗ n (cid:55)→ τ M,N ( R. ( m ⊗ n )) , (3.22) θ Rep Q M : M → M , m (cid:55)→ v − .m . (3.23)We will show in Lemma 3.5 below that the linear maps (3.21)–(3.23) are indeed morphismsin Rep Q , that they are natural and that α Rep Q gives an associator, c Rep Q a braiding and θ Rep Q a twist in Rep Q . Definition 3.4.
Let C be a monoidal category and D a category with a tensor productfunctor ⊗ D : D × D → D and tensor unit D . A functor F : C → D is called multiplicative if there exists a family of natural isomorphisms Θ
U,V : F ( U ⊗ C V ) → F ( U ) ⊗ D F ( V ) and anisomorphism Θ : F ( ) → .If D in the above definition is a monoidal category as well (i.e. equipped with associatorand unit isomorphisms), a multiplicative functor is also called a quasi-tensor functor , see e.g.[EGNO, Def. 4.2.5], i.e. we dropped the coherence conditions with the associators from thedefinition of the tensor functor. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 13 Let F : C → D be a multiplicative equivalence. We now describe how F can be used totransport structure from C to D . • By transporting the associator from C to D along F we mean seeking a natural family α D X,Y,Z , for
X, Y, Z ∈ D , such that for all
U, V, W ∈ C , the diagram(3.24) F (cid:0) U ⊗ C ( V ⊗ C W ) (cid:1) Θ U,V ⊗C W (cid:15) (cid:15) F ( α C U,V,W ) (cid:47) (cid:47) F (cid:0) ( U ⊗ C V ) ⊗ C W (cid:1) Θ U ⊗C V,W (cid:15) (cid:15) F ( U ) ⊗ D F ( V ⊗ C W ) id ⊗ Θ V,W (cid:15) (cid:15) F ( U ⊗ C V ) ⊗ D F ( W ) Θ U,V ⊗ id (cid:15) (cid:15) F ( U ) ⊗ D (cid:0) F ( V ) ⊗ D F ( W ) (cid:1) α DF ( U ) , F ( V ) , F ( W ) (cid:47) (cid:47) (cid:0) F ( U ) ⊗ D F ( V ) (cid:1) ⊗ D F ( W )commutes. Since F is an equivalence such an α D exists. Moreover, because α D isnatural and F is essentially surjective, α D is unique. By construction α D satisfies thepentagon condition. The tensor unit structure (i.e. left and right unit morphisms) canbe transported in the same way which makes D a monoidal category. • Suppose C is in addition braided. Transporting the braiding from C to D meansdetermining a family c D X,Y such that the following diagram commutes for all
U, V ∈ C :(3.25) F ( U ⊗ C V ) F ( c C U,V ) (cid:47) (cid:47) Θ U,V (cid:15) (cid:15) F ( V ⊗ C U ) Θ V,U (cid:15) (cid:15) F ( U ) ⊗ D F ( V ) c DF ( U ) , F ( V ) (cid:47) (cid:47) F ( V ) ⊗ D F ( U )For the same reasons as above this diagram determines the c D X,Y uniquely and turns D into a braided category. • Suppose C is in addition ribbon. By transporting the ribbon twist we mean givinga natural family θ D X such that for all U ∈ C , θ DF ( U ) = F ( θ C U ). Again, the family θ D X is unique and turns D into a ribbon category (with braided monoidal structure asabove). Lemma 3.5.
The linear maps α Rep Q , c Rep Q , θ Rep Q in (3.21) – (3.23) are natural isomor-phisms in Rep Q . There exists a multiplicative equivalence F : SF →
Rep Q which transports • the associator (2.3) of SF to α Rep Q , and the unit isomorphisms of SF to those of theunderlying vector spaces in Rep Q , • the braiding (2.4) of SF to c Rep Q , • the ribbon twist (2.6) of SF to θ Rep Q . The proof of this lemma is lengthy and tedious (and fills half of this paper). It is spreadacross the Appendices A and B. In Appendix A we transport the structure morphisms of SF to an intermediate category of representations of a quasi-bialgebra in Svect . In Appendix Bwe transport the structure morphisms further to a quasi-bialgebra in vect which, finally, weexhibit to be a twisting of Q . Proof of Proposition 3.2.
By Lemma 3.5, α Rep Q fulfils the pentagon identity (since α SF doesand F is multiplicative) and c Rep Q the hexagon identities. Hence, Rep Q is braided monoidal.We conclude that ( Q , · , , ∆ , ε, R ) is a quasi-triangular quasi-bialgebra.We will now show that S, α , β define a quasi-Hopf structure on Q , see I:6.1 for definitions.A straightforward calculation shows that S , as defined on generators in (3.12), is compatiblewith the relations on Q and hence provides an algebra anti-homomorphism on Q . It remainsto show the identities(3.26) (cid:88) ( a ) S ( a (cid:48) ) α a (cid:48)(cid:48) = ε ( a ) α , (cid:88) ( a ) a (cid:48) β S ( a (cid:48)(cid:48) ) = ε ( a ) β , for all a ∈ S and(3.27) (cid:88) (Φ) S (Φ ) α Φ β S (Φ ) = , (cid:88) (Φ − ) (Φ − ) β S ((Φ − ) ) α (Φ − ) = . For example, to see the first equality in (3.27) one computes (cid:88) (Φ) S (Φ ) α Φ β S (Φ ) = e + e (cid:0) β (i K ) N (cid:1) S (cid:0) β (i K ) N (cid:1) (3.28) = e + e K N (( − N K ) N = e + e ( − N − N = . To see the first identity in (3.26) we define the linear map P := µ ◦ ( S ⊗ id) : Q ⊗ Q → Q where µ denotes the multiplication in Q and check that P (cid:0) ∆( f ± i ) (cid:1) = S ( f ± i ) + S (cid:0) ( e ± i e ) K (cid:1) f ± i (3.29) = f ± i K ( e ± ( − N i e ) + ( e ± ( − N i e ) Kf ± i = 0 ,P (cid:0) ∆( K ) (cid:1) = ( e + ( − N e ) K − (1 + ( − N ) e K = . For general basis elements f ε i · · · f ε m i m K n from (3.7) and by defining f := f ε i and f := f ε i · · · f ε m i m K n we get P (cid:0) ∆( f ε i · · · f ε m i m K n ) (cid:1) = P (cid:0) ∆( f )∆( f ) (cid:1) = (cid:88) ( f ) , ( f ) S ( f (cid:48) f (cid:48) ) f (cid:48)(cid:48) f (cid:48)(cid:48) (3.30) = (cid:88) ( f ) S ( f (cid:48) ) P (cid:0) ∆( f ) (cid:1) f (cid:48)(cid:48) (3.29) = 0 ,P (cid:0) ∆( K n ) (cid:1) = (cid:88) ( K n − ) S (cid:0) ( K n − ) (cid:48) (cid:1) P (cid:0) ∆( K ) (cid:1) ( K n − ) (cid:48)(cid:48) = P (cid:0) ∆( K n − ) (cid:1) = , where the last equality follows by induction in n . The second identity in (3.26) can be shownin a similar way. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 15 Having a quasi-Hopf structure we can now conclude from Lemma 3.5 that v defines aribbon element in Q . In particular, S ( v ) = v follows from the duality property of the twist θ U ∗ = ( θ U ) ∗ (one can verify that if this equality holds for one choice of left duality ( − ) ∗ onthe category in question it holds for all choices of left duality). (cid:3) Another important consequence of Lemma 3.5 is the following theorem.
Theorem 3.6. SF (N , β ) is ribbon equivalent to Rep Q (N , β ) for all choices of N and β asin (2.1) . The precise definition of the equivalence functor F : SF →
Rep Q is given in Section B.8. Remark 3.7.
For fixed N, the quasi-triangular quasi-Hopf algebras Q (N , β ) for the fourpossible choices of β differ by abelian 3-cocycles for Z . The 3rd abelian group cohomologyof Z is H ab ( Z , C × ) = Z and it describes possible braided monoidal structures on Rep Z ,up to braided monoidal equivalence (see [JS] for details). A generator of H ab ( Z , C × ) is givenby the class of ( ω, σ ), where ω is a 3-cocycle for group-cohomology with (writing Z = { , } additively) ω (1 , ,
1) = − σ is a 2-cochain with only non-trivial value σ (1 ,
1) = i. Multiplying the coassociator Φ by (cid:80) a,b,c ∈ Z ω ( a, b, c ) · e a ⊗ e b ⊗ e c and the R -matrix by (cid:80) a,b ∈ Z σ ( a, b ) · e a ⊗ e b is equivalent to changing β to i β . Repeating this makes β run through its four possibilities.3.2. Factorisability of Q . A finite-dimensional quasi-triangular quasi-Hopf algebra is called factorisable if its representation category is factorisable in the sense of Definition I:6.5. Adirect definition in terms of the data of Q can be found in Remark I:6.6. It is shown inCorollary I:7.6 that this definition is equivalent to the one given in [BT, Def. 2.1]. We obtainthe following corollary to Theorem 3.6: Corollary 3.8. Q is a factorisable ribbon quasi-Hopf algebra. Below in Lemma 4.2 we will give an alternative proof by verifying non-degeneracy of theHopf pairing of the universal Hopf algebra in
Rep Q by direct calculation.3.3. Q (2 n, as a Drinfeld double. In Remark 3.1 we saw that for N = 2 n and β = 1, Q (N , β ) is a Hopf algebra, not only a quasi-Hopf algebra. Combining this with Corollary 3.8shows that Q (2 n, ±
1) is a factorisable ribbon Hopf algebra. It turns out that as a quasi-triangular Hopf algebra, Q (2 n,
1) is isomorphic to a Drinfeld double, as we now explain.Let H = H (N) be the algebra generated by the elements k and f i , i = 1 , . . . , N, subject tothe relations { f i , f j } = 0 , { f i , k } = 0 , k = . (3.31) We define the coproduct, counit and antipode on H (N) as ( i = 1 , . . . , N)∆( f i ) = f i ⊗ k + ⊗ f i , ∆( k ) = k ⊗ k ,ε ( f i ) = 0 , ε ( k ) = 1 , S ( f i ) = − f i k , S ( k ) = k . (3.32)One can easily verify that we get an injective Hopf-algebra homomorphism H (N) → Q (N , k (cid:55)−→ ω − = ( e − i e ) K , f i (cid:55)−→ f − i ω − , i = 1 , . . . , N . This embedding also proves that H (N) is indeed a Hopf algebra. Remark 3.9.
Above we defined H (N) for even N, but the definition works just as well forodd N. In this case, the map (3.33) defines an embedding H (N) → Q (N + 1 ,
1) (or intoany Q (2 n,
1) with 2 n >
N) and therefore H (N) is a Hopf algebra for all N ∈ N . Note that H (1) is Sweedler’s 4-dimensional Hopf algebra, so that for N > H (N) is theHopf algebra associated to (or bosonisation of) the super-group algebra Λ C N , see e.g. [AEG,Sec. 3.4]. Proposition 3.10.
For N ≥ , the Drinfeld double of H (N) is isomorphic to Q (N , β ) as a C -algebra. For even N and β = ± the Drinfeld double of H (N) is isomorphic to Q (N , β ) asa Hopf algebra. Moreover, if β = 1 this isomorphism is an isomorphism of quasi-triangularHopf algebras. The proof of this proposition is given in Appendix C.
Remark 3.11. (1) The double of H (N) has also been constructed in [BN]. It appears in [GS] in theclassification of factorisable tensor categories which contain Rep H (N) as a Lagrangiansubcategory (and of which SF (N , β ) provide four of the 16 possible cases).(2) Following Remark 3.7, the quasi-triangular quasi-Hopf algebras Q (2 n, β ) for the otherthree choices of β are simple modifications of the Drinfeld double of H (2 n ) by the 3rdabelian cohomology classes of Z .(3) It is shown in [DR3, Thm. 6.11] that SF (N , β ) contains a Lagrangian algebra L iff N iseven and β = 1 (see [DR3] for definition and references). This implies that precisely inthese cases SF (N , β ) is equivalent as a braided category to the Drinfeld centre Z ( D ) ofsome other (non-unique) finite tensor category D [DMNO, Cor. 4.1], e.g. one may choose D = L -mod. Proposition 3.10 shows that D can also be taken to be Rep H (N).3.4. Some special elements of Q . The Drinfeld twist f of a quasi-Hopf algebra expressesthe deviation of the antipode from being an anti-coalgebra map via(3.34) f ∆( S ( a )) = ( S ⊗ S )(∆ op ( a )) f , a ∈ A .
YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 17 Its expression in terms of quasi-Hopf algebra structure maps is given in I:(6.25) following [Dr].When evaluated for Q , one quickly checks that the general expression reduces to f = e ⊗ + e ⊗ K N e + β ( − i K ) N e ⊗ e . (3.35)As reviewed in Section I:6.3, the canonical Drinfeld element u and the correspondingelement ˜ u with inverse braiding defined as u = (cid:88) (Φ) , ( R ) S (cid:0) Φ β S (Φ ) (cid:1) S ( R ) α R Φ , (3.36) ˜ u = (cid:88) (Φ) , ( R − ) S (cid:0) Φ β S (Φ ) (cid:1) S (cid:0) ( R − ) (cid:1) α ( R − ) Φ . Lemma 3.12. In Q the elements u , ˜ u and u − take the form u = (cid:16) e K + e β ( − i K ) N (cid:17) · N (cid:89) i =1 ( − f + i f − i ) , (3.37) ˜ u = u − = (cid:0) e K + e β − ( − i K ) N (cid:1) · N (cid:89) i =1 ( + 2 f + i f − i K ) . Proof.
We start with the expression for u in (3.37). We give the details for sector , thecomputation in sector is similar.In sector , the first equality in (3.36) reduces to u · e = S ( R ) R · e . Define the linearmap T : Q ⊗ Q → Q , a ⊗ b (cid:55)→ S ( b ) a , so that u · e = T ( R ) · e . The last factor of R in (3.15)can be written as X := N (cid:89) i =1 ( ⊗ − f − i ω − ⊗ f + i ) = ⊗ + (cid:88) ≤ m ≤ N1 ≤ i <...
For even N the element g equals ω ± for β = ∓
1, while for N odd g = K ± for β = ∓ i. Given a pivotal structure on a monoidal category with left duals, all other pivotalstructures are obtained by composing the given one with natural monoidal automorphismsof the identity functor. In Rep Q , these are given by acting with group-like elements in thecentre of Q . It follows from Proposition 3.15 below that these are precisely { , K } . Modifyingthe pivotal structure by K has the effect of replacing β by − β in (3.45).3.5. Integrals. A two-sided integral of a quasi-Hopf algebra A is an element c ∈ A suchthat [HN, Def. 4.1](3.46) c a = ε ( a ) c = a c , for all a ∈ A .
In [BT, Sec. 6] it is shown that factorisable quasi-Hopf algebras are unimodular. Togetherwith [HN, Thm. 4.3] this shows that the space of two-sided integrals is one-dimensional. It iseasy to see that every element in Q of the form c = ν N β f +1 f − . . . f +N f − N e (1 + K ) , ν ∈ C , (3.47) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 19 satisfies (3.46). Indeed, c K = c = K c and c f ± i = 0 = f ± i c . The prefactor ν N β will beconvenient later, when a normalisation condition will require the constant ν to be a sign (seeProposition 4.4 below).3.6. The centre of Q . We define(3.48) e ± := e (cid:0) ∓ i K N (cid:89) i =1 ( − f + i f − i ) (cid:1) . Lemma 3.14.
The elements e ± are central orthogonal idempotents. Moreover, the ribbontwist acts on e ± by a scalar: (3.49) v − · e ± = ± β − e ± . Proof.
It is easy to see the commutativity property of e ± . For example, since(3.50) K ( − f + i f − i ) f + i e = − Kf + i e = f + i K ( − f + i f − i ) e e ± commutes with f + i .The orthogonality and idempotent property follow immediately from e ( − f + i f − i ) = e ,see (3.42). In order to prove (3.49) we express e ± in terms of v from (3.16) as(3.51) e ± = e ( ± β − v ) . Together with β − ve = β v − e , this gives(3.52) v − e ± = e ( v − ± β − ) = ± β − e ( ± β v − + ) = ± β − e ± . (cid:3) Proposition 3.15.
The centre of Q is Z ( Q ) = Z ⊕ Z , where Z := span C (cid:110) e k (cid:89) j =1 f ε j i j (cid:12)(cid:12) k ≤ N / , ≤ i j ≤ N , ε j = ± (cid:111) ⊕ C K e (cid:89) i =1 f + i f − i , (3.53) Z := span C (cid:8) e +1 , e − (cid:9) . It has dimension − .Proof. From (3.5) we know that the ideal Q is a direct sum of two matrix algebras, soits centre is two-dimensional and is therefore spanned by the central idempotents e ± fromLemma 3.14. It remains to compute the centre Z of Q . For an element of Q to commutewith K , it must be a sum of monomials in f ± i e of even degree, where each monomial can bemultiplied by K , that is,(3.54) Z ⊂ span C (cid:110) K δ k (cid:89) j =1 f ε j i j e (cid:12)(cid:12) k ≤ N / , ≤ i j ≤ N , ε j = ± , δ ∈ { , } (cid:111) . Any monomial from RHS of (3.54) with δ = 0 obviously commutes with f ± i , for 1 ≤ i ≤ N,while a monomial with δ = 1 commutes with all f ± i iff it is annihilated by all f ± i . This gives theexpression for Z in (3.53). Since the C -linear span of the even degree monomials (multipliedby e ) has the dimension 2 − , the overall dimension of the centre is(3.55) dim C Z ( Q ) = 3 + 2 − . (cid:3) Simple and projective Q -modules. Recall the decomposition (3.4) of Q onto thedirect sum of two algebras Q ⊕ Q where the first is the Graßmann algebra times Z , whichis non-semisimple, while the second is the Clifford algebra times Z , which is semisimple.Therefore, the algebra Q has up to an isomorphism only four simple modules that we willdenote as X ± s , with s = 0 ,
1, and where X ± ∈ Rep Q while X ± ∈ Rep Q . They are ofhighest-weight type: X ± are one-dimensional of weights ± K and with zeroaction of f ± i , i.e. they are spanned by v ± such that(3.56) K .v ± = ± v ± , f ± i .v ± = 0 ; X ± are of the highest weights ± i, i.e. they are generated by v ± such that(3.57) K .v ± = ± i v ± , f − i .v ± = 0 . A basis of X ± is given by the set(3.58) (cid:110) v ± i := N (cid:89) k =1 ( f + k ) i k .v ± (cid:12)(cid:12)(cid:12) i = ( i , . . . , i N ) , i k ∈ { , } (cid:111) . In particular, the dimension of X ± is 2 N . In terms of the primitive central idempotents e ± from (3.48), the modules X ± are Q e ± and the generating vector v ± can be obtained within Q as (cid:81) N i =1 f − i e ± . The modules X ± are therefore projective.We note that(3.59) e ± = ( ± K ) e are primitive (non-central) idempotents and K e ± = e ± K = ± e ± . The module P ± := Q e ± istherefore a projective cover for X ± . It is indecomposable but reducible and has the basis(3.60) P ± : span (cid:110)(cid:16) N (cid:89) k =1 ( f + k ) i + k ( f − k ) i − k (cid:17) e ± (cid:12)(cid:12)(cid:12) i + k , i − k ∈ { , } (cid:111) . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 21 The dimension of P ± is thus 2 and each of them has 2 − copies of X ± in its compositionseries. We can finally conclude that the Cartan matrix C ( Q ) is(3.61) C ( Q ) = − − − − . One can check now the dimension of Q by decomposing it as the left regular representation: Q = P +0 ⊕ P − ⊕ N X +1 ⊕ N X − and this indeed gives dim Q = 2 .3.8. Basic algebra.
The basic algebra of Q is E := End Q ( G Q ) where G Q is the minimalprojective generator(3.62) G Q = P +0 ⊕ P − ⊕ X +1 ⊕ X − . In what follows, we will need a description of this algebra. Recall from (3.5) that Λ := Λ isthe subalgebra in Q generated by f ± i e . Lemma 3.16. E op = ( Λ ⊕ C e X ) (cid:111) CZ , where • e X denotes the idempotent corresponding to X +1 ⊕ X − , i.e. it acts as identity on X +1 ⊕ X − and as zero otherwise, • a ∈ Λ acts on P +0 ⊕ P − = Q by right multiplication, • the generator κ of the group algebra CZ acts on P +0 ⊕ P − by right multiplication with K which is ± id on P ± ; on X ± it equally acts by ± id , • the algebra structure is that of Λ ⊕ C e X with κa = − aκ for all a = f ± i e , κe X = e X κ .Proof. We have the decomposition End( G Q ) = E ⊕ E where E = End Q ( P +0 ⊕ P − ) and E =End Q ( X +1 ⊕ X − ). The latter algebra is a direct sum of two matrix algebras of dimension 1 each,and is isomorphic to the algebra C e X (cid:111) CZ from the statement. We then note that P +0 ⊕ P − is equal to the left regular representation of Q , with the action by left multiplication. Thealgebra centralising this action is given by Q op0 which acts by right multiplication. Therefore E op0 = Q . We then recall from (3.5) that Q = Λ (cid:111) CZ , with the same Z action as inthe statement. The fact that K acts by ± id on the direct summands P ± follows from theidentification P ± := Q e ± , recall (3.59). This finally proves the lemma. (cid:3) Remark 3.17.
In the above lemma we give E op instead of E as we describe the endomorphismalgebra via a right action. However, from the defining relations of Q it is easy to give analgebra isomorphism E op → E , e.g. via the antipode S . We also note that Rep Q is equivalentto Rep E as abelian categories, i.e. Q and E are Morita equivalent, and E is the minimalalgebra with such a property. Its matrix elements are the multiplicities of the simple Q -module V in the composition series of theprojective cover P U of U , i.e. C ( Q ) U,V = Hom Q ( P V , P U ). We recall then the equivalence stated in Theorem 3.6 (we need only the equivalence ofabelian C -linear categories) and given by the functor F : SF →
Rep Q . Under this functor,the projective covers are mapped as F ( P ) = P +0 , F ( P Π ) = P − , F ( T ) = X +0 , and F (Π T ) = X − , recall (2.5). In particular, the minimal projective generator in SF (3.63) G SF := P ⊕ P Π ⊕ T ⊕ Π T = ( Λ ⊕ T ) ⊗ C | goes to F ( G SF ) ∼ = G Q .We denote the functor inverse to F , as a C -linear functor, by J : Rep Q → SF and J = E ◦ H , where E is given in the proof of Proposition A.3 and H in the proof of Proposition B.3.To describe End SF ( G SF ) explicitly, consider the isomorphism ψ : Λ ⊗ C | ∼ −→ J ( Q ) given by(3.64) ψ : f ⊗ v (cid:55)→ f · (cid:0) v e +0 + v e − (cid:1) , f ∈ Λ , v ∈ C | , where v / ∈ C is the even/odd component of v . Using this isomorphism, we have that theendomorphism R a of Q from Lemma 3.16 given by the right multiplication with a ∈ Λ goesunder the functor J to ψ − ◦ J ( R a ) ◦ ψ : Λ ⊗ C | → Λ ⊗ C | , (3.65) f ⊗ v (cid:55)→ f · a ⊗ Π deg a v . Indeed, the shift of the degree part is due to(3.66) ψ − ( R a ( f e ± )) = ψ − ( f e ± · a ) = ψ − ( f · a e ∓ ) = f · a ⊗ Π( − ) , for odd a . Then, as a corollary to Lemma 3.16 we get: Corollary 3.18.
For the minimal projective generator G SF the opposite algebra of End SF ( G SF ) is ( Λ ⊕ C e T ) (cid:111) CZ where • e T is the idempotent corresponding to T ⊕ Π T , • the generator κ acts by id on P and T and by − id on P Π and Π T , • the element a ∈ Λ acts as in (3.65) , • the algebra structure is that of Λ ⊕ C e T with κa = − aκ for odd a , κe T = e T κ . Properties of the coend in
Rep Q In this section we investigate the universal Hopf algebra L of Rep Q , which can be expressedas a coend of a certain functor. We refer to Sections I:2 and I:3 for general background andreferences on the universal Hopf algebra of a braided monoidal category with duals, and toSection I:7 for the particular case of categories of representations of quasi-triangular quasi-Hopf algebras.We give explicitly the Hopf algebra structure and Hopf pairing on the universal Hopfalgebra L of Rep Q , we verify by direct calculation that the Hopf pairing is non-degenerate,and we describe the integrals and cointegrals of L . In section 4.4, we also calculate the centralelements corresponding to internal characters of L . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 23 The universal Hopf algebra L . By Proposition I:7.1, universal Hopf algebra – de-scribed by a coend L in Rep Q – can be chosen to be the object L = Q ∗ equipped with thecoadjoint action(4.1) Q ⊗ Q ∗ → Q ∗ , a ⊗ ϕ (cid:55)→ (cid:88) ( a ) ϕ (cid:0) S ( a (cid:48) )( − ) a (cid:48)(cid:48) (cid:1) , and the dinatural transformation(4.2) ι M : M ∗ ⊗ M → Q ∗ , ϕ ⊗ m (cid:55)→ (cid:0) a (cid:55)→ ϕ ( a.m ) (cid:1) , M ∈ Rep Q (see Figure I:2 for a string diagram representation).We define the contraction maps (cid:104)− , −(cid:105) : Q ∗ ⊗ Q → C , (cid:104) ϕ, a (cid:105) = ϕ ( a ) , (4.3) (cid:104)− , −(cid:105) : Q ∗ ⊗ Q ∗ ⊗ Q ⊗ Q → C , (cid:104) ϕ ⊗ ψ, a ⊗ b (cid:105) = ϕ ( b ) ψ ( a ) . As in Section I:7.2, we will express the Hopf algebra structure maps of L in terms of theirduals as follows ( f, g ∈ Q ∗ , a, b ∈ Q ): (cid:10) µ L ( f ⊗ g ) , a (cid:11) = (cid:10) f ⊗ g , ˆ µ L ( a ) (cid:11) , ˆ µ L : Q → Q ⊗ Q , (4.4) (cid:10) ∆ L ( f ) , a ⊗ b (cid:11) = (cid:10) f , ˆ∆ L ( a ⊗ b ) (cid:11) , ˆ∆ L : Q ⊗ Q → Q ,η L (1) = (cid:0) a (cid:55)→ ˆ η L ( a ) (cid:1) , ˆ η L : Q → C ,ε L ( f ) = f (ˆ ε L ) , ˆ ε L ∈ Q , (cid:10) S L ( f ) , a (cid:11) = (cid:10) f , ˆ S L ( a ) (cid:11) , ˆ S L : Q → Q ,ω L ( f ⊗ g ) = (cid:10) f ⊗ g , ˆ ω L (cid:11) , ˆ ω L ∈ Q ⊗ Q . Proposition 4.1.
Via the dualisation in (4.4) , the Hopf algebra structure maps and the Hopfpairing on the coend L = Q ∗ are given by ˆ µ L ( a ) = (cid:88) ( R ) , ( a ) ,n,m ∈ Z (cid:0) K N nm R K N m (cid:1) (cid:46) a (cid:48) ⊗ ( K N n (1 − m ) (cid:46) a (cid:48)(cid:48) ) R · e n ⊗ e m , (4.5) ˆ∆ L ( a ⊗ b ) = ba , ˆ η L ( a ) = ε ( a ) , ˆ ε L = , ˆ S L ( a ) = (cid:88) ( R ) S ( aR ) ˜ u R , ˆ ω L = (cid:88) ( M ) S ( M ) ⊗ M e + S ( K − N M K N ) ⊗ M e , where h (cid:46) a := (cid:80) ( h ) S ( h (cid:48) ) ah (cid:48)(cid:48) defines an action of Q op on Q , and ˜ u was defined in (3.37) . Proof.
Applying Theorem I:7.3 the only non-trivial equalities in (4.5) are the first two andlast one. Note that for the calculation of ˆ η L we used ε ( β ) = 1. By the same theorem weknow that ˆ µ L ( a ) = (cid:88) (Φ) , (Ψ) , ( ˜Ψ) , ( R ) (cid:104) S (Φ Ψ R (cid:48) ˜Ψ (cid:48) ) ⊗ S (Φ ˜Ψ ) (cid:105) · f (4.6) · ∆( a Φ ) · (cid:104) (Ψ R (cid:48)(cid:48) ˜Ψ (cid:48)(cid:48) ) ⊗ (Ψ R ˜Ψ ) (cid:105) , where Ψ = Φ − , ˜Ψ is another copy of Φ − , and f is the Drinfeld twist from (3.35). Wecompute the result sector by sector. For this, it is useful to expand f and Φ ± sector bysector first:(4.7) f ⊗ ⊗ ⊗ K N β ( − i) N ⊗ K N Φ ± ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ K N β ( ± i) N ⊗ ⊗ K N The elements f and Φ ± are recovered from these tables by summing over the idempotents e a ⊗ e b (resp. e a ⊗ e b ⊗ e c ) multiplied with the corresponding entry of the table.We now give the contribution of each sector to ˆ µ L ( a ). In doing so, we indicate whichsectors of f , Φ , Ψ , ˜Ψ , R contribute to the expression (the equalities in sectors ij are true upthe multiplication with the corresponding idempotents e i ⊗ e j , here we omit them for brevity): f : 00 , Φ : 000 , Ψ : 000 , ˜Ψ : 000 , R : 00 (cid:88) ( R ) [ S ( R (cid:48) ) ⊗ ] · ∆( a ) · [ R (cid:48)(cid:48) ⊗ R ] = (cid:88) ( R ) , ( a ) R (cid:46) a (cid:48) ⊗ a (cid:48)(cid:48) R f : 01 , Φ : 101 , Ψ : 001 , ˜Ψ : 110 , R : 10 (cid:88) ( R ) (cid:2) S ( R (cid:48) K N ) ⊗ (cid:3) · ∆( a ) · (cid:2) R (cid:48)(cid:48) K N ⊗ R (cid:3) = (cid:88) ( R ) , ( a ) ( R K N ) (cid:46) a (cid:48) ⊗ a (cid:48)(cid:48) R f : 10 , Φ : 011 , Ψ : 110 , ˜Ψ : 000 , R : 00 (cid:88) ( R ) [ S ( R (cid:48) ) ⊗ ] · ⊗ K N · ∆( a ) · (cid:2) R (cid:48)(cid:48) ⊗ ( K N R ) (cid:3) = (cid:88) ( R ) , ( a ) R (cid:46) a (cid:48) ⊗ K N a (cid:48)(cid:48) K N R = (cid:88) ( R ) , ( a ) R (cid:46) a (cid:48) ⊗ ( K N (cid:46) a (cid:48)(cid:48) ) R YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 25 f : 11 , Φ : 110 , Ψ : 111 , ˜Ψ : 110 , R : 10 (cid:0) β ( − i) N (cid:1) (cid:88) ( R ) (cid:2) S ( R (cid:48) K N ) ⊗ (cid:3) · K N ⊗ · ∆( a K N ) · (cid:2) R (cid:48)(cid:48) K N ⊗ K N R (cid:3) = (cid:88) ( R ) , ( a ) S ( R (cid:48) K N ) K N a (cid:48) K N R (cid:48)(cid:48) K N ⊗ a (cid:48)(cid:48) K R = (cid:88) ( R ) , ( a ) S ( K N R (cid:48) K N ) a (cid:48) K N R (cid:48)(cid:48) K N ⊗ a (cid:48)(cid:48) R = (cid:88) ( R ) , ( a ) ( K N R K N ) (cid:46) a (cid:48) ⊗ a (cid:48)(cid:48) R . Combining the four sectors above results in the expression for ˆ µ L ( a ) given in (4.5).In order to determine ˆ∆ L we have to calculate(4.8) ˆ∆ L ( a ⊗ b ) = (cid:88) ( D ) S ( D ) bD S ( D ) aD where D = (id ⊗ id ⊗ ∆)(Φ) · ⊗ Φ − · ⊗ β ⊗ ⊗ , see I:(7.15) and I:(7.16). Note that weneed D only in sectors and which are ⊗ and ⊗ K N ⊗ K N ⊗ , respectively. Itis then immediate that (4.8) reduces to ˆ∆ L ( a ⊗ b ) = ba as claimed in (4.5).The Hopf pairing is given by (see I:(7.15) and I:(7.16))ˆ ω L = (cid:88) ( W ) S ( W ) W ⊗ S ( W ) W (4.9)with W = ( ⊗ α ⊗ ⊗ α ) · ( ⊗ Φ − ) · ( ⊗ M ⊗ ) · ( ⊗ Φ) · (id ⊗ id ⊗ ∆)(Φ − ). Recallfrom (3.12) and (4.7), that α = and that Φ is non-trivial only in the third tensor factor.Thus W simplifies to(4.10) W = ( ⊗ M ⊗ ) · (id ⊗ id ⊗ ∆)(Φ − ) . To compute ˆ ω L we only need the following four sectors of W :(4.11) W ⊗ M ⊗ ⊗ M ⊗ ⊗ M ⊗ ) · ( ⊗ ⊗ K N ⊗ K N )1111 ( ⊗ M ⊗ ) · ( ⊗ ⊗ K N ⊗ K N )From this it is straightforward to read off the expression for ˆ ω L in (4.5). (cid:3) Non-degeneracy of the monodromy matrix.
We compute the monodromy matrix(or double braiding) M = R R ∈ Q ⊗ Q . For this, we need the identities (for any n, m ∈ Z )(4.12) f ± ⊗ f ∓ ω − · ρ n,m · e n ⊗ e m = ( − m +1 ρ n,m · f ± ω − ⊗ f ∓ · e n ⊗ e m (note that ω − = ( e − i e ) K appears in different tensor factors) and(4.13) ( ρ m,n ) · ρ n,m · e n ⊗ e m = ( − nm K m ⊗ K n · e n ⊗ e m , where ( ρ m,n ) stands for the flip of ρ m,n from (3.14). Then, using the expression (3.15) forthe R -matrix the computation is straightforward:(4.14) M = R R = (cid:88) n,m =0 (cid:0) − β (cid:1) nm K m e n ⊗ K n e m × N (cid:89) j =1 (cid:110)(cid:0) ⊗ + 2( − m f + j ω − ⊗ f − j (cid:1)(cid:0) ⊗ − f − j ω − ⊗ f + j (cid:1)(cid:111) . Lemma 4.2.
The M -matrix is non-degenerate, i.e., it can be written as M = (cid:80) I ∈ X g I ⊗ f I ,where { g I } I ∈ X and { f I } I ∈ X are two bases of Q , for some indexing set X .Proof. Using the expression (4.14), we can rewrite M as M = (cid:88) n,m =0 1 (cid:88) s ,t =0 . . . (cid:88) s N ,t N =0 (cid:0) − β (cid:1) nm (cid:80) i ( s i + t i ) ( − (cid:80) i ( mt i + s i ) × K m e n N (cid:89) j =1 (˜ f + j ) t j (˜ f − j ) s j ⊗ K n e m N (cid:89) j =1 ( f − j ) t j ( f + j ) s j (4.15)where (cid:80) i stands for (cid:80) N i =1 and ˜ f ± j = f ± j ω − . This expression suggests to introduce two basesin Q , f I = K n e m N (cid:89) j =1 ( f − j ) t j ( f + j ) s j ,g I = (cid:0) − β (cid:1) nm (cid:80) i ( s i + t i ) ( − (cid:80) i ( mt i + s i ) K m e n N (cid:89) j =1 (˜ f + j ) t j (˜ f − j ) s j , (4.16)where the indices I run over the set(4.17) X = (cid:8) ( n, m, s , t , . . . , s N , t N ) (cid:12)(cid:12) n, m, s j , t j ∈ Z , j = 1 , . . . , N (cid:9) . We then have M = (cid:80) I ∈ X g I ⊗ f I and thus the statement of the lemma is proven. (cid:3) As a consequence of Lemma 4.2, we have that ˆ ω L is also non-degenerate (the antipode S and the conjugation with K map a basis to another basis) and this is a direct proof ofCorollary 3.8. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 27 Integrals and cointegrals for the coend.
Since
Rep Q is factorisable (Corollary 3.8), L = Q ∗ has a one-dimensional space of two-sided integrals Λ L : C → L (see Proposition I:4.10,due to [Ly1]). This space of integrals contains an element (unique up to a sign) normalisedsuch that ω L ◦ (Λ L ⊗ Λ L ) = id C . From Lemma I:2.3 (due to [Ke2]) we furthermore know thatthere is a two-sided cointegral Λ co L : L → such that Λ co L = ω L ◦ (Λ L ⊗ id L ).In this section we give Λ L and Λ co L for L = Q ∗ in the above normalisation. We refer toSection I:2.2 for our conventions for integrals and cointegrals in braided categories.By Proposition I:7.8 the integrals on Q from (3.47) define cointegrals on L = Q ∗ by(4.18) Λ co L = (cid:104)− , c (cid:105) ∈ Q ∗∗ . To describe the integrals of L , let ˜ B m = { K m , f +1 K m , f − K m , f +1 f − K m , . . . , f +1 f − · · · f +N f − N K m } , sothat B = (cid:83) m =0 ˜ B m is a basis in Q . We use the basis dual to B to define the element ˆΛ L ∈ Q ∗ as(4.19) ˆΛ L = ( − N νβ − N (cid:16) N (cid:89) i =1 f + i f − i (cid:17) ∗ , where ν ∈ C is the same constant as in the definition of the integral for Q in (3.47). In whatfollows, we will use several times the following simple lemma (whose proof we omit). Lemma 4.3.
Let { w I } I ∈ X , for some indexing set X , be a linearly independent subset of Q such that each w I is a monomial in the generators f ± i and K . Assume that a central element z ∈ Z ( Q ) can be written as (4.20) z = (cid:88) I ∈ X α I w I , α I ∈ C , α I (cid:54) = 0 . Then each w I commutes with K . Proposition 4.4.
The linear map Λ L : C → L given by Λ L (1) = ˆΛ L is a two-sided integralfor L . For ν ∈ {± } this integral satisfies (4.21) ω L ◦ (Λ L ⊗ Λ L ) = id C , Λ co L ◦ Λ L = id C . Proof.
We know that Λ co L in (4.18) is a cointegral for L . To show that Λ L is a two-sidedintegral, by Lemma I:2.3 it is enough to verify the identity Λ co L = ω L ◦ (Λ L ⊗ id L ). In thepresent setting, this is equivalent to (cid:104)− , c (cid:105) = ω L ( ˆΛ L ⊗ − ) or(4.22) c = (cid:88) ( ω L ) (ˆ ω L ) ˆΛ L (cid:0) (ˆ ω L ) (cid:1) = (cid:88) ( M ) ˆΛ L ( M ) S ( M ) , where for the last equality we used Lemma 4.3 (each non-zero term in the sum commuteswith K ).RHS of (4.22) = (cid:88) m,n =0 ( − β (cid:1) mn ( − N( m +1) = δ m, νβ − N (cid:122) (cid:125)(cid:124) (cid:123) ˆΛ L ( K m e n f +1 ω − f − ω − . . . f +N ω − f − N ω − ) S ( K n e m f − f +1 . . . f − N f +N )= ( − N νβ (cid:0) S ( e f − f +1 . . . f − N f +N ) + S ( e f − f +1 . . . f − N f +N K ) (cid:1) = 2 N νβ e f +1 f − . . . f +N f − N (1 + K ) = c . It remains to show that the normalisation condition holds. Indeed, we have ω L ( ˆΛ L ⊗ ˆΛ L ) = (cid:104) ˆΛ L ⊗ ˆΛ L , ˆ ω L (cid:105) (4.22) = ˆΛ L ( c )(4.23) = 2 N νβ ˆΛ L ( e f +1 f − . . . f +N f − N ) = ν = 1 . (cid:3) Internal characters and φ M . We first recall that the internal character of V ∈ Rep Q is the intertwiner from the trivial representation to the coend L given by (see [FSS, Sh1])(4.24) χ V = (cid:2) (cid:103) coev V −−−→ V ∗ ⊗ V ι V −→ L (cid:3) , where we follow conventions Section I:5.2. Via the isomorphism Hom Q ( , L ) → End( Id Rep Q )given in I:(5.22), the internal characters χ V correspond to natural endomorphisms φ V ofthe identity functor. For C = Rep Q , under the categorical modular S -transformation S C : End( Id C ) → End( Id C ) in I:(5.14) the images φ V of the internal characters are mapped tonatural endomorphisms S C ( φ V ) which are the “Hopf link operators” in I:(5.23). We also recallfrom Corollary I:5.5 and Theorem I:5.6 that the assignments [ V ] (cid:55)→ φ V and [ V ] (cid:55)→ S C ( φ V )are injective linear maps Gr C ( C ) → End( Id C ) (the second map is actually an algebra map).Both, φ V and S C ( φ V ) will be useful in the computation of the S -transformation on the centre Z ( Q ) in Section 5 and when comparing SL (2 , Z )-actions in Section 6.Let us identify χ V with their images χ V (1) ∈ Q ∗ . In I:(7.40), we expressed these linearforms via the traces(4.25) χ V ( − ) = Tr V ( κ · − ) , κ = u − v S ( β ) , where explicitly(4.26) κ = ( e − i β e ) K . The φ V ∈ End( Id Rep Q ) correspond to central elements φφφ V ∈ Z ( Q ), see Section I:7.6. Wenow compute the φφφ V for all simple modules V . Lemma 4.5.
We have (4.27) φφφ V = (cid:88) ( c ) c (cid:48) ⊗ χ V ( S ( c (cid:48)(cid:48) )) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 29 and, in particular, (4.28) φφφ X ± = ν N β ( K ± ) e (cid:89) i =1 f + i f − i , φφφ X ± = ± ν N+1 e ± . Proof.
Recall from Section I:7.6 that(4.29) φφφ V = (cid:88) ( F ) F χ V ( F ) , where(4.30) F = ε ( β ) (cid:88) (Ψ) , (Φ) , ( c ) Ψ c (cid:48) Φ ⊗ S (Ψ c (cid:48)(cid:48) Φ ) α Ψ Φ and Ψ = Φ − . It is straightforward to see that for Q we have(4.31) F = (cid:88) ( c ) c (cid:48) ⊗ S ( c (cid:48)(cid:48) ) , which together with (4.29) proves (4.27).Next we compute from (3.47) that∆( c ) = a (cid:0) ⊗ + K ⊗ K (cid:1)(cid:0) ⊗ + ∆( K ) (cid:1) (4.32) × N (cid:89) i =1 (cid:0) f + i f − i ⊗ + f + i ω − ⊗ f − i − f − i ω + ⊗ f + i + K ⊗ f + i f − i (cid:1) , where a = ν N − β .We note that for any x ∈ Q we get(4.33) Tr X ± i ( e j x ) = δ i,j Tr X ± i ( x ) , i, j ∈ { , } . Recall in Section 3.7 that X ± have the trivial action of f ± i . Therefore, in the traces over X ± only the basis elements K n , 0 ≤ n ≤
3, have non-zero contribution. Then for V = X ± andusing (4.25) together with (4.26) we get(4.34) φφφ X ± = (cid:88) ( c ) Tr X ± ( K c (cid:48)(cid:48) ) c (cid:48) = 2 a ( K ± ) e (cid:89) i =1 f + i f − i , where we used that only the components with c (cid:48)(cid:48) = K n contribute to the expression, and that S ( K n e ) = K n e . This gives the first expression in (4.28).For the computation of φφφ X ± we first recall that X ± are simple projective, as discussed inSection 3.7. It then follows from [GR3, Eq. (7.5)] that φφφ X ± = b ± e ± for some non-zero b ± ∈ C .To determine b ± it is enough to act on the highest-weight vector v ± ∈ X ± defined by (3.57).We then note that in the expression(4.35) φφφ X ± .v ± = − i β (cid:88) ( c ) Tr X ± (cid:0) K S ( c (cid:48)(cid:48) ) (cid:1) c (cid:48) .v ± only the term a ( ⊗ + K ⊗ K )( ⊗ + ∆( K )) (cid:0) K ⊗ (cid:81) N i =1 f + i f − i (cid:1) in ∆( c ) gives a non-zerocontribution: the other terms either have f − i in c (cid:48) , and therefore are zero on v ± , or have f + k without f − k in c (cid:48)(cid:48) and therefore zero in the trace. Within the trace Tr X ± ( − ), we replace e S ( (cid:81) N i =1 f + i f − i ) = e (cid:81) N i =1 f − i f + i . Therefore, we need to calculate the trace of the operator K n (cid:81) N i =1 f − i f + i . For any n , this operator in the basis v ± i from (3.58) is given by a diagonalmatrix with one-dimensional eigenspace of non-zero eigenvalue – spanned by v ± . We thushave Tr X ± (cid:0) K n (cid:81) N i =1 f − i f + i (cid:1) = ( ± i) n . Using this, a simple calculation finally gives(4.36) φφφ X ± = ± ν N+1 e ± , which agrees with the second expression in (4.28). (cid:3) The Hopf link operators S C ( φ V ) ∈ End( Id Rep Q ) correspond to central elements χχχ V ∈ Z ( Q )given by(4.37) χχχ V = (cid:88) (Φ) , (Ψ) , ( M ) Tr V (cid:16) κ S (cid:0) Ψ M Φ (cid:1) α Ψ Φ (cid:17) Ψ M Φ , see I:(7.42). We now compute the χχχ V for all simple Q -modules V . Lemma 4.6.
We have (4.38) χχχ V = (cid:88) ( M ) Tr V (cid:0) κ S ( M ) (cid:1) M and on simple V : (4.39) χχχ X ± = e ± e , χχχ X ± = ± β N K e (cid:89) j =1 f + j f − j + 2 N ( e +1 − e − ) . Proof.
It is straightforward to see that for Q the expression in (4.37) reduces to(4.40) χχχ V = (cid:88) ( M ) Tr V (cid:0) κ S ( M )) M . We first compute(4.41) χχχ X ± = (cid:88) ( M ) Tr X ± (cid:0) K S ( M )) M = e ± e , where we used that only the Cartan part of M in (4.14) contributes into the trace over X ± .Next, to compute χχχ X ± we study the action of these central elements on all the four projectivecovers, actually on generating vectors of the covers. Recall that they are v ± for X ± and e ± for P ± , recall (3.59). The point is that only very few terms in an expansion of χχχ X ± will contributeto the action. To start we act on v α , α = ± ,(4.42) χχχ X ± .v α = − i β (cid:88) ( M ) Tr X ± (cid:0) K S ( M )) M .v α = α N v α , YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 31 where again only the Cartan part of M (actually just the n = m = 1 term) contributed intothe non-zero action. Next, we have(4.43) χχχ X ± . e α = − i β (cid:88) ( M ) Tr X ± (cid:0) K S ( M )) M . e α = ± αβ N N (cid:89) j =1 f + j f − j e α , where we also noticed that only one term in the expansion of M :(4.44) M = ( − N N ( K e ⊗ e ) N (cid:89) j =1 f + j f − j ⊗ N (cid:89) j =1 f − j f + j + . . . contributes, otherwise the trace over X ± is zero. We used here(4.45) Tr X ± (cid:0) K (cid:89) k terms f + i f − i (cid:1) = ± i( − k δ N ,k . Combining (4.42) and (4.43), we finally get(4.46) χχχ X ± = ± β N K e (cid:89) j =1 f + j f − j + 2 N ( e +1 − e − ) . (cid:3) SL (2 , Z ) -action on the centre of Q In this section, we specialise the projective SL (2 , Z )-action on the centre Z of a generalfactorisable ribbon quasi-Hopf algebra obtained in Section I:8 to the symplectic fermion quasi-Hopf algebra Q . The result is summarised in Theorem 5.3.By Theorem I:8.1, in general the S - and T -transformations are given by the followinginvertible linear endomorphisms of Z , for z ∈ Z , S Z ( z ) = (cid:88) (Ψ) , (ˆ ω L ) Ψ β S (Ψ ) (ˆ ω L ) Ψ ˆΛ L (cid:16) ˆ∆ L (cid:0) (ˆ ω L ) ⊗ α z (cid:1)(cid:17) , (5.1) T Z ( z ) = v − z , (5.2)where Ψ = Φ − , the dual structure maps ˆ∆ L , ˆ ω L are defined by (4.4). We evaluate (5.1)for the quasi-Hopf algebra Q using the map ˆ∆ L and the explicit form of ˆ ω L computed inProposition 4.1:(5.3) S Z ( z ) = (cid:88) (ˆ ω L ) (ˆ ω L ) ˆΛ L (cid:0) z (ˆ ω L ) (cid:1) = (cid:88) ( M ) ˆΛ L ( M z ) S ( M ) , where we also used the fact that in the sum all non-zero summands commute with K , recallLemma 4.3 and that S Z ( z ) is central. We also recall that ˆΛ L is given in (4.19). Our aim is to give a decomposition of the SL (2 , Z )-action (5.2), (5.3) on Z = Z ( Q ). InProposition 3.15, we described a decomposition of the centre Z = Z ⊕ Z and a basis in it.In what follows, we will need a slightly different decomposition:(5.4) Z ( Q ) = Z P ⊕ Z Λ with Z P := span C (cid:8) φφφ P +0 , φφφ X +1 , φφφ X − (cid:9) , (5.5) Z Λ := span C (cid:110) e k (cid:89) j =1 f ε j i j | k ≤ N / , i j ∈ { , . . . , N } , ε j = ± (cid:111) , where Z P is spanned by the internal characters φφφ V for V projective, while Z Λ is the centreof Λ from (3.5). The central elements φφφ X ± are given in (4.28). For φφφ P +0 , recall that themap V (cid:55)→ φφφ V factors through the Grothendieck ring (as reviewed in Section 4.4 and see alsoSection I:5.2) and thus using the Cartan matrix in (3.61) we can write(5.6) φφφ P +0 = 2 − φφφ X +0 + 2 − φφφ X − = ν β K e (cid:89) i =1 f + i f − i . This shows that (5.4) is indeed a decomposition of the centre Z ( Q ) as computed in Proposi-tion 3.15.From [GR3, Prop. 7.1 & Cor. 8.5] we know that Z P is an invariant subspace of Z ( Q ) for the SL (2 , Z )-action. The next lemma gives the action of S Z and T Z on Z P . Lemma 5.1.
The restriction of the linear maps S Z , T Z from (5.1) and (5.2) to Z P in thebasis (5.5) is given by the matrices (5.7) S Z P = ν − N − − N N − / / − N − / / , T Z P = β −
00 0 − β − . Proof.
To compute the S -transformation on Z P we use the relation stated in Corollary I:8.2,(5.8) S Z ( φφφ V ) = χχχ V , where the central elements χχχ V are given by (4.38) and were computed for simple V inLemma 4.6. Applying this formula for V = X ± and using (4.39) together with (5.6), wecan write(5.9) S Z ( φφφ X ± ) = χχχ X ± = ν (cid:0) ± − N φφφ P +0 + ( φφφ X +1 + φφφ X − ) (cid:1) . This gives the second and third column in (5.7).As discussed in Section 4.4 (see also Section I:5.2), the map V (cid:55)→ χχχ V factors throughGr C ( Rep Q ) and therefore we can write χχχ P +0 = 2 − χχχ X +0 + 2 − χχχ X − . Using (4.39) together YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 33 with the expression for φφφ X ± in (4.28), recall that e = e +1 + e − , the relation (5.8) for V = P +0 is(5.10) S Z ( φφφ P +0 ) = χχχ P +0 = ν N − ( φφφ X +1 − φφφ X − ) . This finally gives the first column in (5.7) and it finishes our calculation of S Z P .For the T -transformation (5.2) we use the expression (3.17) for v − . This immediately givesthe diagonal matrix for T Z with the entries { , β − , − β − } in the basis (cid:8) φφφ P +0 , φφφ X +1 , φφφ X − (cid:9) . (cid:3) Let U i ⊂ Q be the subalgebra U i := span C { e , f − i e , f + i e , f + i f − i e } , with 1 ≤ i ≤ N.Consider the injective linear map ϑ : U ⊗ · · · ⊗ U N → Q given by(5.11) ϑ ( a ⊗ · · · ⊗ a N ) := a · · · a N . Note that this is not an algebra map. Write U + ⊂ U ⊗ · · · ⊗ U N for the subspace spannedby all homogeneous vectors with an even overall number of f ± i ’s. In other words, U + is theeigenspace of ( K ( − ) K − ) ⊗ N of eigenvalue +1. The direct summand Z Λ of the centre in (5.5)is the image of U + ,(5.12) Z Λ = ϑ ( U + ) . In particular, ϑ | U + : U + → Z Λ is a bijection and below we use ϑ − on elements from Z Λ . Withthe help of ϑ we will now describe the S - and T -transformations on Z Λ as a tensor productof linear maps S iZ Λ , T iZ Λ : U i → U i , for i = 1 , . . . , N. Lemma 5.2. S Z and T Z from (5.1) - (5.2) map Z Λ to itself. The restrictions of S Z , T Z to Z Λ are given by S Z (cid:12)(cid:12) Z Λ = ϑ ◦ (cid:16) νβ · (cid:0) S Z Λ ⊗ · · · ⊗ S N Z Λ (cid:1)(cid:12)(cid:12) U + (cid:17) ◦ ϑ − , (5.13) T Z (cid:12)(cid:12) Z Λ = ϑ ◦ (cid:0) T Z Λ ⊗ · · · ⊗ T N Z Λ (cid:1)(cid:12)(cid:12) U + ◦ ϑ − , (5.14) where the individual maps S iZ Λ , T iZ Λ are given in the basis { e , f − i e , f + i e , f + i f − i e } of U i by thematrices (5.15) S iZ Λ = − − , T iZ Λ = . In particular, the restriction of S Z to Z Λ is the identity, and T Z has Jordan blocks of maximumrank N + 1 .Proof.
We first recall that the S -transformation S Z for Q was given in (5.3). Using the explicitexpression of M from (4.15), S Z on a central element z becomes S Z ( z ) = (cid:88) n,m,t ,s ,...,t N ,s N =0 ( − β ) nm (cid:80) i ( s i + t i ) ( − (cid:80) i ( mt i + s i ) ˆΛ L (cid:0) z e n K m N (cid:89) j =1 (˜ f + j ) t j (˜ f − j ) s j (cid:1) (5.16) × e m S (cid:0) K n N (cid:89) j =1 ( f − j ) t j ( f + j ) s j (cid:1) . Our aim is to calculate S Z ( z ) on Z Λ . Recall the basis elements of Z Λ in (5.5), they areall in Z e . Therefore the terms with n = 1 are zero in the sum (5.16). Then, we recall ˆΛ L from (4.19) and as a basis element of Z Λ has even number of f ± i the sum (cid:80) j ( t j + s j ) shouldbe an even number too, otherwise the value of ˆΛ L on the corresponding element is zero. As (cid:80) i ( t i + s i ) is even, the factors of K in e ˜ f ± j = e f ± j K cancel (resulting in signs), and so theterm ˆΛ L ( · · · ) in (5.16) is zero for m = 1. Combining all these observations, for z ∈ Z Λ wehave(5.17) S Z ( z ) = (cid:88) t ,s ,...,t N ,s N =0 (cid:80) i ( s i + t i ) ( − (cid:80) i ( s i +( t i + s i ) / ˆΛ L (cid:16) z e (cid:89) j =1 ( f + j ) t j ( f − j ) s j (cid:17) e S (cid:16) N (cid:89) j =1 ( f − j ) t j ( f + j ) s j (cid:17) , where the sign ( − (cid:80) i ( t i + s i ) / arises when cancelling the factors of K contained in ˜ f ± j (it helpsto rewrite the product as in (5.18) below to see this). It is clear that we can write any basiselement of Z Λ from (5.5) in the form(5.18) z = (cid:16) k (cid:89) j =1 f ε j i j (cid:17)(cid:16) M (cid:89) j =1 f + k j f − k j (cid:17) e , for some 1 ≤ k ≤ N / M, i j , k j ∈ { , . . . , N } such that i < · · · < i k , and ε j ∈ {±} . Theessential part of the calculation is to analyse the coefficients ˆΛ L (cid:0) z e (cid:81) j ( f + j ) t j ( f − j ) s j (cid:1) in thesum (5.17). For a given z as in (5.18), the 2N-tuple { t , s , . . . , t N , s N } here is unique because z (cid:81) N j =1 ( f + j ) t j ( f − j ) s j should be proportional to the support of ˆΛ L , recall (4.19). We thus getthat ˆΛ L ( · · · ) is non-zero iff, for the same index choice as in (5.18),(5.19) N (cid:89) j =1 ( f + j ) t j ( f − j ) s j = (cid:16) k (cid:89) j =1 f − ε j i j (cid:17) · (cid:16) N − M − k (cid:89) n =1 f + l n f − l n (cid:17) , where the l n take the N − M − k values in { , . . . , N } which are different from all i j , k j in(5.18). Note that this set of l n ’s is unique. After reordering the generators f ± j , the elementwithin ˆΛ L ( . . . ) is ( − (cid:80) kj =1 ε j e (cid:81) N j =1 f + j f − j . We also note that the 2N tuple correspondingto (5.19) gives(5.20) (cid:88) i ( t i + s i ) = 2(N − M − k ) , (cid:88) i s i = N − M − k + k (cid:88) j =1 ( ε j + 1) . To get the sign, first we write (cid:81) kj =1 f ε j i j · (cid:81) kj =1 f − ε j i j = ( − k (cid:81) kj =1 f ε j i j f − ε j i j , where f − ε i is commuted pastan odd number of fermions, so we get −
1, then f − ε i is commuted past an even number of fermions, so weget +1, etc. – this explains ( − k . Then to reorder f ε j i j f − ε j i j we do nothing if ε j = +1 and get − − (cid:80) kj =1 1 − εj . This together with ( − k gives the sign ( − (cid:80) kj =1 ε j . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 35 We thus get for (5.17) the following expression, for z as in (5.18), S Z ( z ) = 2 N − M − k ) ˆΛ L (cid:0) e (cid:89) j =1 f + j f − j (cid:1) e S (cid:32)(cid:16) k (cid:89) j =1 f ε j i j (cid:17) · (cid:16) N − M − k (cid:89) n =1 f − l n f + l n (cid:17)(cid:33) (5.21) = ( − M νβ N − M + k ) e (cid:16) k (cid:89) j =1 f ε j i j (cid:17) · (cid:16) N − M − k (cid:89) n =1 f + l n f − l n (cid:17) , where the l n are as in (5.19), and where we used our assumption that i < · · · < i k and thushad to reorder f ε j i j ’s after the application of the antipode S . It is now clear that the image of Z Λ under S Z is Z Λ itself. We also note by a direct calculation that S Z ( z ) = z .Our aim is now to check the expression in (5.13) together with (5.15). Recall that thebijection ϑ was defined in (5.11). Then having z as in (5.18), its image under ϑ − has theform (of course here for a particular choice of i j , k j )(5.22) ϑ − ( z ) = e ⊗ . . . ⊗ e ⊗ e f ε i ⊗ . . . ⊗ e f + k f − k ⊗ . . . ⊗ e . The action of each tensor component S iZ Λ , as defined in (5.15), replaces e in the i ’th factorby 2 f + i f − i e and f + i f − i e by − e , while does nothing to the factors e f ε i . With this, it is nowstraightforward to see the equality in (5.13) using the final expression in (5.21).For the T -transformation on Z Λ , we use the expression (3.17) that can be written as(5.23) v − e = N (cid:89) k =1 ( + 2 f + k f − k ) e . It is therefore clear that T Z acts on Z Λ . Recall that the injective linear map (5.11) restrictsto an isomorphism ϑ (cid:12)(cid:12) U + : U + → Z Λ . Note that(5.24) ϑ − ( v − e ) = ( e + 2 f +1 f − e ) ⊗ ( e + 2 f +2 f − e ) ⊗ . . . ⊗ ( e + 2 f +N f − N e ) . It is easy to check that, even though ϑ (cid:12)(cid:12) U + : U + → Z Λ is itself not an algebra map, whenrestricted to the subalgebra of Q generated by f + i f − i , 1 ≤ i ≤ N, it does become an algebramap. In particular for v − e we have ϑ − ( v − z ) = ϑ − ( v − e ) · ϑ − ( z ) for all z ∈ Z Λ .Thus T Z | Z Λ acts on U + by multiplication with (5.24). From this, we immediately get (5.14)with (5.15).Finally, from the T Z ( e ) we see that T Z has Jordan blocks of maximum rank N + 1. Indeed,recall that Jordan blocks of a given dimension behave under tensoring like irreducible sl (2)-modules of that same dimension. The matrix T iZ Λ in (5.15) has one rank-2 Jordan block andtwo rank-1 blocks (use the basis { f + i f − i e , e , f − i e , f + i e } ). Thus the maximal rank Jordancell in the tensor product arises from from tensoring the N rank-2 blocks. This correspondsto the tensor product of N fundamental sl (2)-modules, which decomposes into a direct sumwith the largest irreducible summand being of dimension N + 1. (cid:3) As a corollary of the two last lemmas we have the following theorem.
Theorem 5.3.
The projective SL (2 , Z ) -action on the centre Z of Q given by the linear maps S Z and T Z from (5.1) - (5.2) is (5.25) S Z = S Z P ⊕ S Z Λ , T Z = T Z P ⊕ T Z Λ , with the constituent maps as defined in Lemmas 5.1 and 5.2. We have S Z = id Z and T Z hasJordan blocks of ranks up to and including N + 1 . Equivalence of the two projective SL (2 , Z ) -actions In this final section, we review the modular-group action on the symplectic fermion pseudo-trace functions and compare it with the SL (2 , Z ) action computed in the previous sectionon the centre of Q . The main result of this paper is that these two actions are projectivelyequivalent (Theorem 6.11).6.1. Modular properties of symplectic fermion pseudo-trace functions.
Here, wereview the computation of the symplectic fermion pseudo-trace functions and of their modularproperties carried out in [GR2].We define two affine Lie super-algebras, (cid:98) h and (cid:98) h tw , in terms of a symplectic C -vector space h of dimension 2N with symplectic form ( − , − ). The underlying super-vector spaces are(6.1) (cid:98) h = h ⊗ C [ t ± ] ⊕ C ˆ k , (cid:98) h tw = h ⊗ t C [ t ± ] ⊕ C ˆ k , where t ± and ˆ k are parity-even, and h is parity-odd. For u ∈ h and m ∈ Z (resp. m ∈ Z + ),abbreviate u m := u ⊗ t m . The Lie super-bracket is given by taking ˆ k central and setting, for u, v ∈ h and m, n ∈ Z (resp. m, n ∈ Z + ),(6.2) [ u m , v n ] = ( u, v ) m δ m + n, ˆ k . By convention for the bracket of a Lie super-algebra, this is an anti-commutator as u m , v n are parity odd. We refer to [Ru] for more on (cid:98) h (tw) and its representations.To make the connection to Section 2, choose a basis { a i , b i | i = 1 , . . . , N } of h which satisfies(6.3) ( a j , b k ) = i πδ j,k . With this basis as generators, we have Λ = Λ( h ), that is, the 2 -dimensional Graßmannalgebra defined in Section 2.1 is the exterior algebra of h . The reason to put the factor of i π in the above normalisation is that we will also work with a rescaled basis of h , namely(6.4) α = a , α = π b , . . . , α − = a N , α = π b N . The α i are a symplectic basis in the sense that ( α , α ) = 1, etc. The basis { α i } is a naturalchoice when working with the affine Lie algebra (cid:98) h (tw) , and it is used e.g. in [Ru] and [GR2,Sec. 6]. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 37 For later use we recall the action of the Virasoro zero-mode L on highest weight modulesof (cid:98) h (tw) from [Ru, Rem. 2.5 & 2.7]: (cid:98) h -module : L = N (cid:88) k ∈ ( α k α k − − α k − α k ) + H , (6.5) (cid:98) h tw -module : L = − N8 + H tw . where H = (cid:88) m ∈ Z > N (cid:88) k ∈ (cid:0) α k − m α k − m − α k − − m α km (cid:1) , (6.6) H tw = (cid:88) m ∈ Z ≥ + N (cid:88) k ∈ (cid:0) α k − m α k − m − α k − − m α km (cid:1) . One verifies that H (tw) acts as grading operator on a highest-weight module of (cid:98) h (tw) , assigninggrade zero to the space of ground states.The braiding on SF was originally computed in [Ru] with respect to the basis in (6.4) and,for example, for X, Y ∈ SF the result was(6.7) c X,Y = τ s . v .X,Y ◦ exp (cid:16) − i π N (cid:88) k =1 ( α k ⊗ α k − − α k − ⊗ α k ) (cid:17) , see [Ru, Eqn. (6.1)]. This formula, and many others, have factors of i π , which can be absorbedinto a suitably rescaled copairing. Indeed, with respect to the basis { a k , b k } , the abovebraiding produces the one presented in (2.14).Given a module M ∈ SF = Rep s . v . Λ , we can use induction to construct an (cid:98) h -module (cid:99) M as follows. We take u m for u ∈ h , m >
0, to act as zero on M , ˆ k to act by 1, and the zeromodes ( a i ) and ( b i ) to act as a i and b i , respectively. Then(6.8) (cid:99) M := U ( (cid:98) h ) ⊗ U ( (cid:98) h ≥ ⊕ C ˆ k ) M , where U ( − ) denotes the universal enveloping algebra (in super-vector spaces) of a Lie super-algebra, and (cid:98) h ≥ is the subalgebra spanned by non-negative modes. Similarly, for V ∈ SF = Svect , we consider the induced (cid:98) h tw -module(6.9) (cid:98) V := U ( (cid:98) h tw ) ⊗ U (( (cid:98) h tw ) > ⊕ C ˆ k ) V , which is defined as above, except that in (cid:98) h tw there are no zero modes to take care of.The symplectic fermion vertex operator super-algebra V is defined in [Ab], see also [DR3,Sec. 3.1] which uses the present notation (except that V is denoted by V ( h ) there). Theunderlying super-vector space is V = (cid:98) , for ∈ SF the tensor unit as given in (2.4). Write V ev for the parity-even subspace of V – this is a vertex operator algebra. It is shown in [Ab] (and stated this way in [GR2, Cor. 6.4]) that the functor of first inducinga (cid:98) h (tw) -module and then restricting to its even subspace defines a V ev -module: Proposition 6.1. (cid:0) (cid:98) − (cid:1) ev is a faithful C -linear functor SF →
Rep V ev . Next we compute the image of the endomorphisms of the minimal projective generator G SF of SF given in (3.63). This requires a bit of preparation. Define(6.10) G := (cid:98) Λ ⊕ (cid:98) T .
Note that we do not take the even part. Instead we consider the super-vector space on theRHS simply as a vector space. In terms of the functor ( (cid:98) − ) ev this means(6.11) G ∼ = (cid:0) (cid:98) G SF (cid:1) ev . We will need this isomorphism explicitly. We first describe the isomorphism(6.12) : (cid:98) Λ ∼ −→ (cid:92) ( Λ ⊗ C | ) ev of V ev -modules. Recall by (6.8) that (cid:98) Λ = U ( (cid:98) h ) ⊗ U ( (cid:98) h ≥ ⊕ C ˆ k ) Λ . We then define for homogeneous u ∈ U ( (cid:98) h ) and f ∈ Λ :(6.13) : u ⊗ f (cid:55)→ u ⊗ f ⊗ (cid:40) (1 , , if u ⊗ f even , (0 , , if u ⊗ f odd , with the inverse map given by (note that v has to be homogeneous too)(6.14) − : u ⊗ f ⊗ v (cid:55)→ v | u | + | f | · u ⊗ f , v ∈ C | . The isomorphism tw : (cid:98) T ∼ −→ (cid:92) ( T ⊗ C | ) ev is defined analogously. Lemma 6.2.
The image E under (cid:0) (cid:98) − (cid:1) ev of End SF ( G SF ) in End V ev ( G ) is generated by • the action of the zero modes α i , i = 1 , . . . , , • (cid:99) id T , the idempotent corresponding to (cid:98) T , i.e. the identity map on (cid:98) T and zero on (cid:98) Λ , • the parity involution ω G on G .Proof. We first refer to Corollary 3.18 where End SF ( G SF ) was described. The functor (cid:0) (cid:98) − (cid:1) ev maps a morphism g to id U ( (cid:98) h ) ⊗ g . We first calculate the image (under the functor) of actionof a ∈ h from (3.65):(6.15) u ⊗ f ⊗ v (cid:55)→ u ⊗ f · a ⊗ Π( v ) = ( − | u | + | f | a · u ⊗ f ⊗ Π( v ) , were the latter equality is by the definition of the induced representation (6.8). Then com-posing with the isomorphism − from (6.14) we get the corresponding endomorphism on (cid:98) Λ :(6.16) v | u | + | f | u ⊗ f (cid:55)→ v | a | + | u | + | f | +1 ( − | u | + | f | a · u ⊗ f = v | u | + | f | a · ω G ( u ⊗ f ) . We finally note that the image of the κ generator is given by ω G . Therefore the image ofEnd SF ( G SF ) is generated by ω G , by a ◦ ω G for a ∈ h (or, equivalently, by a ), and by theidempotent for (cid:98) T . (cid:3) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 39 We note that the algebra from Lemma 6.2 was also introduced in [AN] in order to describepseudo-trace functions for V ev . Let us briefly review this construction. For a k -algebra A , thespace of central forms on A is defined as(6.17) C ( A ) = (cid:8) ϕ : A → k (cid:12)(cid:12) ϕ ( ab ) = ϕ ( ba ) for all a, b ∈ A (cid:9) . By definition, an element ϕ ∈ C ( A ) induces a symmetric pairing ( a, b ) → ϕ ( ab ) on A .Recall the algebra E ⊂
End V ev ( G ) from Lemma 6.2 and note that G is an E -module. Toeach ϕ ∈ C ( E ) one can assign a pseudo-trace function [AN](6.18) ξ ϕ G : V ev × H −→ C . Explicitly, ξ ϕ G is written as, for v ∈ V ev , τ ∈ H ,(6.19) ξ ϕ G ( v, τ ) = t ϕ G (cid:0) o ( v ) e π i τ ( L − c/ (cid:1) , where t ϕ G : G → C is a Hattori-Stallings trace, o : V ev → End C ( G ) is the zero mode linear mapand c = −
2N is the central charge of V ev . We refer to [AN] and [GR2, Sec. 4] for details (seealso [GR2, Prop. 5.2] on how the above definition relates to [AN]).The pseudo-trace functions are generalisations of the characters of VOA modules. Indeed,for each simple V ev -module M there is a unique ϕ M ∈ C ( E ) such that (6.20) ξ ϕ M G ( v, τ ) = Tr M (cid:0) o ( v ) e π i τ ( L − c/ (cid:1) , i.e. the pseudo-trace function for ϕ M equals the usual trace over M . In particular, there is aunique ϕ V ev corresponding to the vacuum character. Remark 6.3.
The general background for the above definition is the theory – due to [Zh] –of torus one-point functions C ( V ) for a vertex operator algebra V . Torus one-point functionsclose under modular transformations, and for a certain class of vertex operator algebraswith semisimple representation theory, the space of torus one-point functions is spanned bycharacters [Zh]. These results were extended in [Mi] to certain vertex operator algebras withnon-semisimple representation theory. It is proved in [Mi] that in this case the torus one-pointfunctions are spanned by a variant of pseudo-trace functions. Pseudo-trace functions in thesense of [Mi] are hard to work with, and we use here an easier notion introduced in [AN].However, it is not known if those simpler functions span the torus one-point functions (evenin the case of V ev ).Pseudo-trace functions provide a linear map ξ G : C ( E ) → C ( V ev ). This map is injective[AN, Thm. 6.3.2]. We need in addition: Conjecture 6.4 ([AN, Conj. 6.3.5], [GR2, Conj. 5.8]) . ξ G : C ( E ) → C ( V ev ) is an isomor-phism. Existence of ϕ M is proven in [GR2, Prop. 5.5] (which also provides the simple expression ϕ M = Tr ˜ M ( − )where ˜ M = Hom V ev ( G , M ) is the corresponding E -module). Uniqueness follows from injectivity of ξ G asproven in [AN] for the choice of G given in (6.10). Under the above conjecture, the following statement is verified for V ev in [GR2, Sec. 6] andLemma 6.6 below by explicit calculation. Proposition 6.5.
Assuming Conjecture 6.4, we have • the action of the modular S - and T -transformation defines linear isomorphisms (6.21) S V ev , T V ev : C ( E ) → C ( E ) ; • the element δ := S V ev ( ϕ V ev ) ∈ C ( E ) defines a non-degenerate pairing on E via ( f, g ) (cid:55)→ δ ( f ◦ g ) ; the assignment ˆ δ : Z ( E ) → C ( E ) , z (cid:55)→ δ ( z · ( − )) therefore is a linearisomorphism. In [GR2, Sec. 5] this statement is conjectured to hold in general for the class of vertexoperator algebras considered there. As a consequence of Proposition 6.5, we obtain uniquelinear isomorphisms(6.22) S Z = ˆ δ − ◦ S V ev ◦ ˆ δ , T Z = ˆ δ − ◦ T V ev ◦ ˆ δ . Next we give the explicit results for ˆ δ ( ϕ M ) and S Z , T Z . Setting ϕ X := ϕ ( (cid:98) X ) ev , for X ∈ SF ,the calculation in [GR2, Sec. 6] can be expressed asˆ δ − ( ϕ ) = (2 π ) N α · · · α ( ω G + 1) , ˆ δ − ( ϕ T ) = 2 N (cid:99) id T ( ω G + 1) , (6.23) ˆ δ − ( ϕ Π ) = (2 π ) N α · · · α ( ω G − , ˆ δ − ( ϕ Π T ) = 2 N (cid:99) id T ( ω G − . The centre of E is computed in [AN] to be(6.24) Z ( E ) = Z P ( E ) ⊕ Z Λ ( E ) , where Z Λ ( E ) is spanned by even monomials in the zero modes α i , and Z P ( E ) has the basis(6.25) Z P ( E ) = span C (cid:8) ˆ δ − ( ϕ + ϕ Π ) , ˆ δ − ( ϕ T ) , ˆ δ − ( ϕ Π T ) (cid:9) . The form of the centre also follows from Proposition 3.15 via the equivalence in Theorem 3.6.The linear map S Z from (6.22) is computed explicitly in [GR2, Cor. 6.9]. For T Z we givethe explicit expression in the next lemma. Lemma 6.6.
For z ∈ Z Λ ( E ) we have (6.26) T Z ( z ) = e π iN / · e π i (cid:80) N k =1 α k α k − z , while on Z P ( E ) in the basis (6.25) we get the matrix representation (6.27) T Z (cid:12)(cid:12) Z P ( E ) = e π iN / · e − π iN /
00 0 − e − π iN / . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 41 Proof.
We start by computing T V ev . By definition, for all ϕ ∈ C ( E ),(6.28) ξ ϕ G ( v, τ + 1) = ξ T V ev ( ϕ ) G ( v, τ ) . Recall the direct sum decomposition (6.10) of G . There are no V ev -intertwiners between (cid:98) Λ and (cid:98) T , nor between the different parity subspaces of (cid:98) T , and so (as we have already seen inthe explicit description above)(6.29) E = E ⊕ E +1 ⊕ E − , E = End V ev ( (cid:98) Λ ) , E +1 = C id ( (cid:98) T ) ev , E − = C id ( (cid:98) T ) odd . From the definition of the pseudo-trace functions in (6.19) and from the explicit form in [GR2,Eqn. (6.16) & App. A] it is a straightforward exercise to show that(6.30) ξ ϕ G ( v, τ + 1) = t ϕ G (cid:0) e π i( L − c/ o ( v ) e π i τ ( L − c/ (cid:1) = t ϕ (cid:48) G (cid:0) o ( v ) e π i τ ( L − c/ (cid:1) , where we used that the zero mode o ( v ) commutes with L and set(6.31) ϕ (cid:48) ( f ) = e π i( L − c/ f ; f ∈ E e π i( − N / − c/ f ; f ∈ E +1 − e π i( − N / − c/ f ; f ∈ E − and, for f ∈ E ,(6.32) e π i( L − c/ f = exp (cid:110) π i (cid:16) N (cid:88) k =1 (cid:0) α k α k − − α k − α k (cid:1) + N12 (cid:17)(cid:111) f .
The above computation makes use of the explicit form of L as given in (6.5), and of the factthat H has eigenvalues in Z ≥ , and so vanish in exp(2 π i( · · · )). On the other hand, H tw haseigenvalues in Z ≥ on the even part of an (cid:98) h tw -module, and in Z ≥ + on the odd part. (cid:3) Similar to the construction in (5.11) we let W i ⊂ E be the subalgebra with basis(6.33) W i = span C (cid:8) id E , α i − , α i , α i α i − (cid:9) , and consider the linear map (cid:36) : W ⊗ · · · ⊗ W N → E given by(6.34) (cid:36) ( f ⊗ · · · ⊗ f N ) := f · · · f N . Let W + ⊂ W ⊗ · · · ⊗ W N be the subspace spanned by products with an even total numberof α k ’s. The map (cid:36) restricts to an isomorphism (cid:36) : W + → Z Λ ( E ). Let σ k , τ k : W k → W k bethe linear maps whose matrix representations in the basis (6.33) are(6.35) σ k = − π ) − − i 0 00 0 − i 0 − π , τ k = π i 0 0 1 . Theorem 6.7.
Assume Conjecture 6.4 holds. The linear maps S Z , T Z : Z ( E ) → Z ( E ) pre-serve the direct sum decomposition (6.24) : (6.36) S Z = S Z P ⊕ S Z Λ , T Z = T Z P ⊕ T Z Λ . The linear endomorphisms S Z P , T Z P of Z P ( E ) are given by the matrices, with respect to thebasis (6.25) , (6.37) S Z P = N − N − N − − − − − N − − − , T Z P = e π iN / · e − π iN /
00 0 − e − π iN / . The linear endomorphisms S Z Λ , T Z Λ of Z Λ ( E ) are given by S Z Λ = (cid:36) ◦ (cid:16)(cid:0) σ ⊗ · · · ⊗ σ N (cid:1)(cid:12)(cid:12) W + (cid:17) ◦ (cid:36) − , (6.38) T Z Λ = e π iN / · (cid:36) ◦ (cid:16)(cid:0) τ ⊗ · · · ⊗ τ N (cid:1)(cid:12)(cid:12) W + (cid:17) ◦ (cid:36) − . Proof.
The expression for S Z is computed in [GR2, Cor. 6.9]. The expression for T Z is imme-diate from Lemma 6.6. (cid:3) We conclude this section by a conjectural refinement of the C -linear inclusion in Proposi-tion 6.1 to a ribbon equivalence which we need in the next section. For details on how todefine the necessary coherence isomorphisms we refer to [DR3, Conj. 7.4], and for the specificvalue of β to [Ru, Thm. 6.4]. Conjecture 6.8 ([DR3, Conj. 7.4]) . For the choice (6.39) β = e − i π N / , (cid:0) (cid:98) − (cid:1) ev is an equivalence of C -linear ribbon categories. Comparison of SL (2 , Z ) -actions. To compare the results in Theorems 5.3 and 6.7 wefirst need to correct a mismatch in conventions for the ribbon twist and then relate Z ( Q )from (3.53) to Z ( E ) from (6.24).We start by remarking that in [Ly1, Ly2] – which we use to compute the SL (2 , Z )-actionon Z ( Q ) – the convention is that the T generator acts by composing with the ribbon twist θ .However, the modular T -transformation acts by composition with θ − . Below we will showthat the agreement with the categorical T -transformation can be achieved by changing thevalue of β in Conjecture 6.8 to its inverse. The reason for this mismatch is that by the convention chosen in [Ru], following e.g. [FRS], the ribbontwist acts by e − π i L while for Rep V ev the modular T -action is (up to a constant) given by e π i L . Oneof course could redefine the Lyubashenko’s SL (2 , Z )-action, but we find it more convenient to adapt theconventions from [Ru] to the present context. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 43 According to Conjecture 6.8, (cid:0)
Rep V ev , θ = e − π i L (cid:1) ∼ = SF (cid:0) N , β = e − i π N / (cid:1) as ribboncategories. This is equivalent to(6.40) (cid:0) Rep V ev , θ = e π i L (cid:1) (cid:39) SF (cid:0) N , β = e − i π N / (cid:1) rev . Here, given a ribbon category C , C rev denotes the reversed ribbon category , where braidingand twist are replaced by their inverses. On the LHS of (6.40) we now have the conventionfor θ that matches our quasi-Hopf computation of the T -transformation. To reformulate theRHS, we need the following lemma. Lemma 6.9.
For all N , β we have SF (N , β ) rev (cid:39) SF (N , β − ) as ribbon categories.Proof. For the sake of the proof we will write SF (N , β ; Λ co Λ ( β ) , C ), where Λ co Λ ( β ) is the coin-tegral in (2.12) and C refers to the copairing in (2.9). We stress the β -dependence in thenotation Λ co Λ ( β ) as our normalisation convention for the cointegral is β -dependent. By [DR1,Prop. 4.12] there is a ribbon equivalence(6.41) SF (N , β ; Λ co Λ ( β ) , C ) rev (cid:39) SF (N , β − ; Λ co Λ ( β ) , − C ) , whose underlying functor is the identity (with non-trivial coherence isomorphisms). In [DR1,Prop. 4.12] the cointegral stays the same, hence Λ co Λ ( β ) appears also on the RHS instead ofΛ co Λ ( β − ).It remains to show that on the RHS of (6.41) one can replace Λ co Λ ( β ) by Λ co Λ ( β − ) and − C by C . Let ϕ : Λ → Λ be the Hopf algebra isomorphism given on the generators in (2.2) as ϕ ( a i ) = a i , ϕ ( b i ) = − b i , i = 1 , . . . , N. The isomorphism ϕ satisfies(6.42) Λ co Λ ( β ) ◦ ϕ = ( − N Λ co Λ ( β ) = Λ co Λ ( β − ) , ( ϕ ⊗ ϕ )( C ) = − C .
We can now apply [DR1, Prop. 4.11] to conclude that(6.43) SF (N , β − ; Λ co Λ ( β ) , − C ) (cid:39) SF (N , β − ; Λ co Λ ( β − ) , C )as ribbon categories. The underlying functor of this equivalence is the identity functor in SF , and it is given by ϕ ∗ , the pullback of representations along ϕ , on SF . Combining(6.41) and (6.43) proves the statement of the lemma. (cid:3) Remark 6.10.
Thanks to Lemma 6.9 and Theorem 3.6, the equivalence in Conjecture 6.8can be restated as(6.44) (cid:0)
Rep V ev , θ = e π i L (cid:1) (cid:39) SF (cid:0) N , β = e i π N / (cid:1) (cid:39) Rep Q (cid:0) N , β = e i π N / (cid:1) . More generally, the ribbon category SF is constructed from a symplectic vector space h , a cointegraland a choice of β , see [DR1, Sec. 5.2] for details. C is the copairing for the symplectic form. Including thecointegral and the copairing in the notation makes these choices explicit. Denote by J : Rep Q (N , β ) → SF (N , β ) the ribbon equivalence from Proposition 3.6 andconstructed in Appendices A and B, see also the inverse equivalence in Section B.8. We needto compute the following chain of algebra isomorphisms for β = e i π N / :(6.45) Υ : Z ( Q ) (1) −→ End( Id Rep Q ) (2) −→ End( Id SF ( β ) ) (3) −→ End( Id SF ( β − ) rev ) (4) −→ Z ( E ) , and use this to compare the results of Theorems 5.3 and 6.7. Let z ∈ Z ( Q ). Arrow (1) issimply given by acting with z on a module. Arrow (2) maps η to the unique η (cid:48) such that(6.46) η (cid:48)J ( X ) = J ( η X ) for all X ∈ Rep Q . Arrow (3) is described in Lemma 6.9 and arrow (4) maps ψ to ( (cid:99) ψ G ) ev .We compute the composition of these maps separately for Z ( Q ) and Z ( Q ). Let us startwith z ∈ Z ( Q ). The functor F : SF (N , β ) → Rep Q (N , β ) which is inverse to J is givenexplicitly in Appendix B.8. Conversely, the functor J , as C -linear functor, is J = E ◦ H ,where E is given in the proof of Proposition A.3 and H in the proof of Proposition B.3. Thenfor X ∈ SF , η (cid:48) X is given by the action of z , where now f + k acts as a k , f − k acts as b k and K acts as parity involution ω X . This gives arrow (2). Arrow (3) is pullback of representations(see the proof of Lemma 6.9) with the result that f + k still acts as a k but now f − k acts as − b k .Finally, for arrow (4) the relation between a k , b k and the zero modes ( α j ) is given in (6.4).Altogether, for ν j , ε j ∈ { , } , (cid:80) N j =1 ( ν j + ε j ) even,Υ (cid:0) e ( f +1 ) ν ( f − ) ε · · · ( f +N ) ν N ( f − N ) ε N (cid:1) = ( a ) ν ( − b ) ε · · · ( a N ) ν N ( − b N ) ε N , (6.47) = ( − π i) ε + ··· + ε N · ( α ) ν ( α ) ε · · · ( α − ) ν N ( α ) ε N , where the factors on the RHS are zero modes of (cid:98) h acting on G . Furthermore,Υ (cid:0) K e f +1 f − · · · f +N f − N (cid:1) = ( a ) ( − b ) · · · ( a N ) ( − b N ) ω G , (6.48) = ( − π i) N · α α · · · α ω G . For Z ( Q ) one quickly finds thatΥ( e +1 ) = id ( (cid:98) T ) ev = · (cid:99) id T ◦ (id + ω G ) , (6.49) Υ( e − ) = id ( (cid:98) T ) odd = · (cid:99) id T ◦ (id − ω G ) . After these preparations, we can finally compare the elements φφφ V from (4.28) to (6.23) andthe action of the generators of SL (2 , Z ) as given in Theorems 5.3 and 6.7. Theorem 6.11.
If one chooses the normalisation constant of the integral for the coend L in (4.19) to be ν = 1 , then Υ( φφφ X +0 ) = ˆ δ − ( ϕ ) , Υ( φφφ X +1 ) = ˆ δ − ( ϕ T ) , (6.50) Υ( φφφ X − ) = ˆ δ − ( ϕ Π ) , Υ( φφφ X − ) = ˆ δ − ( ϕ Π T ) . Furthermore, S Z ◦ Υ = Υ ◦ S Z and T Z ◦ Υ = e π iN / · Υ ◦ T Z . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 45 Proof.
That the identities in (6.50) hold is immediate from comparing (4.28) and (6.23) viathe explicit form of Υ given above. For example,(6.51) Υ( φφφ X +0 ) = ν N β ( − π i) N ( ω G + id) α · · · α , which is equal to ˆ δ ( ϕ ) in (6.23) since ν = 1 and β is fixed as in (6.39).The basis (5.5) gets mapped to(6.52) (cid:8) φφφ P +0 , φφφ X +1 , φφφ X − (cid:9) Υ (cid:55)−→ (cid:8) · ˆ δ − ( ϕ + ϕ Π ) , ˆ δ − ( ϕ T ) , ˆ δ − ( ϕ Π T ) (cid:9) , which differs from (6.25) by a coefficient in the first basis vector. Taking this into accountone arrives at S Z P ◦ Υ = Υ ◦ S Z P and T Z P ◦ Υ = e π iN / · Υ ◦ T Z P .The basis for U k used in Lemma 5.2 is transported to a basis of W k (via (cid:36) − ◦ Υ ◦ ϑ ) as(6.53) (cid:8) e , f − k e , f + k e , f + k f − k e (cid:9) (cid:55)−→ (cid:8) id , − π i α k , α k − , − π i α k − α k (cid:9) . This differs from the basis used in (6.33), namely { id , α k − , α k , α k α k − } , in the order ofbasis factors and by coefficients. Note that for the present choice of β , the factor β in (5.13) isequal to i N , which can be distributed over the individual factors of S kZ Λ . One then verifies thatthe above change of basis transports i S kZ Λ to σ k and T kZ Λ to τ k as given in (6.35). Comparing(5.13) and (6.38), we see that this proves S Z Λ ◦ Υ = Υ ◦ S Z Λ and T Z Λ ◦ Υ = e π iN / · Υ ◦ T Z Λ . (cid:3) This also proves Theorem 1.1 from the introduction, which states that the linear SL (2 , Z )-action π E on Z ( E ) and the projective SL (2 , Z )-action π Q on Z ( Q ) agree projectively. Namely,there is a function γ : SL (2 , Z ) → C × such that the linear isomorphism Υ : Z ( Q ) → Z ( E )satisfies(6.54) π E ( g ) ◦ Υ = γ ( g ) · Υ ◦ π Q ( g ) for all g ∈ SL (2 , Z ) . Appendix A. Equivalence between SF and Rep S We give here the first part of the proof of Lemma 3.5. Fix N ∈ N and β ∈ C with β = ( − N , see (2.1). In this section we introduce a quasi-bialgebra S = S (N , β ) in Svect ,define a braiding on its category
Rep S of finite-dimensional representations in Svect , andshow that
Rep S is equivalent to SF (N , β ) as a braided monoidal category.A.1. A quasi-bialgebra in Svect.
The unital associative algebra S = S (N , β ) over C hasgenerators x ± i , i = 1 , . . . , N and L , subject to the relations x ± i L = Lx ± i , { x + i , x − j } = δ i,j (1 − L ) , { x ± i , x ± j } = 0 , L = . (A.1)We have dim S = 2 . Next we turn S into an algebra in Svect by giving it a Z -gradingsuch that x ± i have odd degree and L has even degree.Define the central idempotents e := (1 + L ) , e := (1 − L ) . (A.2) Using these, S decomposes as S = S ⊕ S , S := e S , S := e S . (A.3)From the defining relations it is immediate that S is a 2 -dimensional Graßmann algebraand S is a 2 -dimensional Clifford algebra.In the following we will often use the tensor product A ⊗ B of algebras A, B in Svect . Asa super-vector space, A ⊗ B is the tensor product A ⊗ Svect B of the underlying super-vectorspaces. The unit is 1 ⊗ µ A ⊗ B includes parity signs:(A.4) µ A ⊗ B := ( µ A ⊗ µ B ) ◦ (id A ⊗ τ s . v .B,A ⊗ id B ) : A ⊗ B ⊗ A ⊗ B −→ A ⊗ B , where τ s . v . is the symmetric braiding in Svect , see (2.7). In terms of homogeneous elements,this reads ( a ⊗ b ) · ( a (cid:48) ⊗ b (cid:48) ) = ( − | b || a (cid:48) | ( aa (cid:48) ) ⊗ ( bb (cid:48) ).The notion of a quasi-bialgebra in vect is given e.g. in Definition I:6.1 (that definition is forquasi-Hopf algebras – just omit the antipode and the corresponding conditions). In Svect ,the definition is basically the same (and works in general symmetric monoidal categories):
Definition A.1.
A quasi-bialgebra in
Svect is a tuple (
A, ε, ∆ , Φ), where • A ∈ Svect is a unital associative algebra, such that the unit ∈ A is even and theproduct respects the Z -grading. • ε : A → C | and ∆ : A → A ⊗ A are even linear maps and algebra homomorphisms.(The algebra structure on A ⊗ A involves the symmetric braiding in Svect as describedin (A.4).) • Φ ∈ A ⊗ A ⊗ A is an even element.These data is subject to the conditions I:(6.2)–I:(6.5) in Definition I:6.1, with products in A ⊗ A ⊗ A involving parity signs as in (A.4).In presenting the quasi-bialgebra structure for S , we first list the data and will then provein Proposition A.2 below that these indeed define a quasi-bialgebra in Svect .The (non-coassociative) coproduct ∆ S : S → S ⊗ S and counit (cid:15) : S → C are given by∆ S ( x ± i ) = x ± i ⊗ + ( e − i e ) ⊗ x ± i ± i e ⊗ e ( x + i − x − i ) , ε S ( x ± i ) = 0 , (A.5) ∆ S ( L ) = L ⊗ L , ε S ( L ) = 1 . It is straightforward to check that ∆ S is well-defined on S , i.e. that the above definition interms of generators is compatible with the relations in (A.1).The coassociator Λ ∈ S ⊗ S ⊗ S is given byΛ = e ⊗ e ⊗ e (A.6) + e ⊗ e ⊗ e + Λ e ⊗ e ⊗ e + e ⊗ e ⊗ e + Λ e ⊗ e ⊗ e + Λ e ⊗ e ⊗ e + e ⊗ e ⊗ e YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 47 + Λ e ⊗ e ⊗ e , where(A.7) Λ abc = β abc N (cid:89) k =1 Λ abc ( k ) (i.e. there is a factor of β in sector ) withΛ k ) = ⊗ ⊗ + (1 + i) x − k ⊗ ⊗ x + k − (1 − i) x + k ⊗ ⊗ x − k − x + k x − k ⊗ ⊗ x + k x − k , (A.8)Λ k ) = ⊗ ⊗ − i ⊗ ( x + k + x − k ) ⊗ ( x + k + x − k ) , Λ k ) = ⊗ ⊗ + i ⊗ ( x + k + i x − k ) ⊗ ( x + k + i x − k ) + (1 − i)( x + k − x − k ) ⊗ x + k x − k ⊗ (i x + k − x − k )+ (1 + i) x − k ⊗ x + k ⊗ − (1 − i) x + k ⊗ x − k ⊗ + (1 + i) ⊗ x + k x − k ⊗ − x + k x − k ⊗ x + k x − k ⊗ , Λ k ) = (i − ⊗ ⊗ + (1 − i) ⊗ ⊗ x + k x − k + (1 + 2i) ⊗ x − k ⊗ x − k + ⊗ x − k ⊗ x + k + ⊗ x + k ⊗ x − k + ⊗ x + k ⊗ x + k + (1 − i) ⊗ x + k x − k ⊗ − (2 − ⊗ x + k x − k ⊗ x + k x − k − (1 − i) x − k ⊗ ⊗ x − k + (1 + i) x − k ⊗ ⊗ x + k − (1 − i) x − k ⊗ x − k ⊗ x + k x − k + (1 + i) x − k ⊗ x + k ⊗ x + k x − k + (1 − i) x − k ⊗ x + k x − k ⊗ x − k − (1 + i) x − k ⊗ x + k x − k ⊗ x + k + (1 − i) x + k ⊗ x − k ⊗ − (1 − i) x + k ⊗ x − k ⊗ x + k x − k − (1 + i) x + k ⊗ x + k ⊗ + (1 + i) x + k ⊗ x + k ⊗ x + k x − k − (1 − i) x + k ⊗ x + k x − k ⊗ x − k + (1 + i) x + k ⊗ x + k x − k ⊗ x + k + 2 x + k x − k ⊗ ⊗ − x + k x − k ⊗ ⊗ x + k x − k − x + k x − k ⊗ x − k ⊗ x − k − x + k x − k ⊗ x + k ⊗ x + k − x + k x − k ⊗ x + k x − k ⊗ + 4 x + k x − k ⊗ x + k x − k ⊗ x + k x − k . Note the product of the Λ abc ( k ) is taken in the tensor product of super-algebras S ⊗ S ⊗ S , whichincludes parity signs, as defined in (A.4). The Λ abc ( k ) is parity-even, and one quickly checksthat for k (cid:54) = l the elements Λ abc ( k ) and Λ abc ( l ) commute in S ⊗ S ⊗ S , so that the order in whichone takes the product does not matter.Finally, we define an element r ∈ S ⊗ S which is given by(A.9) r = r · e ⊗ e + r · e ⊗ e + r · e ⊗ e + r · e ⊗ e where r nm = β nm (cid:81) N k =1 r nm ( k ) and r k ) = ⊗ − x − k ⊗ x + k , (A.10) r k ) = ⊗ − (1 + i) x − k ⊗ x + k − (1 + i) x + k ⊗ x − k − (1 + i) x + k x − k ⊗ + 2i x + k x − k ⊗ x + k x − k ,r k ) = ⊗ − (1 − i) x − k ⊗ x + k − (1 − i) x + k ⊗ x − k − (1 − i) ⊗ x + k x − k + 2i x + k x − k ⊗ x + k x − k ,r k ) = − i x − k ⊗ x − k − x − k ⊗ x + k − x + k ⊗ x − k + i x + k ⊗ x + k + 2 x + k x − k ⊗ x + k x − k . Again, for k (cid:54) = l the elements r mn ( k ) and r mn ( l ) commute in S ⊗ S . We will use r later to define abraiding in Rep S (but r is not an R-matrix for S as the braiding will involve an extra paritymap). For N = 1, the quasi-bialgebra algebra S was defined in [GR1, Sec. 4]. The proof that S (N , β ) is a quasi-bialgebra for general N relies on reducing the problem to the N = 1 case.To do so, choose β , . . . , β N such that β i = − β = β · · · β N . Define A to be the N-foldtensor product(A.11) A := S (1 , β ) ⊗ · · · ⊗ S (1 , β N )of quasi-bialgebras in Svect . Thus A is itself a quasi-bialgebra in Svect . The product andcoproduct of A are defined with parity signs (see (A.4) for the product) and the coassociatorΛ A of A is the product in A ⊗ A ⊗ A of those in the individual factors S (1 , β i ). Consider thetwo-sided ideal I in A generated by(A.12) I := (cid:10) L i − L j (cid:12)(cid:12) ≤ i, j ≤ N (cid:11) . where L i stands for the element with L on the i -th tensor factor and else. One verifies thatthis ideal satisfies ∆ A ( I ) ⊂ I ⊗ A + A ⊗ I and ε A ( I ) = 0, i.e. it is a (quasi-)bialgebra ideal. Proposition A.2.
For N , β as in (2.1) , the data ( S , · , , ∆ S , ε S , Λ) defines a quasi-bialgebrain Svect and S ∼ = A /I as quasi-bialgebras.Proof. We write x + , x − and L for the generators of S (1 , β j ). Consider the surjective evenlinear map ϕ : A → S (N , β ) given by ϕ (cid:16) ( x + ) ε ( x − ) δ L m ⊗ · · · ⊗ ( x + ) ε N ( x − ) δ N L m N (cid:17) (A.13) = ( x +1 ) ε ( x − ) δ · · · ( x +N ) ε N ( x − N ) δ N L m + ··· + m N . The map ϕ satisfies(A.14) ϕ ( ⊗ N ) = , ε S ◦ ϕ = ε A , ϕ ( ab ) = ϕ ( a ) ϕ ( b ) , ( ϕ ⊗ ϕ ) ◦ ∆ A = ∆ S ◦ ϕ Since ϕ is surjective, this proves that ∆ S is an algebra homomorphism. Furthermore, ϕ ⊗ maps the coassociator of A to that of S :(A.15) ϕ ⊗ (Λ A ) = Λ . This follows from the product form of Λ ∈ S ⊗ and the fact that β = β · · · β N . Using oncemore that S is surjective, it follows that Λ satisfies the defining relations of a coassociator(see Definition A.1).Thus S is a quasi-bialgebra. Finally, it is clear from the definition of ϕ that ker( ϕ ) = I . (cid:3) A.2.
An equivalence from SF to Rep S . Due to (A.3),
Rep S can be decomposed intotwo parts:(A.16) Rep S = Rep S ⊕ Rep S . In this section we construct an equivalence of C -linear categories D : SF →
Rep S . Thefunctor has two components due to the decompositions in (2.3) and (A.16). On SF we have(A.17) D : SF → Rep S , D ( U ) = U where x + i u = a i u , x − i u = b i u ( u ∈ U ) , YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 49 and D ( f ) = f on morphisms.For the second component we need the non-central idempotent b := N (cid:89) i =1 x − i x + i e ∈ S . (A.18)To see that indeed bb = b note that for each i separately we have x − i x + i x − i x + i e = x − i x + i e .The idempotent b generates a S -submodule B ⊂ S : B := Sb = S b = span (cid:8) ( x +N ) i N · · · ( x +1 ) i b | i , . . . , i N ∈ { , } (cid:9) . (A.19)Since S is a Clifford algebra, B is a simple (and projective) S -module in Svect . There are upto isomorphism two distinct simple S -modules, namely B and B ⊗ Svect C | , the parity-shiftedcopy of B .The second component of D is defined as D : SF → Rep S , D ( U ) = B ⊗ Svect
U , (A.20)where the S -action on D ( U ) is the left multiplication on B and where U ∈ SF (cid:39) Svect .On morphisms we set D ( f ) = id B ⊗ f . Proposition A.3.
The functor D is an equivalence of C -linear categories.Proof. We will give an inverse functor E : Rep S → SF to D . The functor E will again bedefined separately in the two sectors.Given V ∈ Rep S , the Λ -module E ( V ) has V as the underlying super-vector space withthe action of a i and b i given by the action of x + i and x − i , respectively. On morphisms f in Rep S we set E ( f ) = f . Clearly, D and E are inverse to each other as functors between SF and Rep S .For a given object M ∈ Rep S , we set E ( M ) = Hom vect S ( B , M ). This means we considerall S -linear maps from B to M , not just the parity-even maps. The decomposition intoparity-even and parity-odd maps turns Hom vect S ( B , M ) into a super-vector space. Since B issimple, we get a canonical isomorphism Hom vect S ( B , B ⊗ Svect V ) ∼ = V of super-vector spaceswhich is natural in V , that is, E ( D ( V )) ∼ = V . Conversely, since every S -module is isomorphicto a direct sum of copies of B and B ⊗ Svect C | , that is M ∼ = B ⊗ V for some V ∈ Svect , weget M ∼ = B ⊗ Hom vect S ( B , M ). (cid:3) A.3. D as a multiplicative functor. Below we will make an ansatz for isomorphisms(A.21) ∆
U,V : D ( U ∗ V ) → D ( U ) ⊗ Rep S D ( V ) . which will be given sector by sector. Both, the expression for ∆ U,V and the proof that it isan isomorphism mimics the corresponding construction for N = 1 in [GR1, Sec. 6.3]. δ U VU V B U VU B Vδ b BB U V b U V B S U b bB U V B V Figure 1.
String diagram notation for ∆
U,V in the four sectors.The multiplication map S ⊗ S → S is denoted by µ S , and for an S -module U the expression ρ U : S ⊗ U → U stands for the S -action on U . We will also need the constants δ = N (cid:89) i =1 (cid:0) ⊗ + x − i ⊗ x + i (cid:1) ∈ S ⊗ S , δ = N (cid:89) i =1 (cid:0) + x + i x − i (cid:1) ∈ S . (A.22)Note that they are multiplicative invertible with inverses δ − = N (cid:89) i =1 (cid:0) ⊗ − x − i ⊗ x + i (cid:1) ∈ S ⊗ S , δ − = N (cid:89) i =1 (cid:0) − x + i x − i (cid:0) e + e (cid:1)(cid:1) ∈ S . (A.23)We denote the right multiplication with a ∈ S by R a .The following list defines ∆ U,V for each of the four possibilities to choose U ∈ SF i and V ∈ SF j , i, j ∈ { , } , which we refer to as “sector ij ”. The underlines indicate on whichfactors S acts. In Figure 1 we give string diagram expressions for ∆ U,V . • sector: ∆ U,V : U ⊗ V → U ⊗ V is given by(A.24) ∆ U,V = (cid:0) ρ U ⊗ ρ V (cid:1) ◦ (cid:0) id S ⊗ τ s . v . S ,U ⊗ id V (cid:1) ◦ (cid:0) δ ⊗ id U ⊗ id V (cid:1) , where τ s . v . is defined in (2.7). YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 51 • sector: ∆ U,V : B ⊗ U ⊗ V → U ⊗ B ⊗ V is given by(A.25) ∆ U,V = (cid:0) ρ U ⊗ R b ⊗ id V (cid:1) ◦ (cid:0) R δ ⊗ τ s . v . S ,U ⊗ id V (cid:1) ◦ (cid:0) ∆ S ⊗ id U ⊗ id V (cid:1) . The image of R b on S is B , that is, the target is indeed U ⊗ B ⊗ V . • sector: ∆ U,V : B ⊗ U ⊗ V → B ⊗ U ⊗ V is given by(A.26) ∆ U,V = (cid:0) R b ⊗ id U ⊗ ρ V (cid:1) ◦ (cid:0) id S ⊗ τ s . v . S ,U ⊗ id V (cid:1) ◦ (cid:0) ∆ S ⊗ id U ⊗ id V (cid:1) . • sector: ∆ U,V : S ⊗ U ⊗ V → B ⊗ U ⊗ B ⊗ V is given by(A.27) ∆ U,V = (cid:0) id B ⊗ τ s . v . B ,U ⊗ id V (cid:1) ◦ (cid:0) [( R b ⊗ R b ) ◦ ∆ S ] ⊗ id U ⊗ id V (cid:1) . Here, in the source S -module we have identified S and Λ . Lemma A.4.
The linear maps ∆ U,V are intertwiners of S -modules.Proof. In all sectors beside this clear since ∆ S is an algebra map, and since the right-multiplications R δ and R b are left-module intertwiners. In sector we have to check that( e ⊗ e ) · ∆ S ( a ) · δ = ( e ⊗ e ) · δ · ∆ S ( a ) holds for all a ∈ S . This is clear for a = L , andfor a = x ± i it is an easy check. (cid:3) Lemma A.5.
The ∆ U,V are isomorphisms.Proof.
In sectors
00, 01 and the proof is similar as that in [GR1, Lemma 6.5]. Indeed,for sector , invertibility of ∆ U,V follows from that of δ . For sector and the inverseof ∆ U,V is given by the same string diagram as that in [GR1, Lemma 6.5]. The proof thatthe expressions in [GR1, Lemma 6.5] are indeed inverses rests on the two identities( b ⊗ ) · ((id ⊗ ω S ) ◦ ∆ S ( b )) · ( b ⊗ ) = b ⊗ ( δ e ) , (A.28) ( b ⊗ ) · (cid:0) (id ⊗ ω S ) ◦ τ s . v . ◦ ∆ S ( b ) (cid:1) · ( b ⊗ ) = b ⊗ ( δ − e ) , where ω S denotes the parity involution on the super-vector space S as defined in (2.13) . Weneed to show that these identities remain true for general N, and we will do so for the firstidentity.Namely, for i (cid:54) = j and by defining b ( i ) = x − i x + i e and δ ( i )1 = + x + i x − i we get b ( i ) b ( j ) ⊗ δ ( i )1 δ ( j )1 e = ( b ( i ) ⊗ ( δ ( i )1 e )) · ( b ( j ) ⊗ ( δ ( j )1 e ))(A.29) ( ∗ ) = (cid:0) ( b ( i ) ⊗ )((id ⊗ ω S ) ◦ ∆ S ( b ( i ) ))( b ( i ) ⊗ ) (cid:1) · (cid:0) ( b ( j ) ⊗ )((id ⊗ ω S ) ◦ ∆ S ( b ( j ) ))( b ( j ) ⊗ ) (cid:1) ( ∗∗ ) = ( b ( i ) ⊗ ) · ( b ( j ) ⊗ ) · ((id ⊗ ω S ) ◦ ∆ S ( b ( i ) ) · (id ⊗ ω S ) ◦ ∆ S ( b ( j ) )) · ( b ( i ) ⊗ ) · ( b ( j ) ⊗ ) Note that the antipode in [GR1] coincides with ω S in sector 0. = ( b ( i ) b ( j ) ⊗ ) · ((id ⊗ ω S ) ◦ ∆ S ( b ( i ) b ( j ) )) · ( b ( i ) b ( j ) ⊗ ) , where we used that b ( i ) is an even element and in step (*) that the corresponding identity holdsfor N = 1, then in step (**) that ω S , ∆ S are morphisms in Svect and so (id ⊗ ω S ) ◦ ∆ S ( b ( i ) ) iseven and moreover does not contain x ± j and therefore it commutes with b ( j ) ⊗ . Using thenthe fact that b = (cid:81) N i =1 b ( i ) and δ = (cid:81) N i =1 δ ( i )1 , this analysis shows the first identity in (A.28).The second identity there is established in a similar way.In sector , we have to show thatΘ : S → B ⊗ Svect B (A.30) a (cid:55)→ ∆ S ( a ) · b ⊗ b is an isomorphism. Note that ∆ S ( x ± i ) · e ⊗ e = ( x ± i ⊗ − i ⊗ x ∓ i ) · e ⊗ e and x − i b = 0.We get Θ( e ) = ∆ S ( e ) · b ⊗ b = b ⊗ b (A.31) Θ( x + i x + j e ) = ∆ S ( x + i ) · ( x + j b ) ⊗ b = ( x + i x + j b ) ⊗ b Θ( x − i x − j e ) = − ∆ S ( x − i ) · i b ⊗ ( x + j b ) = ( − i) b ⊗ ( x + i x + j b )Θ( x + i x − j e ) = − ∆ S ( x + i ) · i b ⊗ ( x + j b ) = − (i x + i b ) ⊗ ( x + j b ) + δ i,j i b ⊗ ( x − i x + j b )= − (i x + i b ) ⊗ ( x + j b ) + δ i,j b ⊗ b . Now it is straightforward to see that for i < . . . < i k and j < . . . < j m we haveΘ( x + i · · · x + i k x − j · · · x − j m e ) = ∆ S ( x + i · · · x + i k ) · ( − i) m b ⊗ ( x + j · · · x + j m b )(A.32) = ( − i) m ( x + i · · · x + i k b ) ⊗ ( x + j · · · x + j m b ) + ˜ w, where ˜ w ∈ (cid:76) k + m − i =0 ˜ W i and ˜ W i is the span of elements ( x + i · · · x + i u b ) ⊗ ( x + j · · · x + j v b ) with i = u + v . So we can show by induction over m + k that Θ is surjective and since S and B ⊗ B have dimension 2 , Θ is even bijective. (cid:3) From the definition of ∆
U,V it is immediate that these maps are natural in U and V .Together with Lemmas A.4 and A.5 this proves the following proposition: Proposition A.6.
With the isomorphisms ∆ U,V , the functor D : SF →
Rep S is multiplica-tive. A.4.
Compatibility with associator and unit isomorphisms.
In this section we verifythat the functor D from Proposition A.6 is monoidal, i.e. that it is compatible with associatorsand unit isomorphisms.In fact, since the unit isomorphisms in SF and Rep S are just those of the underlyingsuper-vector spaces, after choosing the isomorphism → D ( ) to be the identity on C | ,compatibility of D with the unit isomorphisms is immediate. YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 53 : Λ · ( ⊗ δ ) · (id ⊗ ∆ S )( δ )= ( e ⊗ e ⊗ e ) · ( δ ⊗ ) · (∆ S ⊗ id)( δ ) : Λ · (cid:0) (id ⊗ ∆ S ) ◦ ( R δ ⊗ R b ) ◦ ∆ S ( v ) (cid:1) · ( ⊗ δ ⊗ b )= ( e ⊗ e ⊗ ) · ( δ ⊗ ) · (cid:0) (∆ S ⊗ id) ◦ ( R δ ⊗ R b ) ◦ ∆ S ( v ) (cid:1) : Λ · (cid:0) (id ⊗ ∆ S ) ◦ ( R δ ⊗ R b ) ◦ ∆ S ( v ) (cid:1) · ( ⊗ b ⊗ )= (cid:0) (∆ S ⊗ id) ◦ ( R b ⊗ id) ◦ ∆ S ( v ) (cid:1) · ( δ ⊗ b ⊗ ) · γ (13) : Λ · ( ⊗ δ ) · (cid:0) (id ⊗ ∆ S ) ◦ ( R b ⊗ id) ◦ ∆ S ( v ) (cid:1) = (cid:0) (∆ S ⊗ id) ◦ ( R b ⊗ id) ◦ ∆ S ( v ) (cid:1) · ( b ⊗ ⊗ ) : Λ · (cid:0) (id ⊗ ∆ S ) ◦ ( R b ⊗ R b ) ◦ ∆ S ( h ) (cid:1) · ( ⊗ b ⊗ )= (cid:8) (∆ S ⊗ id) (cid:0) δ · ∆ S ( h ) (cid:1)(cid:9) · ( b ⊗ b ⊗ ) : Λ · (cid:8) (id ⊗ ∆ S ) (cid:0) ∆ S ( h ) · b ⊗ b (cid:1)(cid:9) · ( ⊗ δ ⊗ b )= ( R b ⊗ µ S ⊗ id) ◦ (∆ S ⊗ τ s . v . B , S ) ◦ (cid:0)(cid:8) ( R b ⊗ R b ) ◦ ∆ S ◦ µ S (cid:9) ⊗ id (cid:1) ( h ⊗ γ ) : Λ · (cid:0) (id ⊗ ∆ S )( δ ) (cid:1) · (cid:0) ⊗ (∆ S ( h ) · b ⊗ b ) (cid:1) = (cid:8)(cid:0) ω S ◦ µ S ◦ (id ⊗ ω S ) (cid:1) ⊗ id ⊗ id (cid:9) ◦ (cid:8) id ⊗ (cid:0) ( R δ ⊗ R b ) ◦ ∆ S (cid:1) ⊗ id (cid:9) ◦ (cid:8) id ⊗ (cid:0) ( R b ⊗ R b ) ◦ ∆ S (cid:1)(cid:9) ◦ ∆ S ( h ) : Λ · (cid:8)(cid:0) id ⊗ (∆ S ◦ µ S ) (cid:1)(cid:0) ∆ S ( v ) ⊗ h (cid:1)(cid:9) · b ⊗ b ⊗ b = (cid:8)(cid:0) ( R b ⊗ R b ) ◦ ∆ S ◦ µ S (cid:1) ⊗ id (cid:9) ◦ (id ⊗ τ s . v . B , S ) ◦ (cid:8)(cid:0) ( R δ ⊗ R b ) ◦ ∆ S ( v ) (cid:1) ⊗ φ ( h ) (cid:9) Table 1.
Evaluation of the compatibility of associators as in (3.24) for D and ∆ U,V in each of the eight sectors. The constant γ is defined as γ :=exp( C ) · e ⊗ e with C as in (2.9) and we use the identification x + k = a k , x − k = b k as in (A.17). The map φ : S → S is defined as in (2.11) under thesame identification. The above conditions have to hold for all h ∈ S and v ∈ B .The main effort lies in showing that the diagram in (3.24) commutes for D : SF →
Rep S and ∆ U,V . In a calculation similar to that in [GR1, Sec. 7.2] one can evaluate (3.24) for eachof the eight possibilities of choosing U ∈ SF a , V ∈ SF b , W ∈ SF c , which we refer to as“sector abc ”. We define(A.33) Λ abc = Λ abc · ( e a ⊗ e b ⊗ e c ) with Λ abc as in (A.6) (it is understood that the Λ abc not spelled out explicitly in (A.6) are setto ⊗ ). Then (3.24) for D and ∆ U,V is equivalent to the eight conditions in Table 1.To verify the eight identities in Table 1, we reduce them to the N = 1 case which has beenchecked in [GR1, Prop. 7.3]. To do so, define S ( k ) to be the subalgebra of S generated by x + k , x − k and L . By definition of S , for k (cid:54) = l elements from S ( k ) and S ( l ) super-commute.The aim is now to rewrite each condition in Table 1 as a product of terms k = 1 , . . . , Nwhich use only elements in S ( k ) and note that since by [GR1, Prop. 7.3] the correspondingidentity holds for each k separately, it holds for the product.For Λ abc the product decomposition is given in (A.7). The constants b , δ and δ werealready defined as products over elements in S ( k ) and S ( k ) ⊗ S ( k ) in (A.18) and (A.22). For γ in Table 1 set C ( i ) = x − i ⊗ x + i − x + i ⊗ x − i and use(A.34) γ = exp (cid:16) N (cid:88) i =1 C ( i ) (cid:17) = N (cid:89) i =1 γ ( i ) with γ ( i ) := exp C ( i ) . Now sector directly decomposes into products over elements in ( S ( k ) ) ⊗ and henceholds by [GR1, Prop. 7.3]. For sector note that, for c ( k ) , d ( k ) ∈ S ( k ) even and v ( k ) ∈ B ( k ) we have, for i (cid:54) = j ,(id ⊗ ∆ S ) ◦ ( R c ( i ) c ( j ) ⊗ R d ( i ) d ( j ) ) ◦ (∆ S ( v ( i ) v ( j ) ))(A.35) = (id ⊗ ∆ S ) ◦ ( R c ( i ) ⊗ R d ( i ) ) ◦ ( R c ( j ) ⊗ R d ( j ) ) (cid:0) ∆ S ( v ( i ) ) · ∆ S ( v ( j ) ) (cid:1) = (id ⊗ ∆ S ) ◦ ( R c ( i ) ⊗ R d ( i ) ) (cid:0) ∆ S ( v ( i ) ) · ( R c ( j ) ⊗ R d ( j ) )(∆ S ( v ( j ) )) (cid:1) = (id ⊗ ∆ S ) (cid:0) ( R c ( i ) ⊗ R d ( i ) )(∆ S ( v ( i ) )) · ( R c ( j ) ⊗ R d ( j ) )(∆ S ( v ( j ) )) (cid:1) = (id ⊗ ∆ S ) (cid:0) ( R c ( i ) ⊗ R d ( i ) )(∆ S ( v ( i ) )) (cid:1) · (id ⊗ ∆ S ) (cid:0) ( R c ( j ) ⊗ R d ( j ) )(∆ S ( v ( j ) )) (cid:1) . This allows one to write the LHS of the equation for sector as a product over elementsin ( S ( k ) ) ⊗ . For the RHS one uses a similar statement where id ⊗ ∆ S is replaced by ∆ S ⊗ id.Analogous arguments work in all sectors, except that in sector we need to deal with φ ( h ) separately.The linear map φ : S → S is defined in (2.11), where by abuse of notation we identify S and Λ via x + k e = a k , x − k e = b k . Following [DR1, Sec. 5.2] φ can be written explicitly as, for u ∈ S , φ ( u ) = (cid:88) m =0 i m ( m +1) (cid:88) i ≤ ... ≤ i m ε i · · · ε i m Λ co S ( x ε i i · · · x ε im i m · u ) x − ε i i · · · x − ε im i m e , (A.36)where ε i n ∈ {± } such that i n = i n +1 ⇒ ε i n > ε i n +1 . Furthermore, Λ co S : S → C is definedas in (2.12), again using the identification between S and Λ . Explicitly, it is zero everywhereon S except in the top degree, where it takes the value(A.37) Λ co S (cid:0) x +1 x − · · · x +N x − N e (cid:1) = β − . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 55 As in the definition of A in (A.11) choose β , . . . , β N such that β i = − β = β · · · β N .On each of the subalgebras S ( k )0 we can define the “N = 1 version of φ ” as follows:(A.38) φ ( k ) ( e ) = β k x + k x − k e , φ ( k ) ( x ± k e ) = β k x ± k e , φ ( k ) ( x + k x − k e ) = − β k e . Indeed, for N = 1 we have φ = φ (1) (see also [GR1, Eqn. (33)]). Lemma A.7.
We have, for v ( k ) ∈ S ( k )0 , (A.39) φ ( v (1) · · · v (N) ) = φ (1) ( v (1) ) · · · φ (N) ( v (N) ) . Proof.
We start by describing φ in a different way: Let T = { , . . . , N } × {− , } be a totallyordered set whose ordering is given by, for ( i , ε ) , ( i , ε ) ∈ T ,(A.40) ( i , ε ) < ( i , ε ) ⇐⇒ i < i ∨ ( i = i ∧ ε < ε ) . Let K ⊂ T be a subset of the form K = { ( j , ε j ) , . . . , ( j k , ε j k ) } , where the indexing agreeswith the ordering in the sense that ( j a , ε a ) < ( j a +1 , ε a +1 ). We introduce the elements u K := x ε j j · · · x ε jk j k e , u − K := x − ε j j · · · x − ε jk j k e , u ∅ := e , β K = β j · · · β j k . (A.41)For two subsets K ⊂ L ⊂ T we define the linear map φ L ( u K ) := β L i | L | +2 | K | u − L \ K . (A.42)For n = 1 , . . . , N let L n = { ( n, +) , ( n, − ) } . We claim that(a) φ L n ( u K ) = φ ( n ) ( u K ) for K ⊂ L n , (b) φ ∪ rn =1 L n ( u K ) = φ ∪ r − n =1 L n ( u K \ L r ) · φ L r ( u K ∩ L r ) , (c) φ T = φ , where for (b) we have r = 1 , . . . , N and K ⊂ (cid:83) rn =1 L n . Properties (a)–(c) together imply thestatement of the lemma.Properties (a) and (b) are immediate from the definition. It remains to show property (c).Abbreviate k = | K | . The cointegral Λ co S in (A.36) is non-zero only in top-degree, so that for agiven u K , the only non-zero summand in (A.36) is that with { ( i , ε i ) , . . . , ( i m , ε i m ) } = T \ K .Thus φ ( u K ) = ± u − L \ K . To determine the sign, we define η ( K ) = i (2N − k )(2N − k +1) = ( − N i ( k − k , (A.43) η ( K ) = (cid:89) ( i,ε i ) ∈ T \ K ε i , η ( K ) = (cid:16) (cid:89) ( n,ε n ) ∈ K − ε n (cid:17) · (cid:16) k (cid:89) m =1 ( − − k − m (cid:17) . The factors in φ ( u K ) from (A.36) correspond to η , η , η as follows. We have β = β T andthe product β − T · η corresponds to the factor i m ( m +1) , for m = | T \ K | = 2N − k , timesthe normalisation of Λ co S from (2.12). The sign η corresponds to ε i · · · ε i m where the i a runover values in the complement T \ K , and η corresponds to the sign we get when bringing u T \ K · u K in Λ co S ( · · · ) into the form x +1 x − · · · x +N x − N e . Altogether, the coefficient in front of u − L \ K is β − T η ( K ) · η ( K ) · η ( K ). It is not hard to verify that η ( K ) · η ( K ) = (cid:16) (cid:89) ( i,ε i ) ∈ T \ K ε i (cid:17) (cid:16) (cid:89) ( n,ε n ) ∈ K − ε n (cid:17) i ( k − k = i ( k − k · ( − N+ k . (A.44)and hence η ( K ) · η ( K ) · η ( K ) = ( − k . Thus φ ( u K ) = β − T ( − k u − T \ K = β T ( − N+ k u − T \ K = φ T ( u K ), as claimed. (cid:3) Using the above lemma, one can also write the two sides of sector in Table 1 as productsover elements in ( S ( k ) ) ⊗ and conclude its validity from the N = 1 case.Combining the above result with Propositions A.3 and A.6 we obtain: Proposition A.8.
The functor D : SF →
Rep S , together with the isomorphisms ∆ U,V and id C | : → D ( ) , is C -linear monoidal equivalence. A.5.
Transporting the braiding.
Recall the definition of the element r ∈ S ⊗ S in (A.9).We use r to define a family of natural isomorphisms(A.45) ψ M,N : M ⊗ N → N ⊗ M , M, N ∈ Rep S in Rep S as a sum over sectors: (A.46) ψ M,N = (cid:88) a,b ∈{ , } ψ abM,N , ψ abM,N = τ s . v .M,N ◦ (cid:0) r ab · e a ⊗ e b (cid:1) ◦ (cid:0) id M ⊗ ω aN (cid:1) , where ω N = id N and ω N is the parity involution, and composition with r ab · e a ⊗ e b denotesthe action of this element of S ⊗ S on M ⊗ N (with the corresponding parity signs resultingfrom braiding one copy of S past M ).We will show that ψ is the result of transporting the braiding from SF to Rep S via thefamily of isomorphisms ∆ U,V introduced in Section A.3, that is, it is the unique natural familyof isomorphisms that makes the diagram (3.25) commute:(A.47) D ( U ∗ V ) D ( c U,V ) (cid:47) (cid:47) ∆ U,V (cid:15) (cid:15) D ( V ∗ U ) ∆ V,U (cid:15) (cid:15) D ( U ) ⊗ D ( V ) ψ D ( U ) , D ( V ) (cid:47) (cid:47) D ( V ) ⊗ D ( U )This then proves that ψ is a braiding on Rep S and that D is a braided monoidal functor. Lemma A.9.
For all
M, N ∈ Rep S , ψ M,N is a morphism in
Rep S . Furthermore it isinvertible, natural in M, N and makes the diagram (A.47) commute. The appearance of the parity involution is the reason that r is not a universal R -matrix for S . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 57 : τ s . v .U,V (cid:16) τ s . v .S,S ( δ ) . γ − . ( u ⊗ v ) (cid:17) = ψ U,V (cid:16) δ . ( u ⊗ v ) (cid:17) : τ s . v .U, B ⊗ V (cid:16)(cid:2) τ s . v . B ,U (cid:0) ∆ S ( a ) . ( b ⊗ κ ) . ( ⊗ u ) (cid:1)(cid:3) ⊗ v (cid:17) = ψ U, B ⊗ V (cid:16)(cid:2) ∆ S ( a ) . ( δ ⊗ b ) . ( u ⊗ ) (cid:3) ⊗ v (cid:17) : τ s . v . B ⊗ U,V ◦ (id B ⊗ id U ⊗ ρ V ) ◦ (id B ⊗ τ s . v .S,U ⊗ ω V ) (cid:16)(cid:2) ( τ s . v .S,S ◦ ∆ S ( a )) · ( b ⊗ ( δ κ )) (cid:3) ⊗ u ⊗ v (cid:17) = ψ B ⊗ U,V ◦ (id B ⊗ id U ⊗ ρ V ) ◦ ( R b ⊗ τ s . v . S ,U ⊗ id V ) (cid:16) ∆ S ( a ) ⊗ u ⊗ v (cid:17) : β · τ s . v . B ⊗ U, B ⊗ V ◦ (id B ⊗ τ s . v . B ,U ⊗ ω V ) (cid:16) τ s . v . B , B (cid:2) ∆ S ( hκ − ) · ( b ⊗ b ) (cid:3) ⊗ u ⊗ v (cid:17) = ψ B ⊗ U, B ⊗ V ◦ (id B ⊗ τ s . v . B ,U ⊗ id V ) (cid:16)(cid:2) ∆ S ( h ) · ( b ⊗ b ) (cid:3) ⊗ u ⊗ v (cid:17) Table 2.
Conditions on ψ U,V such that (A.47) commutes in each of the foursectors. The conditions have to hold for all h ∈ S , a ∈ B , u ∈ U , v ∈ V andall U ∈ SF i , V ∈ SF j . The element γ is defined as in Table 1 (and its inverseis given in (A.48)) and κ is defined in (A.49). Proof.
Given morphisms f : M → M (cid:48) and g : N → N (cid:48) in Rep S , it is immediate fromthe definition that ψ M (cid:48) ,N (cid:48) ◦ ( f ⊗ g ) = ( g ⊗ f ) ◦ ψ M,N . The lemma follows once we provedthat (A.47) commutes. Indeed, since the top path in (A.47) is a morphism in
Rep S , so is ψ D ( U ) , D ( V ) . And since the top path is invertible, so is ψ .We now show that (A.47) commutes. In Table 2, the conditions on ψ M,N are given in eachof the four sectors. These conditions use the inverse of γ ,(A.48) γ − = exp( − C ) = N (cid:89) i =1 ( ⊗ − x − i ⊗ x + i + x + i ⊗ x − i − x + i x − i ⊗ x + i x − i ) · e ⊗ e , and the constant(A.49) κ := exp (cid:0) ˆ C (cid:1) = N (cid:89) i =1 ( − x + i x − i ) e . Substituting the definition of ψ in (A.46), the condition in sector can be rewritten as(A.50) r · e ⊗ e = τ s . v .S,S ( δ ) · γ − · δ − , where δ and its inverse are given in (A.22) and (A.23). Note that all ingredients in theabove equality are written as products over elements in S ( k ) for k = 1 , . . . , N. This reduces the verification of (A.50) to the case N = 1, in which case it is a short calculation combining(A.22), (A.23), (A.48) and the definition of r in (A.9).In the remaining three sectors the strategy is the same: we show that the required identitycan be written as a product over elements in S ( k ) . This implies that if the equality holds forN = 1, it holds for all N. The case N = 1 can then be checked by hand or by computeralgebra (which is what we did). Below we only explain the reduction to N = 1 and we omitthe details of the verification for N = 1.The condition on ψ in the -sector can be expressed via r as the following equation in S ⊗ S :(A.51) τ s . v . S , S (cid:2) r · ∆ S ( a ) · ( ⊗ b ) (cid:3) = ∆ S ( a ) · ( b ⊗ κδ − ) , a ∈ B . The elements b , δ and κ are all given in product form, see (A.18), (A.22) and (A.49). Thisreduces the verification in sector to the case N = 1.We see from (A.46) and Table 2 that the situation in the - and -sectors is differentbecause of the presence of the parity-involution ω . In sector the condition on ψ is expressedin terms of r as the following equation in S ⊗ S :(A.52) τ s . v . S , S (cid:2) r · (id ⊗ ω ) (cid:0) ∆ S ( a ) (cid:1) · ( b ⊗ ) (cid:3) = ∆ S ( a ) · ( δ κ ⊗ b ) , a ∈ B . As in sector all elements are given in factorised form, and the above equality thus followsfrom the fact that it holds for N = 1.To rewrite the condition on ψ in sector in terms of r , we have to relate ω B ⊗ V , whichappears in ψ on the RHS of the condition in sector , recall (A.46), and ω V , which appearson the LHS of the condition. This can be done as follows. Note that(A.53) ω B ⊗ ω V = ω B ⊗ V . We recall then the basis B = span (cid:110) b I =( i ,...,i N ) = N (cid:89) k =1 ( x − k ) i k x + k e | i k ∈ { , } (cid:111) , (A.54)with ω ( b I ) = ( − N+ (cid:80) k i k b I and so ω B = Ω . ( − ) with(A.55) Ω = N (cid:89) i =1 ( x − i x + i − x + i x − i ) e . The braiding ψ can be then expressed as ψ B ⊗ U, B ⊗ V = τ s . v . B ⊗ U, B ⊗ V ◦ r ◦ (cid:0) id B ⊗ U ⊗ ω B ⊗ V (cid:1) (A.56) = τ s . v . B ⊗ U, B ⊗ V ◦ ( r · ⊗ Ω) ◦ (id B ⊗ U ⊗ id B ⊗ ω V ) . In the last line, the element r · ⊗ Ω ∈ S ⊗ S acts on the two tensor factors B in B ⊗ U ⊗ B ⊗ V .Substituting this into the condition in Table 2 gives the following condition on r :(A.57) τ s . v . S , S (cid:2) ( r · ⊗ Ω) · ∆ S ( h ) · ( b ⊗ b ) (cid:3) = β · ∆ S ( hκ − ) · ( b ⊗ b ) , h ∈ S . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 59 Again one can write this equality as a product over elements in S ( k ) , k = 1 , . . . , N, reducingthe verification to N = 1. (cid:3)
Since SF is a braided monoidal category, the above lemma shows that ψ defines a braidingon Rep S . Altogether we have shown: Proposition A.10.
The functor D in Proposition A.8 is braided monoidal. Appendix B. Equivalence between SF and Rep Q Here, we present the second part of the proof of Lemma 3.5. We begin with introducinga quasi-bialgebra ˆ Q in vect and show a braided monoidal equivalence between Rep S and Rep ˆ Q . Then we present a twisting of ˆ Q into Q , and therefore Rep Q is braided monoidallyequivalent to Rep S and thus to SF . Finally, we use this equivalence to transport the ribbontwist from SF to Rep Q .B.1. The quasi-bialgebra ˆ Q . In this section we introduce the quasi-bialgebra ˆ Q = ˆ Q (N , β )which is equal to Q as an algebra. It has a different quasi-bialgebra structure, namelyˆ∆( K ) = K ⊗ K − (1 + ( − N ) e ⊗ e · K ⊗ K , ε ( K ) = 1 , (B.1) ˆ∆( f ± i ) = f ± i ⊗ + K − ⊗ f ± i − (1 + ( − N ) e ⊗ e · K − ⊗ f ± i , ε ( f ± i ) = 0 , where we use the central idempotents e i ∈ ˆ Q ( i = 0 ,
1) defined as in (3.2). The coassociatorfor this coproduct is (we will check the axioms below)ˆΦ = e ⊗ e ⊗ e + e ⊗ e ⊗ e + e ⊗ e ⊗ e + e ⊗ e ⊗ e (B.2) + e ⊗ e ⊗ e + ˆΦ e ⊗ e ⊗ e + ˆΦ e ⊗ e ⊗ e + ˆΦ e ⊗ e ⊗ e , where the non-trivial components are given byˆΦ = N (cid:89) k =1 ˆΦ k ) , (B.3) ˆΦ =( − K ) N − ⊗ K N − ⊗ · (cid:16) N (cid:89) k =1 ˆΦ k ) (cid:17) · K N − ⊗ K ⊗ , ˆΦ = − i N β K N − ⊗ ⊗ · (cid:16) N (cid:89) k =1 ˆΦ k ) (cid:17) · K N − ⊗ K N ⊗ , withˆΦ k ) = (cid:16) ⊗ ⊗ + (1 + i) f + k K ⊗ K ⊗ f − k (cid:17)(cid:16) ⊗ ⊗ + (1 − i) f − k K ⊗ K ⊗ f + k (cid:17) , (B.4)ˆΦ k ) = (cid:16) ⊗ ⊗ + (1 + i) ⊗ f + k K ⊗ f − k + (1 − i) f − k K ⊗ f + k ⊗ (cid:17) × (cid:16) ⊗ ⊗ + (1 + i) f + k K ⊗ f − k ⊗ + (1 − i) ⊗ f − k K ⊗ f + k (cid:17) , ˆΦ k ) = (cid:16) ⊗ ⊗ + (i − (cid:0) ⊗ f + k K ⊗ f − k + f + k K ⊗ K ⊗ f − k − f + k K ⊗ f − k ⊗ + ⊗ f − k f + k ⊗ (cid:1)(cid:17) × (cid:16) ⊗ ⊗ − (i − (cid:0) ⊗ f − k K ⊗ f + k + f − k K ⊗ K ⊗ f + k − f − k K ⊗ f + k ⊗ − ⊗ f − k f + k ⊗ (cid:1)(cid:17) × (cid:16) ⊗ ⊗ − ⊗ f − k f + k ⊗ (cid:17) . The quasi-bialgebra ˆ Q can also be equipped with an R -matrix which is defined as(B.5) ˆ R = ˆ R e ⊗ e + ˆ R e ⊗ e + ˆ R e ⊗ e + ˆ R e ⊗ e , where ˆ R i = ρ ( K ) · N (cid:89) k =1 ˆ R i ( k ) , i = 0 , , (B.6) ˆ R = ρ ( K ) · N (cid:89) k =1 ˆ R k ) · ⊗ K , (B.7) ˆ R = ( − N i β ρ ( K ) · ⊗ K N − · (cid:16) N (cid:89) k =1 ˆ R k ) (cid:17) · K N ⊗ (B.8)with the Cartan part(B.9) ρ ( K ) = ( ⊗ + ω ⊗ + ⊗ ω − ω ⊗ ω ) , ω = ( e − i e ) K , and ˆ R k ) = ⊗ − f − k K ⊗ f + k , (B.10)ˆ R k ) = ⊗ − (1 + i) f − k K ⊗ f + k − (1 + i) f + k K ⊗ f − k + (1 − i) f − k f + k ⊗ + 2i f − k f + k ⊗ f − k f + k , ˆ R k ) = ⊗ + (1 + i) f − k K ⊗ f + k + (1 + i) f + k K ⊗ f − k + (1 + i) ⊗ f − k f + k − f − k f + k ⊗ f − k f + k , ˆ R k ) = ⊗ − f − k K ⊗ f + k + (i − ⊗ f − k f + k − (1 + i) f − k f + k ⊗ + 2 f − k f + k ⊗ f − k f + k . Remark B.1.
For N = 1 we haveˆΦ = ˆΦ k =1) , ˆΦ = ˆΦ k =1) · ⊗ K ⊗ , ˆΦ = β i ˆΦ k =1) · ⊗ K ⊗ (B.11)and they coincide with the components of the associator in [GR1, Sec. 7.4]. So, we havethe simple factorisation (B.3) of the nilpotent (off-diagonal) part of the associator into theproduct of N = 1 components. The Cartan part depends only on the parity of N. We alsonote that the components ˆΦ abc ( i ) commute with each other: ˆΦ abc ( i ) · ˆΦ abc ( j ) = ˆΦ abc ( j ) · ˆΦ abc ( i ) .For the ˆ R element, we note that (B.6) for N = 1 is ˆ R i = ρ ( K ) ˆ R i ( k =1) , and it is the and components of the universal R -matrix in [GR1, Sec. 7.7]. The Cartan part thus does notchange with N while the “off-diagonal” part is just the product of the N = 1 components.Also, ˆ R i ( k =1) , for i ∈ { , } , corresponds to N = 1 case: the component of the universal YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 61 R -matrix in [GR1, Sec. 7.7] is expressed as ˆ R = ρ ( K ) · ˆ R k =1) · ⊗ K while the componentis ˆ R = β i ρ ( K ) · ˆ R k =1) · K ⊗ that agrees with (B.7)-(B.8). Hence, we have again the simplefactorisation (B.7)-(B.8) of the nilpotent (off-diagonal) part of ˆ R into the product of N = 1components.The following lemma can be easily checked by a direct calculation. Lemma B.2.
The map ˆ∆ : ˆ Q → ˆ Q ⊗ ˆ Q defined in (B.1) is an algebra homomorphism. In order to show that ( ˆ Q , · , , ˆ∆ , ε, ˆΦ , ˆ R ) is a quasi-triangular quasi-bialgebra we start withthe braided monoidal category Rep S described in Appendix A and verify that ˆΦ and ˆ R canbe obtained via transport along a certain multiplicative functor from Rep S to Rep ˆ Q , seeSections B.2–B.5. From this it follows that ˆ Q is indeed a quasi-triangular quasi-bialgebra andthat Rep ˆ Q is braided equivalent to Rep S .B.2. A C -linear equivalence from Rep S to Rep ˆ Q . In this section we present a C -linearfunctor G : Rep S → Rep ˆ Q . Recall that ω U denotes the parity involution on the super-vectorspace U . For a given U ∈ Rep S , G ( U ) is the underlying vector space U with ˆ Q -action givenby, for u ∈ U , K .u := z .ω U ( u ) = ω U ( z .u ) , where z = e + i e , (B.12) f ± i .u := x ± i .u . Note that K acts as L , that is, e i ∈ ˆ Q ( i = 0 ,
1) acts as e i ∈ S . For a morphism f : U → V in Rep S we set G ( f ) = f . Proposition B.3.
The functor G is an equivalence of C -linear categories.Proof. Since the proof is very similar to the proof of [GR1, Prop. 5.2] we omit the details here.However, for later reference we introduce a functor H : Rep ˆ Q → Rep S which is inverse to G .By inverting the relation in (B.12) we define an involution map on an object V ∈ Rep ˆ Q as(B.13) ω V ( v ) := ( e − i e ) K .v , v ∈ V .
Then the S -module H ( V ) has V as the underlying super-vector space with the Z -gradingdefined by the eigenvalues of ω V as ω V ( v ) = ( − | v | v , for an eigenvector v of ω V . Moreover, L acts on H ( V ) by e − e ∈ ˆ Q and x ± i acts on H ( V ) by f ± i . (cid:3) Since Q and ˆ Q have the same algebra structure we in fact have shown a C -linear equivalenceof Rep S and Rep Q . B.3. G as multiplicative functor. In order to show that G is multiplicative, we define thefamily of isomorphismsΓ U,V : G ( U ⊗ Rep S V ) → G ( U ) ⊗ Rep ˆ Q G ( V ) ,u ⊗ v (cid:55)→ u ⊗ v + e .u ⊗ ( ξ − e .v , (B.14)where ξ is defined as(B.15) ξ = i N(N − / (cid:89) k =1 ξ k with ξ k = x + k + x − k . Invertibility is easy to see since (Γ
U,V ) = id U ⊗ V , which follows from ξ = e . Naturality isalso clear. It remains to prove the following lemma. Lemma B.4. Γ U,V is an intertwiner of ˆ Q -modules.Proof. We need to show that for all a ∈ ˆ Q , u ∈ U , v ∈ V we have(B.16) Γ U,V ( a ˆ . ( u ⊗ v )) = a. Γ U,V ( u ⊗ v ) , where the notation ˆ . emphasises that the action of ∆( a ) ∈ S ⊗ S on U ⊗ V is in Svect andinvolves parity signs.Since ∆ S and ˆ∆ are algebra maps, it is enough to verify this on the generators K , f ± k . If U / ∈ Rep S or V / ∈ Rep S , Γ U,V is just the identity, and the verification is straightforward.As an example for the sector case let a = f + k and assume N to be even. Then the twosides of the above identity are f + k . Γ U,V ( u ⊗ v ) = ( f + k ⊗ + K − ⊗ f + k − K − e ⊗ f + k e ) . ( e .u ⊗ ξ e .v )(B.17) = x + k e .u ⊗ ξ e .v + i( − | u | e .u ⊗ x + k ξ e .v , Γ U,V (cid:0) f + k ˆ . ( u ⊗ v ) (cid:1) = Γ U,V (cid:0) x + k ˆ . ( u ⊗ v ) (cid:1) = Γ U,V (cid:0) ( x + k ⊗ − i ⊗ x − k )ˆ . ( u ⊗ v ) (cid:1) = x + k e .u ⊗ ξ e .v − i( − | u | e .u ⊗ ξ x − k e .v . Since x ± k ξ e = ( − N − ξ x ∓ k e both sides are equal. The calculations for the other generatorsand for odd N are equally straightforward. (cid:3) Altogether, we have shown:
Proposition B.5.
With the isomorphisms Γ U,V as in (B.14) , the functor G : Rep S → Rep ˆ Q is multiplicative. B.4.
Transporting the associator.
In this section we transport the associator from
Rep S to Rep ˆ Q along the lines explained around the diagram (3.24). As Rep ˆ Q is the category of(finite-dimensional) ˆ Q -modules in vector spaces, the associator on Rep ˆ Q takes the form(B.18) α Rep ˆ Q U,V,W ( u ⊗ v ⊗ w ) = ˆΦ . ( u ⊗ v ⊗ w ) , YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 63 where u, v, w are elements of U, V, W ∈ Rep ˆ Q and for some ˆΦ ∈ ˆ Q ⊗ ˆ Q ⊗ ˆ Q . In order tocompute ˆΦ and to see that it agrees with (B.2)-(B.4), we choose U = V = W = ˆ Q and solvethe diagram (3.24) with F replaced by G .Recall the functor H inverse to G from the proof of Proposition B.3. Let us abbreviateˆ Q H := H ( ˆ Q ). The S -module structure on ˆ Q H is as explained in the proof of Proposition B.3.Commutativity of the diagram (3.24) applied for F = G and Θ = Γ then reads for q ∈ ˆ Q ⊗ H :(B.19) (cid:0) (Γ ˆ Q H , ˆ Q H ⊗ id) ◦ Γ ˆ Q H ⊗ ˆ Q H , ˆ Q H (cid:1)(cid:0) Λˆ . q (cid:1) = ˆΦ · (cid:2) (id ⊗ Γ ˆ Q H , ˆ Q H ) ◦ Γ ˆ Q H , ˆ Q H ⊗ ˆ Q H ( q ) (cid:3) , where Γ U,V is defined in (B.14) and the notation ˆ . emphasises that the action of Λ on ˆ Q H ⊗ ˆ Q H ⊗ ˆ Q H is as in Svect , i.e. involves parity signs. During the calculation it is importantto be careful with the parity signs. For example, Λˆ . ( ⊗ ⊗ ) can not be simplified to Λsince ∈ ˆ Q H is not of definite parity. Other examples are the actions of Γ ˆ Q H , ˆ Q H ⊗ ˆ Q H andΓ ˆ Q H ⊗ ˆ Q H , ˆ Q H , which are, for a, b, c ∈ ˆ Q H ,Γ ˆ Q H , ˆ Q H ⊗ ˆ Q H ( a ⊗ b ⊗ c ) = a ⊗ b ⊗ c + e .a ⊗ (cid:2) ∆ S (( ξ − e )ˆ . ( b ⊗ c ) (cid:3) , (B.20) Γ ˆ Q H ⊗ ˆ Q H , ˆ Q H ( a ⊗ b ⊗ c ) = a ⊗ b ⊗ c + ∆ S ( e ) . ( a ⊗ b ) ⊗ (cid:2) ( ξ − e . c (cid:3) . We use below the notations (in the spirit of (A.33))(B.21) ˆΦ abc = ˆΦ abc · ( e a ⊗ e b ⊗ e c )with ˆΦ abc as in (B.2) (it is understood that the ˆΦ abc not spelled out explicitly in (B.2) are setto ⊗ ). We will also use the similar convention for ˆΦ abc ( k ) and Λ abc ( k ) .In the following we will present the calculation of ˆΦ in sector . The other cases aresimilar. For brevity, we write ω instead of ω S for the parity involution in S . Using (B.20)and (B.21), the equality in (B.19) then reduces toˆΦ . (cid:0) S ⊗ ∆ S ( ξ )ˆ . ( ω N ⊗ id ⊗ id)( q ) (cid:1) = S ⊗ S ⊗ ξ . (Λ ˆ . q )(B.22) = ( ω N ⊗ ω N ⊗ id) (cid:0) S ⊗ S ⊗ ξ ˆ . (Λ ˆ . q ) (cid:1) , where the parity involutions appear upon converting the action “ . ” in vect to the action “ˆ . ”in Svect , using that ξ is odd if and only if N is odd. By setting(B.23) q = S ⊗ ∆ S ( ξ )ˆ . ( ω N ⊗ id ⊗ id)( p )for p ∈ ˆ Q ⊗ H (note that since ξ = e the above map is a bijection between p ’s and q ’s), weget ˆΦ . p = ( ω N ⊗ ω N ⊗ id) (cid:16) S ⊗ S ⊗ ξ ˆ · Λ ˆ · S ⊗ ∆ S ( ξ )ˆ . ( ω N ⊗ id ⊗ id)( p ) (cid:17) (B.24) = ( ω N ⊗ ω N ⊗ id) (cid:16) N (cid:89) k =1 (cid:0) S ⊗ S ⊗ ξ k ˆ · Λ k ) ˆ · S ⊗ ∆ S ( ξ k ) (cid:1) ˆ . ( ω N ⊗ id ⊗ id)( p ) (cid:17) , where for the second equality we used the factorisation in (A.7) and (B.15) and the fact thatΛ k ) are even elements, as well as the equality(B.25) ( S ⊗ S ⊗ ξ )ˆ · ( S ⊗ ∆ S ( ξ )) = N (cid:89) k =1 (cid:0) S ⊗ S ⊗ ξ k ˆ · S ⊗ ∆ S ( ξ k ) (cid:1) , which follows from reordering the parity-odd elements ξ k and ∆ S ( ξ k ). We now take ˆΦ inthe form (B.3) and check the above equality. First, we know that this equality holds in theN = 1 case [GR1, Sec. 7.4], which takes the form(B.26) ˆΦ k ) · ⊗ K ⊗ . p = ( ω ⊗ ω ⊗ id) (cid:0) S ⊗ S ⊗ ξ k ˆ · Λ k ) ˆ · S ⊗ ∆ S ( ξ k ) (cid:1) ˆ . (id ⊗ ω ⊗ id)( p )where k = 1 and one has to use (B.4) and the convention in (B.21). We also note that theequality (B.26) holds for general N and k . It can be rewritten as(B.27) ˆΦ k ) · ⊗ K ⊗ . (id ⊗ ω ⊗ id)( p ) = ( ω ⊗ ω ⊗ id) (cid:0) S ⊗ S ⊗ ξ k ˆ · Λ k ) ˆ · S ⊗ ∆ S ( ξ k ) (cid:1) ˆ . p . By choosing p = ⊗ and applying (B.27) multiple times on (B.24) it follows that (recall that ∈ ˆ Q H is not of definite parity)ˆΦ = (cid:0) ω N − ⊗ ω N − ⊗ id (cid:1) (cid:16) ˆΦ · ⊗ K ⊗ . (id ⊗ ω ⊗ id) (cid:16) ˆΦ · ⊗ K ⊗ . (id ⊗ ω ⊗ id)(B.28) · · · (cid:16) ˆΦ − · ⊗ K ⊗ . (id ⊗ ω ⊗ id) (cid:16) ˆΦ · ⊗ K ⊗ . ω N − ( ) ⊗ ⊗ (cid:17) (cid:17) · · · (cid:17) . By identifying ω with the element ( e − i e ) K , the above expression simplifies to (B.3). Notethat ω · e = K · e and hence ⊗ K ⊗ · ⊗ ω ⊗ · e ⊗ e ⊗ e = e ⊗ e ⊗ e .Altogether, we have shown: Proposition B.6.
The natural isomorphism α Rep ˆ Q from (B.18) with ˆΦ as in (B.2) – (B.4) defines an associator on Rep ˆ Q . With respect to this associator, the equivalence G : Rep S → Rep ˆ Q with multiplicative structure Γ U,V defined in (B.14) and id C : → G ( ) , is C -linearmonoidal equivalence. B.5.
Transporting the braiding.
We now similarly transport the braiding along the monoidalequivalence G : Rep S → Rep ˆ Q from Proposition B.6, recall the discussion above (3.25). Lemma B.7.
For all
U, V ∈ Rep ˆ Q , the isomorphisms τ U,V ◦ ˆ R with ˆ R from (B.5) are naturalin U, V and make the diagram (3.25) commute for all
M, N ∈ Rep S : (B.29) G ( M ⊗ Rep S N ) G ( ψ M,N ) (cid:47) (cid:47) Γ M,N (cid:15) (cid:15) G ( N ⊗ Rep S M ) Γ N,M (cid:15) (cid:15) G ( M ) ⊗ Rep ˆ Q G ( N ) τ G ( M ) , G ( N ) ◦ ˆ R (cid:47) (cid:47) G ( N ) ⊗ Rep ˆ Q G ( M ) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 65 Proof.
It is clear that for morphisms f : U → U (cid:48) and g : V → V (cid:48) in Rep ˆ Q we have τ U (cid:48) ,V (cid:48) ◦ ˆ R ◦ ( f ⊗ g ) = ( g ⊗ f ) ◦ τ U,V ◦ ˆ R . The lemma follows once we proved that (B.29)commutes, as it was already argued in the proof of Lemma A.9.Though the monoidal isomorphisms Γ are non-trivial only in the -sector the transportis non-trivial in all the four sectors due to the passage from Svect to vect . We recall firstthe braiding (A.46) in Rep S . The commutativity of the diagram (B.29) in the and sectors then corresponds to the equation τ U,V ◦ ˆ R i . ( u ⊗ v ) = τ s . v .U,V ◦ r i ˆ . ( u ⊗ v ) , i ∈ { , } , u ∈ U, v ∈ V, (B.30)where τ is the symmetric braiding in vector spaces. This equation for N = 1 case holds dueto [GR1], see also Remark B.1. Therefore, using the factorised expression for ˆ R i and r i ,the equation (B.30) has the unique solution as in (B.6), where the first factor ρ ( K ) defined in(B.9) is due to the braiding in super-vector spaces.Recall then the expression of the braiding in the and sectors of Rep S in (A.46).The commutativity of the diagram (B.29) corresponds thus to the equations τ U,V ◦ ˆ R . ( u ⊗ v ) = τ s . v .U,V ◦ r ˆ . (id ⊗ ω ) (cid:2) u ⊗ v (cid:3) , (B.31) τ U,V ◦ ˆ R . ( u ⊗ v ) = (id ⊗ L ξ ) ◦ τ s . v .U,V ◦ r ˆ . (id ⊗ ω ) (cid:2) u ⊗ ξ . v (cid:3) , with the solutions given in (B.7) and (B.8), correspondingly. The derivation of (B.7) isobvious. To derive (B.8), we note that the equation on ˆ R can be rewritten as(B.32) ˆ R . ( u ⊗ v ) = ( − N ρ ( K ) . (id ⊗ ω N − ) (cid:104) (id ⊗ ω ) (cid:0) ξ ⊗ · r · ⊗ ξ (cid:1) ˆ . ( ω N ( u ) ⊗ v ) (cid:105) . We use then the factorised expressions for ξ in (B.15) and for r in (A.9) together with theknown solution for N = 1, recall Remark B.1, and it finally gives the expression in (B.8). Bythe construction of ˆ R , the diagram (B.29) commutes and this finishes the proof. (cid:3) Since
Rep S is a braided monoidal category, the above lemma shows that ˆ R from (B.5) isthe R-matrix of ˆ Q and by the construction of the transport of the braiding the family τ U,V ◦ ˆ R defines the braiding in Rep ˆ Q . Altogether we have shown: Proposition B.8.
The functor G in Proposition B.6 is braided monoidal. B.6.
The quasi-bialgebra Q is the twisting of ˆ Q . We define the twist ζ – an invertibleelement in ˆ Q ⊗ ˆ Q :(B.33) ζ = e ⊗ + ζ · e ⊗ e + ζ · e ⊗ e , where(B.34) ζ = (cid:32) N (cid:89) k =1 ζ k ) (cid:33) · ⊗ K , ζ = (cid:32) N (cid:89) k =1 ζ k ) (cid:33) · ⊗ K N − and ζ k ) = ⊗ + (1 − i) ⊗ f + k f − k + (1 − i) f − k K ⊗ f + k + (1 + i) f + k K ⊗ f − k − f + k f − k ⊗ f + k f − k , (B.35) ζ k ) = ⊗ − (1 + i) ⊗ f + k f − k . Its inverse is(B.36) ζ − = e ⊗ + ( ζ ) − · e ⊗ e + ( ζ ) − · e ⊗ e with( ζ ) − = ⊗ K − · (cid:32) N (cid:89) k =1 ( ζ k ) ) − (cid:33) , ( ζ ) − = ⊗ K − (N − · (cid:32) N (cid:89) k =1 ( ζ k ) ) − (cid:33) , ( ζ k ) ) − = ⊗ + (1 + i) ⊗ f + k f − k − (1 − i) f − k K ⊗ f + k − (1 + i) f + k K ⊗ f − k − f + k f − k ⊗ f + k f − k , ( ζ k ) ) − = ⊗ − (1 − i) ⊗ f + k f − k . Since ( ε ⊗ id)( ζ ) = = (id ⊗ ε )( ζ ) and ζ is invertible, ζ is indeed a twist. This means(see e.g. [Dr] or [CP]) that it defines another quasi-triangular quasi-bialgebra ˆ Q ζ with thesame algebra structure and counit as in ˆ Q , while the new coproduct ∆ ζ , R -matrix R ζ andcoassociator Φ ζ are given by∆ ζ ( x ) = ζ ˆ∆( x ) ζ − , (B.37) R ζ = ζ ˆ R ζ − , (B.38) Φ ζ = ( ζ ⊗ ) · ( ˆ∆ ⊗ id)( ζ ) · ˆΦ · (id ⊗ ˆ∆)( ζ − ) · ( ⊗ ζ − ) . (B.39)The action of the twist defines a multiplicative structure on the identity functor between therepresentation categories of both quasi-bialgebras. In particular, the categories are braidedmonoidally equivalent. Proposition B.9.
We have ˆ Q ζ = Q , that is, Q as a quasi-triangular quasi-bialgebra definedin Section 3.1 is obtained from ˆ Q by twisting via ζ . To prove the proposition, we will need the following lemma.
Lemma B.10.
We have for a, b, c ∈ { , } and ≤ i (cid:54) = j ≤ N the equalities (cid:104) ˆΦ abc ( i ) , ˆΦ abc ( j ) (cid:105) = 0 , (cid:104) (∆ ⊗ id)( ζ i ) ) , ˆΦ j ) (cid:105) = 0 , (B.40) (cid:2) ζ ab ( i ) , ζ ab ( j ) (cid:3) = 0 , (cid:104) ˆΦ i ) , K ⊗ K ⊗ K (cid:105) = 0 and (B.41) (cid:2) ζ i ) ⊗ , (id ⊗ ∆)( ζ j ) ) (cid:3) = 0 . Proof.
This can be easily checked using the anti-commutator relations of Q from (3.3) andby recalling that Q and ˆ Q have the same algebra structure. (cid:3) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 67 Proof of Proposition B.9.
Using (B.37)-(B.39), we have to show that R ζ = R , Φ ζ = Φ and∆ ζ = ∆ with R , Φ and ∆ defined in (3.15), (3.11) and (3.8), respectively. For N = 1 weverified these equalities using computer algebra. The proof for general N will rely on theN = 1 case.We start with the coproduct and show the statement sector by sector. Clearly, ∆ ζ in (B.37)agrees with the coproduct in Q from (3.8) in the sectors and . Since ∆ ζ is an algebramap, it is enough to show the equality ∆ ζ = ∆ on the generators. Below we will use theabbreviation(B.42) ζ ab = ζ ab · e a ⊗ e b , a, b ∈ { , } , and similarly for ζ ab ( k ) . By applying Lemma B.10 and using that for N = 1 the statement istrue we get in sector , for x i ∈ { K , f + i , f − i } , ζ ˆ∆( x i ) ( ζ ) − = (cid:32) N (cid:89) k =1 ζ k ) (cid:33) ⊗ K · ˆ∆( x i ) · ⊗ K − (cid:32) N (cid:89) k =1 ( ζ k ) ) − (cid:33) (B.43) = (cid:32)(cid:89) k (cid:54) = i ζ k ) (cid:33) · ∆( x i ) · (cid:32)(cid:89) k (cid:54) = i ( ζ k ) ) − (cid:33) = ∆( x i ) , where we also used that ζ k ) commutes with ∆( x i ) for i (cid:54) = k . For the sector , we prove itsimilarly.The twisted R -matrix is R ζ = ζ ˆ R ζ − and we begin with the case N = 1. The directcalculation gives (for β = − R ζ = (cid:88) i,j,k =0 k i k ( i − j +1) − ij ( f − ) k K k + i ⊗ ( f +1 ) k K j × ( e ⊗ e + i − i − k e ⊗ e + i j e ⊗ e + i − i − k + j β e ⊗ e ) . We can rewrite R ζ in a more compact form involving the Z -parity ω = K ( e − i e ):(B.45) R ζ = (cid:88) n,m =0 β nm ρ n,m · ( ⊗ − f − ω ⊗ f +1 ) · e n ⊗ e m , with the Cartan factor ρ n,m as in (3.14). In the above calculation we used the identities ρ n, = ρ ( K ) · e n ⊗ e and ρ n, = ( − i) n K ⊗ · ρ ( K ) · e n ⊗ e , with ρ ( K ) introduced in (B.9). Forgeneral N, using the factorised form of ζ we obtain the expression in terms of N = 1 terms:(B.46) R ζ · e n ⊗ e m = β nm ρ n,m · N (cid:89) k =1 ( ⊗ − f − k ω ⊗ f + k ) · e n ⊗ e m , n, m ∈ Z , where now β = ( − N . This proves R = R ζ with the R -matrix for Q given in (3.15) whereone has to use ω − = ω . We now turn to the calculation of the twisted coassociator. We prove the equality Φ ζ = Φin the sector , the proof for the other sectors is similar. Recall the definition of thecoassociators Φ and ˆΦ in (3.11) and (B.2)–(B.4), respectively. We have to show that Φ ζ = ⊗ . The equation (B.39) reduces toΦ ζ · e ⊗ e ⊗ e = ζ ⊗ · ( ˆ∆ ⊗ id)( ζ ) · ˆΦ · (id ⊗ ˆ∆) (cid:0) ( ζ ) − (cid:1) , (B.47)where we used the notations in (B.21) and (B.42), and that ζ = ⊗ in sector . Usingcomputer algebra for N = 1 it turned out that Φ ζ = ⊗ . Taking this into account, theequality (B.47) for N = 1 is then equivalent to(id ⊗ ˆ∆)( ζ k ) ) · ⊗ K ⊗ = ζ k ) ⊗ · ⊗ K ⊗ · ( ˆ∆ ⊗ id)( ζ k ) ) · ˆΦ k ) , (B.48)where k = 1 and we used that K − e = K e . We note that the above equation also holds forgeneral k . Together with Lemma B.10 we get for general N: ζ ⊗ · ( ˆ∆ ⊗ id)( ζ ) · ˆΦ = (cid:0) N (cid:89) i =1 ζ i ) ⊗ (cid:1) · ⊗ K ⊗ · (cid:16) N (cid:89) i =1 ( ˆ∆ ⊗ id)( ζ i ) ) (cid:17) · ⊗ ⊗ K N − (B.49) × ( − K ) N − ⊗ K N − ⊗ · (cid:0) N (cid:89) i =1 ˆΦ i ) (cid:1) · K N − ⊗ K ⊗ = (cid:0) N − (cid:89) i =1 ζ i ) ⊗ (cid:1) · (cid:16) ζ ⊗ · ⊗ K ⊗ · ( ˆ∆ ⊗ id)( ζ ) · ˆΦ (cid:17) × (cid:16) N − (cid:89) i =1 ( ˆ∆ ⊗ id)( ζ i ) ) · ˆΦ i ) (cid:17) · ⊗ K N ⊗ K N − , where the first equality is by definition of ζ and ˆΦ , while we used the relations in (B.40)at the second equality. Next, by applying (B.48) for k = N to the previous expression we get ζ ⊗ · ( ˆ∆ ⊗ id)( ζ ) · ˆΦ = (cid:0) N − (cid:89) i =1 ζ i ) ⊗ (cid:1) · (cid:16) (id ⊗ ˆ∆)( ζ ) · ⊗ K ⊗ (cid:17) (B.50) × (cid:16) N − (cid:89) i =1 ( ˆ∆ ⊗ id)( ζ i ) ) · ˆΦ i ) (cid:17) · ⊗ K N ⊗ K N − . Finally using first the relation (B.41) and then doing the reordering of the terms in theproducts as in (B.49) and (B.50) for i = N − , . . . ,
1, we obtainRHS of (B.50) = (cid:16) N (cid:89) i =1 (id ⊗ ˆ∆)( ζ i ) ) (cid:17) · ⊗ K ⊗ · ⊗ K N ⊗ K N − (B.51) = (id ⊗ ˆ∆)( ζ ) . Hence, Φ ζ = ⊗ = Φ which completes the proof. (cid:3) YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 69 B.7.
Transporting the ribbon twist.
The ribbon twist in SF is described in Section 2.6.Following the same lines as in [GR1, Sec. 7.9] and using that ω B is given by the left-action ofΩ in (A.55) we can write, for M ∈ Rep Q , m ∈ M ,(B.52) θ M ( m ) = (cid:16) e (cid:89) k =1 ( + 2 f + k f − k ) − i β − e K N (cid:89) k =1 ( − f + k f − k ) (cid:17) .m . The action of the ribbon element v ∈ Q on M is by convention equals the inverse twist. Itfollows that(B.53) v = ( e − β i K e ) · N (cid:89) k =1 ( − f + k f − k ) . B.8.
Ribbon equivalence F : SF →
Rep Q . Taking Appendices A and B together (Propo-sitions A.8, A.10 and B.6, B.8, and B.9), we have established a ribbon equivalence F : SF →
Rep Q which is the composition(B.54) F = G ◦ D where the functor D is defined in (A.17) and (A.20), and the functor G is in Section B.2. Themonoidal structure of F is defined by(B.55) F U,V : F ( U ∗ V ) ∼ −→ F ( U ) ⊗ Rep Q F ( V )with(B.56) F U,V = (cid:0) ζ. ( − ) (cid:1) ◦ Γ D ( U ) , D ( V ) ◦ G (∆ U,V ) , where ζ , Γ U,V and ∆
U,V are given in (B.33), (B.14) and in Section A.3, respectively. Thismonoidal equivalence is by construction braided and ribbon and thus finally proves Lemma 3.5.
Appendix C. Proof of Proposition 3.10
In this Appendix, we use for brevity H instead of H (N), for the Hopf algebra introducedin (3.31) and (3.32), and we fix the basis in H :(C.1) H = span C { f u · · · f u m k v | ≤ m ≤ N , ≤ u < u < . . . < u m ≤ N , v ∈ Z } . The coproduct ∆ on the basis elements (C.1) is given by∆( f u · · · f u m k v ) = (cid:32) u m (cid:89) u = u ( f u ⊗ k + ⊗ f u ) (cid:33) · k v ⊗ k v (C.2) = (cid:16) (cid:88) l =( l ,...,l m ) ∈ Z m f l u · · · f l m u m ⊗ f − l u k l · · · f − l m u m k l m (cid:17) · k v ⊗ k v = (cid:16) (cid:88) l ∈ Z m ε l ( m ) f l u · · · f l m u m k v ⊗ f − l u · · · f − l m u m k v + | l | (cid:17) . where ε l ( m ) = ( − (cid:80) m − i =1 l i (( m − i ) − (cid:80) mj = i +1 l j ) and | l | = (cid:80) i l i ∈ N . For the Drinfeld double construction we need the dual Hopf algebra(C.3) ( H op ) ∗ = (cid:0) H ∗ , µ H ∗ = ∆ ∗ , H ∗ = ε ∗ , ∆ H ∗ = (cid:0) µ op (cid:1) ∗ , ε H ∗ = η ∗ , S H ∗ = ( S − ) ∗ (cid:1) , where µ op is the opposite multiplication, and we used the standard isomorphism of vectorspaces ( H ⊗ H ) ∗ ∼ = H ∗ ⊗ H ∗ , so ∆ H ∗ ( ϕ )( a ⊗ b ) = ϕ ( ba ) and ( ϕ · ψ )( a ) = ( ϕ ⊗ ψ ) (cid:0) ∆( a ) (cid:1) .Using (C.2), we note that the canonical duals of the generators of H do not generate ( H op ) ∗ since e.g. f ∗ i · f ∗ j = ( f ∗ i ⊗ f ∗ j ) ◦ ∆ = 0. We then instead use the linear forms κ = ∗ − k ∗ , ϕ i = ( f i k ) ∗ − f ∗ i . (C.4) Lemma C.1.
The algebra ( H op ) ∗ is generated by κ and ϕ i , ≤ i ≤ N , with the definingrelations κ = H ∗ , { ϕ i , ϕ j } = 0 , { ϕ i , κ } = 0 , (C.5) and the set (C.6) { ϕ i . . . ϕ i m κ j | ≤ m ≤ N , ≤ i < i < . . . < i m ≤ N , j ∈ Z } forms a basis. The Hopf-algebra structure of ( H op ) ∗ is ∆ H ∗ ( κ ) = κ ⊗ κ , ∆ H ∗ ( ϕ i ) = ϕ i ⊗ H ∗ + κ ⊗ ϕ i , (C.7) ε H ∗ ( κ ) = 1 , ε H ∗ ( ϕ i ) = 0 ,S H ∗ ( κ ) = κ , S H ∗ ( ϕ i ) = ϕ i κ . Proof.
We begin with the defining relations. The first one from (C.5) is straightforward tocheck using H ∗ = ε = ∗ + k ∗ . The next one follows from the calculation: ϕ i ϕ j = (cid:0) (( f i k ) ∗ − f ∗ i ) ⊗ (( f j k ) ∗ − f ∗ j ) (cid:1) ◦ ∆ = − (( f i k ) ∗ ⊗ f ∗ j + f ∗ i ⊗ ( f j k ) ∗ ) ◦ ∆(C.8) = ( f i f j k ) ∗ + ( f i f j ) ∗ = − (( f j f i k ) ∗ + ( f j f i ) ∗ ) = − ϕ j ϕ i , where we used the coproduct formula (C.2), and similarly for the third relation.Now, we construct a basis in ( H op ) ∗ . Using induction, one can check the relations( − m ϕ i . . . ϕ i m = ( f i . . . f i m ) ∗ + ( − m ( f i . . . f i m k ) ∗ (C.9) ϕ i . . . ϕ i m κ = ( f i . . . f i m ) ∗ − ( − m ( f i . . . f i m k ) ∗ . Since the set of elements ( f u . . . f u m k v ) ∗ , with indices as in (C.1), forms a basis in ( H op ) ∗ ,we conclude from the above relations that the set (C.6) is also a basis in ( H op ) ∗ .For the coproduct we have by definition ∆ H ∗ ( ϕ )( a ⊗ b ) = ϕ ( ba ). This can be written as(C.10) ∆ H ∗ ( ϕ i ) = (( f i k ) ∗ − f ∗ i ) ◦ µ op . For f ∗ i ◦ µ op ( a ⊗ b ) we should find such pairs ( a, b ) that f ∗ i ( ba ) is non-zero. There are foursuch pairs and we get(C.11) f ∗ i ◦ µ op = f ∗ i ⊗ ∗ + ∗ ⊗ f ∗ i − ( f i k ) ∗ ⊗ k ∗ + k ∗ ⊗ f i k ∗ , YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 71 and we get a similar expression for ( f i k ) ∗ ◦ µ op . Combining the eight total terms in (C.10)we get the coproduct in (C.7). For the antipode we have S H ∗ ( ϕ i ) = (( f i k ) ∗ − f ∗ i ) ◦ S − = f ∗ i + ( f i k ) ∗ = ϕ i κ . (C.12)Calculations for the coproduct and antipode for κ , together with the counit, are straightfor-ward. This finally proves the lemma. (cid:3) Next we present D ( H ), the Drinfeld double of H , by following the conventions in [Kas,Chapter IX]. As a vector space D ( H ) is H ∗ ⊗ H . It has a Hopf algebra structure with unit H ∗ ⊗ and counit ε ( φ ⊗ a ) = ε H ∗ ( φ ) ε ( a ), with the multiplication defined as(C.13) ( φ ⊗ a ) · ( ψ ⊗ b ) = (cid:88) ( a ) φ · ψ ( S − ( a (cid:48)(cid:48)(cid:48) )( − ) a (cid:48) ) ⊗ a (cid:48)(cid:48) b , where ψ ( S − ( a (cid:48)(cid:48)(cid:48) )( − ) a (cid:48) ) stands for the map ( x (cid:55)→ ψ ( S − ( a (cid:48)(cid:48)(cid:48) ) xa (cid:48) )). The coproduct and theantipode are given by(C.14) ∆( φ ⊗ a ) = (cid:88) ( φ ) , ( a ) ( φ (cid:48) ⊗ a (cid:48) ) ⊗ ( φ (cid:48)(cid:48) ⊗ a (cid:48)(cid:48) ) , S ( φ ⊗ a ) = (cid:0) H ∗ ⊗ S ( a ) (cid:1) · (cid:0) S H ∗ ( φ ) ⊗ (cid:1) . We will identify an element a in H with H ∗ ⊗ a and an element φ in ( H op ) ∗ with φ ⊗ , inparticular, we write φa = ( φ ⊗ ) · ( H ∗ ⊗ a ). In this notation, the basis of D ( H ) is(C.15) { ϕ i · · · ϕ i m κ u f j · · · f j n k v | ≤ m, n ≤ N , u, v ∈ Z } , where as usual we assume that 1 ≤ i < i < . . . < i m ≤ N and similarly for the j ’s indices.It is well-known that D ( H ) is quasi-triangular: for any basis { b i | i ∈ I } in H the R -matrix isgiven by (using the convention above we interpret b i , b ∗ i ∈ D ( H ))(C.16) R D = (cid:88) i ∈ I b i ⊗ b ∗ i . Proposition C.2. D ( H ) is generated by k, κ, f i , ϕ j with defining relations (3.31) , (C.5) and kκ = κk , ϕ i k = − kϕ i , f i κ = − κf i , [ f i , ϕ j ] = δ i,j ( κ − k ) . (C.17) The Hopf algebra structure is given by (3.32) and (C.7) . The R -matrix for D ( H ) is given by R D = 12 ( ⊗ + ⊗ κ + k ⊗ − k ⊗ κ ) (cid:16) (cid:88) ≤ m ≤ N ,i <...
Using (C.13) it is straightforward to show the relations in (C.17). For example, weget the last equality since (cid:80) ( f ) f (cid:48) i ⊗ f (cid:48)(cid:48) i ⊗ f (cid:48)(cid:48)(cid:48) i = f i ⊗ k ⊗ k + ⊗ f i ⊗ k + ⊗ ⊗ f i and then f i ϕ j = ϕ j ( k − f i ) k + ϕ j ( k − ) f i + ϕ j ( f i k − ) (C.19) = − δ i,j ( ∗ + k ∗ ) k + ( − f ∗ j + ( f j k ) ∗ ) f i + δ i,j ( ∗ − k ∗ ) . The coproduct, counit, and antipode on the generators were already computed.
For the R -matrix in (C.16), we fix the basis { b i } as in (C.1). Then R D = (cid:88) ≤ m ≤ N , ≤ i <...
For any N ≥ , the linear map Ψ( ϕ i · · · ϕ i m κ u f j · · · f j n k v ) = ( − nu i n ( n − m f + i · · · f + i m f − j · · · f − j n ω u + ω v + n − , (C.23) with ω ± from (3.8) , defines an isomorphism of C -algebras D (cid:0) H (N) (cid:1) ∼ −→ Q (N , β ) .Furthermore, for even N this map defines an isomorphism of Hopf algebras between D (cid:0) H (N) (cid:1) and Q (N , ± .Moreover, for β = 1 the R -matrix in Q defined in (3.15) and the image of R D under Ψ ⊗ Ψ coincide, i.e. Ψ is an isomorphism of quasi-triangular Hopf algebras.Proof. In order to show that Ψ is an algebra map it is enough to verify the relations in(3.31), (C.5) and (C.17) for the the image of the generators of D ( H ) under Ψ. Since we haveΨ( ϕ i ) = 2 f + i , Ψ( κ ) = ω + , Ψ( f i ) = f − i ω − , Ψ( k ) = ω − , this is an easy check. For example,Ψ( f i )Ψ( ϕ j ) = ω − ( δ i,j ( K − ) + 2 f + j f − i ) = δ i,j ( ω + − ω − ) + 2 f + j f − i ω − (C.24) = δ i,j (Ψ( κ ) − Ψ( k )) + Ψ( ϕ j )Ψ( f i )Since Ψ( κ + i k ) = K and Ψ( f j κ ) = f − j the image of Ψ clearly generates Q . Moreover, sincethe dimensions of D ( H ) and Q agree Ψ is bijective.To show that Ψ is a coalgebra map we use (3.32) and (C.7), and check(Ψ ⊗ Ψ)(∆( ϕ i )) = (Ψ ⊗ Ψ)( ϕ i ⊗ + κ ⊗ ϕ i ) = 2 f + i ⊗ + 2 ω + ⊗ f + i = ∆(2 f + i ) = ∆(Ψ( ϕ i )) , (C.25)(Ψ ⊗ Ψ)(∆( f i )) = (Ψ ⊗ Ψ)( f i ⊗ k + ⊗ f i ) = f − i ω − ⊗ ω − + ⊗ f − i ω − = ∆( f − i ω − ) = ∆(Ψ( f i )) , (Ψ ⊗ Ψ)(∆( κ )) = (Ψ ⊗ Ψ)( κ ⊗ κ ) = ω + ⊗ ω + ( ∗ ) = ∆( ω + ) = ∆(Ψ( κ )) . YMPLECTIC FERMION RIBBON Q-HOPF ALGEBRA AND SL (2 , Z )-ACTION ON ITS CENTRE 73 Note, the equality ( ∗ ) is in Q and it holds only if N is even. For the antipode, we haveΨ( S ( ϕ i )) = Ψ( ϕ i κ ) = 2 f + i ω + = S (2 f + i ) = S (Ψ( ϕ i )) , (C.26) Ψ( S ( f i )) = − Ψ( f i k ) = − f − i = S ( f − i ω − ) = S (Ψ( f i ))Ψ( S ( κ )) = Ψ( κ ) = ω + = S ( ω + ) = S (Ψ( κ )) . Recall the R -matrices in Q and D ( H ) defined in (3.15) and (C.18), respectively. The imageof R D under Ψ ⊗ Ψ leads to (using β = 1)(Ψ ⊗ Ψ)( R D ) = 12 ( ⊗ + ⊗ ω + + ω − ⊗ − ω − ⊗ ω − ) (cid:16) (cid:88) ( − m f − i ω − . . . f − i m ω − ⊗ f + i . . . f + i m (cid:17) = (cid:88) u,v =0 ρ u,v · N (cid:89) k =1 ( ⊗ − f − k ω − ⊗ f + k ) · e u ⊗ e v = R , where the sum is taken over 1 ≤ i < . . . < i m ≤ N with 0 ≤ m ≤ N. This calculation finishesthe proof of the statement. (cid:3)
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