Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

ZMP-HH/11-9 Hamburger Beitr¨ age zur Mathematik Nr. 4 08 June 2011 MODULAR INV ARIANT FR OBENIUS ALGEBRAS FR OM RIBBON HOPF ALGEBRA A UTOMORPHISMS J ¨ urgen F uc hs a , Christoph Sc hw eigert b , Carl Stigner a a T e o r etisk fysik, Karlstads Universitet Universitetsgatan 21, S – 651 88 Karlstad b Or g a nisationseinheit Mathematik, Universit¨ at Hambur g Ber eich Algebr a und Zahlenthe orie Bundesstr aße 55, D – 20 146 Hambur g Abstract F or any ﬁnite-dimensional f actorizable ribb on Hopf algebra H and an y ribb on auto morphism of H , w e establish the existence of the fo llowing structure: an H -bimodule F ω and a bimo dule morphism Z ω from Lyubashe nko’s Hopf algebra ob ject K for the bimo dule category to F ω . This morphism is in v a r ian t under the natural action of the mapping class g roup of the o ne-punctured torus on the space of bimo dule morphisms f r om K to F ω . W e further sho w that the bimo dule F ω can be endow ed with a natural structure o f a commutativ e symmetric F rob enius algebra in the monoidal catego ry of H -bimo dules, and that it is a sp ecial F rob enius algebra iﬀ H is semisimple. The bimo dules K and F ω can b oth b e c haracterized as co ends of suitable bifunctors. The morphism Z ω is obtained by a pplying a mono dromy op eration to t he copro duct of F ω ; a similar construction for the pro duct of F ω exists as w ell. Our results are motiv ated b y the quest to understand the bulk state space and the bulk partition function in t wo-dimens ional conformal ﬁeld theories with c hiral alg ebras t hat are no t necessarily semisimple. 1 1 In tro ductio n One remark able feature of complex Hopf algebras is their intimate connection with lo w-dimen- sional t o p ology , including in v arian ts of knots, links and three - manifolds. These connections are part icularly w ell understo o d for semisimple Hopf algebras. The represen tatio n category of a semisimple factorizable ﬁnite-dimensional (we a k) Hopf algebra is a mo dular tensor category [NTV] and th us allows one to construct a three-dimens ional top ological ﬁeld theory . As a consequenc e, it provides ﬁnite-dimensional pro jectiv e represen tations o f mapping class groups of punctured surfaces. It has b een sho wn b y Lyubashenk o [Ly1, Ly3] that suc h represen tations of mapping class groups can be constructed for non-semisimple factorizable Hopf algebras H as w ell. This construction is in f a ct purely categorical, in t he sense that it o nly uses the r epresen ta t ion category as an abstract ribb o n category with certain non-degeneracy prop erties. In the presen t pap er w e apply this construction not to the category of left H -mo dules, but rather to the category of H - bimo dules . T o this end w e endow this category H -Bimo d with the structure of a monoidal category using the copro duct of H (rather than b y taking the tensor pro duct ⊗ H o ve r H as an associative algebra). With this tensor pro duct, the category H -Bimo d can b e endo w ed with further structure suc h that it b ecomes a so ve r eign braided mono ida l category . F or o ur presen t purp oses we restrict to the case that the punctured surface in question is a one-punctured to rus. Th us in the absence o f punctures the mapping class group is the mo dular group SL(2 , Z ); if punctures are presen t, then the ma pping class group has additional generators giv en b y Dehn t wists around the punctures and b y bra iding homeomorphisms [Ly1, Sect. 4.3]. W e denote the mapping class group of the o ne-punctured torus by Γ 1;1 . Sp ecializing the results of [Ly1], w e obtain a Hopf algebra ob ject K in the monoidal cat ego ry H -Bimo d. F or an y H -bimo dule X the vector space Hom H | H ( K , X ) of bimo dule morphisms then carries a pro jectiv e repres entation of Γ 1;1 . The main result of this pap er is the follow ing assertion: Theorem L et H b e a (not ne c essarily sem isimple) ﬁnite-dimens ional fa ctorizable ribb on Hopf algebr a over an algebr aic al ly cl o se d ﬁ e ld of char a cteristic zer o, and le t ω : H → H b e an automorphism of H as a ribb on Hopf algebr a. Then ther e is an obje ct F ω in the c ate gory H -Bimo d and a morphism Z ω ∈ Hom H | H ( K , F ω ) that is invariant under the natur al action [Ly1] of the mapping class gr oup Γ 1;1 on Hom H | H ( K , F ω ) . The considerations leading to this result are inspired by structure one hop es to encoun ter in certain tw o-dimensional conformal ﬁeld theories that are ba sed on non-semisimple represen- tation categories. More informatio n ab out this motiv a t io n can b e found in app endix B; here it suﬃces to remark that F ω is a candidate for what in confor ma l ﬁeld theory is called the al- gebr a of bulk ﬁelds , a nd tha t the morphism Z ω is a candidate for a mo dular in v arian t p artition function . Suc h a partitio n function should a lso enjo y in tegrality prop erties. As w e will show elsewhere [FSS], for ω = id H the relev a n t in tegers are closely related to the Cartan matrix of the algebra H . T o ar r iv e a t our result w e sho w in fact ﬁrst that the ob ject F ω actually carries a lot more natural structure: F ω is a comm utative symmetric F rob enius algebra in H -Bimo d. F urthermore, 2 the F rob enius algebra F ω is a sp ecial 1 F rob enius algebra if and only if the Hopf algebra H is semisimple. A F rob enius algebra carries a natural coalgebra structure; the in v ariant morphism Z ω is obtained b y applying a mono dromy op eration to the copro duct of F ω . This pap er is organized a s f o llo ws. In Section 2 w e in tro duce the relev an t structure of a monoidal category on H -Bimo d and construct, fo r the case ω = id H , the bimo dule F = F id H as a F ro b enius algebra in H -Bimo d. In Section 3 we endo w the mono idal category H -Bimo d with a natural braiding and sho w that with resp ect to this braiding the F rob enius algebra F is comm utativ e. In Section 4 it is established that F is symmetric, has trivial t wist, and is sp ecial iﬀ H is semisimple. Mo dular in v ariance of Z id H is prov en in Section 5. Section 6 is ﬁnally dev oted to the case of a general ribb on Hopf algebra automorphism ω of H , whic h can actually be treated b y mo dest mo diﬁcations of the arguments of Sections 2 – 5. In app endix A w e gather some notions from category theory and explain how the bimo dules K and F ω can b e characterize d as co ends of suitable bifunctors. The latter sho ws t ha t the ob j ects in our constructions are canonically asso ciated with the category of H -bimo dules as an abstract category . App endix B contains some motiv ation from (logarithmic) conformal ﬁeld theory . 2 A F rob enius algebra in the b imo dule category 2.1 Finite-dimensional ribb on Hopf algebras In this section w e collect some basic deﬁnitions and nota t io n f or Hopf algebras and recall that ﬁnite-dimensional Ho pf algebras admit a canonical F rob enius alg ebra structure. Throughout this pa p er, k is an algebraically closed ﬁeld of c ha racteristic zero and, unless noted otherwise, H is a ﬁnite-dimensional factor izable ribb o n Hopf algebra ov er k . W e denote b y m , η , ∆, ε and s t he pro duct, unit, copro duct, counit and antipo de of the Hopf algebra H . There exis t plen ty of factorizable ribb o n Hopf algebras (see e.g. [Bu]). F or instance, the Drinfeld double o f a ﬁnite-dimensional Hopf algebra K is factorizable ribb on provided that [KaR, Thm. 3] a certain conditio n for the square o f the an tip o de of K is satisﬁed. Let us recall what it means that a Hopf algebra is factorizable ribb on. Deﬁnition 2.1. (a) A Hopf alg ebra H ≡ ( H , m, η , ∆ , ε, s ) is called quasitriangular iﬀ it is endow ed with an in v ertible elemen t R ∈ H ⊗ H (called the R-matrix ) that in tertwin es the copro duct and opp osite copro duct, i.e. ∆ op = ad R ◦ ∆, and satisﬁes (∆ ⊗ id H ) ◦ R = R 13 · R 23 and ( id H ⊗ ∆) ◦ R = R 13 · R 12 . (2.1) (b) The mono dr omy matrix Q ∈ H ⊗ H of a quasitriangular Hopf algebra ( H, R ) is t he in v ertible elemen t Q := R 21 · R ≡ ( m ⊗ m ) ◦ ( id H ⊗ τ H,H ⊗ id H ) ◦ (( τ H,H ◦ R ) ⊗ R ) . (2.2) (c) A quasitriangula r Hopf algebra ( H , R ) is called a ribb on Hopf algebra iﬀ it is endo w ed with a cen tral in v ertible elemen t v ∈ H , called the ribb on element , that satisﬁes s ◦ v = v , ε ◦ v = 1 and ∆ ◦ v = ( v ⊗ v ) · Q − 1 . 1 A F ro b enius is called sp ecial iﬀ, up to no n-zero scalars, the counit is a left in verse of the unit and the copro duct is a right inverse of the pro duct, see Def. 4.6. 3 (d) A quasitriangular Hopf algebra ( H , R ) is called factorizable iﬀ the mono drom y matrix can b e written as Q = P ℓ h ℓ ⊗ k ℓ with { h ℓ } and { k ℓ } t w o v ector space bases o f H . Here and b elow , the sym b ol ⊗ denotes the tensor pro duct ov er k , a nd for v ector spaces V and W the linear map τ V , W : V ⊗ W ≃ → W ⊗ V is the ﬂip map whic h exc hanges the t wo tensor factors. Also, w e canonically iden tify H with Ho m k ( k , H ) and think of elemen ts o f (tensor pro ducts o v er k of ) H and H ∗ = Hom k ( H , k ) as (m ulti)linear maps. This has e.g. the adv antage that man y o f our considerations still apply directly in the situation that H is a Hopf algebra, with adequate additional structure and prop erties, in an ar bit r a ry k -linear ribb on category instead o f V ect k . V arious prop erties of the R-matr ix and of the r ibb on elemen t, as w ell as of some further distinguished elemen t s of H , will b e recalled la ter o n. No t e that we do not a ssume t he Hopf algebra H to b e semisimple; in particular, the ribb on elemen t do es not need to b e semisimple. W e also need a few f ur t her ing redien ts that a re a v ailable f or general ﬁnite-dimensional Hopf algebras, without assuming quasitriangularit y , in particular the not io ns of (co)integrals and of a F rob enius structure for Hopf a lg ebras. Deﬁnition 2.2. A left inte gr al of a Hopf algebra H is a morphism of left H - mo dules from the trivial H -module ( k , ε ) to the r egula r H - mo dule ( H , m ), i.e. an elemen t Λ ∈ H satisfying m ◦ ( id H ⊗ Λ) = Λ ◦ ε . A right c ointe gr al of H is a morphism of right H - como dules from ( k , η ) to ( H , ∆), i.e. an elemen t λ ∈ H ∗ satisfying ( λ ⊗ id H ) ◦ ∆ = η ◦ λ . Righ t in tegrals and left coin tegrals are deﬁned a nalogously . Recall [LS] that for a ﬁnite-dimensional Hopf k -algebra the antipo de is inv ertible and that H has, up to normalizatio n, a unique non-zero left integral Λ ∈ H and a unique non-zero right coin tegral λ ∈ H ∗ . The num b er λ ◦ Λ ∈ k is inv ertible. A factorizable ribb on Hopf a lgebra is unimo dular [Ra3, Prop. 3(c)], i.e. the left in tegral Λ is also a right in tegr a l, implying that s ◦ Λ = Λ. The in tegral and the cointegral allo w one to endo w Hopf k -algebras with more algebraic structure. The f o llo wing characterization of F rob enius a lgebras will b e conv enient. Deﬁnition 2.3. A F r ob enius algebr a A in V ect k is a vec t o r space A together with (bi)linear maps m A , η A , ∆ A and ε A suc h tha t ( A, m A , η A ) is an (asso ciative , unital) alg ebra, ( A, ∆ A , ε A ) is a (coasso ciative, counita l) coalgebra and ( m A ⊗ id A ) ◦ ( id A ⊗ ∆ A ) = ∆ A ⊗ m A = ( id A ⊗ m A ) ◦ (∆ A ⊗ id A ) , (2.3) i.e. the copro duct ∆ A is a morphism of A -bimo dules. W e hav e Lemma 2.4. A ﬁnite-dimensiona l Hopf k -algebr a ( H, m, η , ∆ , ε, s ) c arries a c anonic al struc- tur e of a F r o b enius algebr a A , with the same algebr a structu r e on A = H , an d with F r ob enius c o p r o duct and F r ob enius c ounit given by ∆ A = ( m ⊗ s ) ◦ ( id A ⊗ (∆ ◦ Λ)) and ε A = ( λ ◦ Λ) − 1 λ . (2.4) 4 This actually holds more generally for ﬁnitely generated pro jectiv e Hopf algebras ov er com- m utative rings (see e.g. [P a, Ka S]), as w ell as for a n y Hopf algebra in a n additiv e ribb on cate- gory C that has an inv ertible antipo de and a left in tegra l Λ ∈ Hom( 1 , H ) and right coin tegra l λ ∈ Hom( H , 1 ) suc h that λ ◦ Λ ∈ End C ( 1 ) is in v ertible (see e.g. app endix A.2 of [F Sc]). The F rob enius alg ebra structure given by (2.4) is unique up to rescaling the integral Λ by an in vertible scalar. In the seq uel, for a giv en choice of (non-zero) Λ, we choose the cointe g ral λ suc h that λ ◦ Λ = 1. 2.2 H -Bimo d as a monoidal category Our fo cus in this pap er is on natural structures on a distinguished H -bimo dule, the coregular bimo dule to b e described b elow. T o form ulate these w e need to endow the ab elian category H -Bimo d of H -bimo dules with t he structure of a sov ereign braided monoidal category . The ob j ects of the k -linear a b elian category H -Bimo d of bimo dules o v er a Ho pf k -algebra H are triples ( X , ρ, ρ ) suc h that ( X , ρ ) is a left H -mo dule and ( X , ρ ) is a rig h t H - mo dule and the left and righ t actions of H comm ute, ρ ◦ ( id H ⊗ ρ ) = ρ ◦ ( ρ ⊗ id H ). Morphisms are k -linear maps comm uting with b oth actions. W e denote the morphism spaces of H -Mo d and H - Bimo d b y Hom H ( − , − ) and Ho m H | H ( − , − ), resp ectiv ely , while Hom( − , − ) ≡ Hom k ( − , − ) is reserv ed for k -linear maps. Just lik e the bimo dules ov er a ny unital asso ciativ e alg ebra, H -Bimo d car r ies a monoidal structure for whic h the tensor pro duct is the one o ve r H , for which the v ector space underlying a tensor pro duct bimo dule X ⊗ H Y is a non-trivial quotien t o f the v ector space tensor pro duct X ⊗ Y ≡ X ⊗ k Y . But fo r our purp oses, w e need instead a diﬀeren t monoida l structure o n H -Bimo d for whic h also the coalgebra structure of H is relev an t. This is obtained by pulling bac k the na t ural H ⊗ H -bimo dule structure on X ⊗ Y along the copro duct t o the structure of an H -bimo dule. Th us if ( X , ρ X , ρ X ) and ( Y , ρ Y , ρ Y ) are H -bimo dules, then their tensor pro duct is X ⊗ Y together with t he left and rig h t a ctions ρ X ⊗ Y := ( ρ X ⊗ ρ Y ) ◦ ( id H ⊗ τ H,X ⊗ id Y ) ◦ (∆ ⊗ id X ⊗ id Y ) and ρ X ⊗ Y := ( ρ X ⊗ ρ Y ) ◦ ( id X ⊗ τ Y , H ⊗ id H ) ◦ ( id X ⊗ id Y ⊗ ∆) (2.5) of H . The monoidal unit for this tensor pro duct is the one-dimensional v ector space k with b oth left and r ig h t H - action given by the counit, 1 H -Bimo d = ( k , ε, ε ). Ob viously , (2.5) is just the standard tensor pro duct ( X , ρ X ) ⊗ H -Mo d ( Y , ρ Y ) = ( X ⊗ Y , ( ρ X ⊗ ρ Y ) ◦ ( id H ⊗ τ H,X ⊗ id Y ) ◦ (∆ ⊗ id X ⊗ id Y ) ) (2.6) of the category H -Mo d of left H - mo dules together with the corresponding tensor pro duct of the category of right H -mo dules. F or b o th monoidal structures the ground ﬁeld k , endow ed with a left, resp ective ly right, action via the counit, is the monoidal unit. If H is a ribb on Hopf algebra, then (see e.g. Section XIV.6 of [K a]) H -Mo d carries the structure o f a ribb on category . Analogous further structure o n H -Bimo d will b ecome relev ant later o n, and w e will in tro duce it in due course: a braiding on H -Bimo d in Section 3, and left and right dualities and a twis t in Section 4. 5 2.3 The coregular bimo dule W e now iden tify an ob ject of the monoidal category H - Bimo d that is distinguished b y the fact (see App endix A.1) that it can b e determined, up to unique isomorphism, b y a univ ersal prop erty for m ulated in H -Bimo d, and thus ma y b e though t of as b eing canonically asso ciated with H -Bimo d as a rigid mo no idal category . Afterwards w e will endow this ob ject F with the structure of a F rob enius algebra in the monoidal category deﬁned b y the tensor pro duct (2.5) . As a v ector space, F is the dual H ∗ of H . Deﬁnition 2.5. The c o r e gular bimo dule F ∈ H -Bimo d is the v ector space H ∗ endo w ed with the dual of the regular left and righ t actions of H on itself. Explicitly , F = ( H ∗ , ρ F , ρ F ) , (2.7) with ρ F ∈ Hom( H ⊗ H ∗ , H ∗ ) and ρ F ∈ Hom( H ∗ ⊗ H , H ∗ ) giv en b y ρ F := ( d H ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ m ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ s ⊗ b H ) ◦ τ H,H ∗ and ρ F := ( d H ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ m ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ id H ⊗ τ H ∗ ,H ) ◦ ( id H ∗ ⊗ b H ⊗ s − 1 ) . (2.8) Expressions in v o lving maps lik e ρ F and ρ F tend to b ecome un wieldy , at least for the presen t authors. It is therefore conv enien t to resort to a pictorial description. W e depict the structure maps of the Hopf alg ebra H as 2 m = H H H η = H ∆ = H H H ε = H s = H H s − 1 = H H (2.9) the in tegral and coin tegral as Λ = H λ = 000 000 000 111 111 111 H (2.10) and the ev a luation and co ev aluation maps, dual maps, and ﬂip maps o f V ect k as d V = V ∗ V b V = V V ∗ f ∨ = W ∗ f V ∗ τ V , W = V W W V = τ − 1 W,V = V W W V (2.11) 2 It is w o rth s tr essing that these pictures refer to the catego ry V ect k of ﬁnite-dimensiona l k -vector spaces . Later on, we will o ccasio nally also work with pictur es for morphisms in more genera l monoidal ca teg ories C ; to av oid confusion we will mar k picture s of the latter t y pe with the symbol C . 6 The left-p ointing arrows in the pictures for the ev a lua tion and co ev aluatio n indicate that they refer to the right dualit y of V ect k . The ev aluation and co ev aluation ˜ d and ˜ b for the left dualit y of V ect k are analogously dra wn with arro ws p ointing to the right. Also, for b etter readabilit y w e indicate the ﬂip by either an o ver- o r underbraiding, ev en t ho ugh in the presen t con text of the symmetric monoidal category V ect k b oth of them describ e the same map. In this graphical description the left a nd righ t actio ns (2.8) o f H on F are giv en b y H ρ F H ∗ H ∗ ρ F H = s H H ∗ H ∗ H s − 1 (2.12) Let us also men tio n that the F r ob enius map Ψ : H → H ∗ and its inv erse Ψ − 1 : H ∗ → H are giv en b y Ψ( h ) = λ ↼ s ( h ) and Ψ − 1 ( p ) = Λ ↼ p , resp ectiv ely (see e.g. [CW3 ]), i.e. Ψ = 00 00 11 11 H s λ H ∗ and Ψ − 1 = 000 000 111 111 H ∗ Λ H (2.13) The statemen t that H is a F rob enius a lgebra (see Lemma 2.4) is equiv alent to the in vertibilit y of Ψ. That the t w o maps (2.1 3) are indeed eac h others’ in ve rses means that 00 00 11 11 000 000 111 111 s H λ Λ H = H H = 000 000 000 111 111 111 00 00 11 11 H H (2.14) 2.4 Morphisms for algebraic stru cture on the bimo dule F W e now in tro duce the morphisms that endo w the ob ject F w it h the structure of a F rob enius algebra in the monoidal category H -Bimo d. V ery m uc h lik e the coregular bimo dule F itself, the algebra structure on F is a consequence of the univ ersal prop erties o f the co end of a functor G H ⊗ k : H -Mo d op × H -Mo d → H -Bimo d (see Appendix A.1). Analogous co ends with similar prop- erties can b e in tro duced fo r any rigid braided monoidal category , so that the F rob enius algebra F can b e thought of as b eing canonically asso ciated with the (abstract) monoidal categor y H -Mo d and the functor G H ⊗ k . 7 Deﬁnition 2.6. F or H a ﬁnite-dimensional Hopf algebra, w e introduce the following linear maps m F : H ∗ ⊗ H ∗ → H ∗ , η F : k → H ∗ , ∆ F : H ∗ → H ∗ ⊗ H ∗ and ε F : H ∗ → k : m F := ∆ ∗ , η F := ε ∗ , ∆ F := [( id H ⊗ ( λ ◦ m )) ◦ ( id H ⊗ s ⊗ id H ) ◦ (∆ ⊗ id H )] ∗ , ε F := Λ ∗ . (2.15) Again the graphical description app ears to b e conv enien t: m F = H ∗ H ∗ H ∗ η F = H ∗ ∆ F = H ∗ λ H ∗ H ∗ ε F = H ∗ Λ (2.16) W e w ould lik e to interpret the maps (2.16) as the structural morphisms of a F rob enius algebra in H -Bimo d. T o this end w e mus t ﬁrst sho w that these maps are actually mo r phisms of bimo dules. W e start with a few general observ ations. Lemma 2.7. (i) F or any Hopf algebr a H we have H H H H = H H H H = H H H H = H H H H = H H H H (2.17) (ii) F urther, if H is unimo dular with inte gr al Λ , we have H H Λ H = H H H Λ and H Λ H H = H Λ H H (2.18) Pr o of. (i) The ﬁrst equalit y holds by the deﬁning prop erties of the antipo de, unit and counit of H . The second equalit y follo ws by asso ciativity and coasso ciativity , the third b y the an ti- coalgebra morphism prop ert y of the an tip o de, and the last by the connecting axiom for pro duct and copro duct o f the bialgebra underlying H . 8 (ii) The ﬁrst equality in (2.18) fo llo ws by comp o sing (2.17) with id H ⊗ Λ and using that Λ is a left in tegral. The second equality in (2 .1 8) fo llo ws b y comp osing the left-right-mirrored v ersion of (2.17) (which is prov en in the same w a y a s in (i)) with Λ ⊗ id H and using t ha t Λ is a rig h t in tegral. W e will refer to the equalit y of the left and right hand sides of (2.17) as the Hopf-F r ob enius trick . Lemma 2.8. The map ∆ F intr o duc e d in (2.16) c an alternatively b e expr esse d as ∆ F = ∆ F ′ with ∆ F ′ := H ∗ λ H ∗ H ∗ (2.19) Pr o of. Using the tw o equalities in (2.14) a nd coassociat ivity of ∆ w e obtain H H H = H H H = H H H = H H H (2.20) Dualizing the expressions on the left and right hand sides of (2.20) establishes the claimed equalit y . Prop osition 2.9. When H -Bimo d i s e ndowe d with the tensor pr o duct (2 .5), k is given the structur e of the trivial H -bimo dule k ε = ( k , ε, ε ) (the monoidal unit of H -Bi m o d) and H ∗ the H -bimo dule structur e (2.8), then the map s (2 . 1 5) a r e morphis m s of H -bimo dules. Pr o of. (i) That m F is a morphism of left H - mo dules is seen a s H H ∗ H ∗ H ∗ = H H ∗ H ∗ H ∗ = H H ∗ H ∗ H ∗ = H H ∗ H ∗ H ∗ = H H ∗ H ∗ H ∗ (2.21) Here the ﬁrst and last equalities just implemen t the deﬁnition (2.12) of the H -action, the second is the connecting axiom of H , and the third the a n ti-a lg ebra morphism prop ert y of the 9 an tip o de. Similarly , that m F is also a right mo dule morphism fo llows as H ∗ H ∗ H H ∗ = H ∗ H ∗ H H ∗ = H ∗ H ∗ H H ∗ = H ∗ H ∗ H H ∗ = H ∗ H ∗ H H ∗ (2.22) (ii) That η F is a left and righ t mo dule morphism f ollo ws with the help of the prop erties ε ◦ m = ε ⊗ ε a nd ε ◦ s = ε of the an tip o de. W e ha ve H H ∗ = H H ∗ = H H ∗ = H H ∗ ρ F (2.23) and H ∗ H = H ∗ H = H ∗ H = H ∗ H = H ∗ ρ F H (2.24) resp ectiv ely . This uses in particular the ho momorphism prop erty of the counit ε of H and the fact that ε ◦ s = ε . (iii) Next w e apply the Hopf- F r o b enius tr ick (2.17), whic h allo ws us to write H H ∗ H ∗ H ∗ (2.17) = H H ∗ H ∗ H ∗ = H H ∗ H ∗ H ∗ = H H ∗ H ∗ H ∗ (2.25) This tells us that ∆ F is a left mo dule morphism. 10 (iv) F or establishing the righ t mo dule morphism prop ert y of ∆ F w e recall from Lemma 2.8 that ∆ F = ∆ F ′ with ∆ F ′ giv en b y (2.19). The follo wing c ha in of equalities sho ws that ∆ F ′ is a morphism of right H -mo dules: H ∗ H ∗ H ∗ H = H ∗ H H ∗ H ∗ = H ∗ H H ∗ H ∗ = H ∗ H H ∗ H ∗ = H ∗ H H ∗ H ∗ = H ∗ H H ∗ H ∗ (2.26) Here the ﬁrst equalit y com bines the an t i- coalgebra morphism prop ert y of the an t ip o de and the connecting axiom, while the second equalit y uses t ha t λ ◦ m = λ ◦ τ H,H ◦ ( id H ⊗ s 2 ) and hence λ ◦ m ◦ (( s ◦ m ) ⊗ id H ) = λ ◦ m ◦ [ s ⊗ ( m ◦ τ H,H ◦ ( s − 1 ⊗ id H ))] , (2.27) whic h can b e sho wn (see [CW3, p. 4306]) b y using that H is unimo dular. (v) Finally , the pro of of the bimo dule morphism prop erty for ε F is similar to the o ne for η F . The sequence of equalities H H ∗ = H H ∗ = H H ∗ = H H ∗ (2.28) sho ws that ε F is a morphism of left mo dules. Here t he second equalit y holds b ecause Λ is a left in tegral. Using that Λ is also a right in tegral, one sho ws analogously that ε F is a morphism of righ t mo dules. 2.5 The F rob enius algebra structure of F Prop osition 2.10. The morphism s (2.15) endow the obje ct F = ( H ∗ , ρ F , ρ F ) with the structur e of a F r ob enius algebr a in H -Bimo d (with tensor p r o duct (2.5 )). That is, ( F , m F , η F ) is a (unital asso ciative) algebr a, ( F , ∆ F , ε F ) is a (c ounital c o ass o ciative) c o algebr a, and the two structur es ar e c onne cte d by ( id H ∗ ⊗ m F ) ◦ (∆ F ⊗ id H ∗ ) = ∆ F ◦ m F = ( m F ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ ∆ F ) . (2.29) 11 Pr o of. (i) That ( F , m F , η F ) = ( H ∗ , ∆ ∗ , ε ∗ ) is a unital a sso ciativ e algebra just follo ws f r o m (and implies) the fact t ha t ( H , ∆ , ε ) is a counital coasso ciative coalgebra. (ii) It follo ws directly f r o m the coassociativity of ∆ that ( id H ∗ ⊗ ∆ F ′ ) ◦ ∆ F = (∆ F ⊗ id H ∗ ) ◦ ∆ F ′ . (2.30) Since, as seen ab o ve , ∆ F ′ = ∆ F , t his sho ws tha t ∆ F is a coasso ciative copro duct. (iii) The coasso ciativity of ∆ also implies directly the ﬁrst of the F rob enius prop erties (2.29), as w ell as ( m F ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ ∆ F ′ ) = ∆ F ′ ◦ m F . (2.31) In view of ∆ F ′ = ∆ F , ( 2 .31) is the second of the equalities (2.29). (iv) That ε F = Λ ∗ is a counit for the copro duct ∆ F follo ws with the help of the in v ertibility (2.14) of the F rob enius map: we ha v e H ∗ H ∗ = H ∗ H ∗ = H ∗ H ∗ = H ∗ H ∗ = H ∗ H ∗ (2.32) Here the left hand side is ( id H ∗ ⊗ ε F ) ◦ ∆ F , while the righ t hand side is ( ε F ⊗ id H ∗ ) ◦ ∆ F . Remark 2.11. The spaces of left- and right-module morphisms, resp ectiv ely , from F to 1 are giv en b y k Λ r and b y k Λ l , resp ectiv ely , with Λ r and Λ l non-zero left and righ t integrals of H . Th us a non-zero bimo dule morphism from F to 1 exists iﬀ H is unimo dular, and in this case it is unique up to a non-zero scalar. In particular, up to a non-zero scalar the F rob enius counit ε F is a lr eady completely determined by the requiremen t t ha t it is a mo r phism of bimo dules. In the situation at hand, the algebra F b eing a F ro b enius algebra is th us a pr op erty ra ther than the c hoice of a structure. 12 3 Comm utativit y The con ven tio nal tensor pro duct ( 2 .6) of bimo dules generically do es not admit a braiding. In con trast, the monoidal category H -Bimo d, with tensor pro duct as deﬁned in (2 .5), ov er a quasitriangular Hopf algebra admits braidings, and in fact can generically b e endo w ed with sev eral inequiv alen t ones. Among these inequiv alen t braidings, one is distinguishe d from the p oin t of view of full lo cal conformal ﬁeld theory . W e will select this particular braiding c and then show that with resp ect to this braiding c the algebra ( F , m F , η F ) is comm utativ e. The R-matrix R ∈ H ⊗ H is equiv alen t to a braiding c H -Mo d on the category H - Mo d of left H -modules, consisting of a natural family of isomorphisms in Hom H ( X ⊗ Y , Y ⊗ X ) for eac h pair ( X , ρ X ) , ( Y , ρ Y ) of H -modules. These braiding isomorphisms are giv en by c H -Mo d X,Y = τ X,Y ◦ ( ρ X ⊗ ρ Y ) ◦ ( id H ⊗ τ H,X ⊗ id Y ) ◦ ( R ⊗ id X ⊗ id Y ) , (3.1) where τ is t he ﬂip map. ( When written in terms of elemen ts x ∈ X and y ∈ Y , this amoun ts to x ⊗ y 7→ P i s i y ⊗ r i x for R = P i r i ⊗ s i , but recall that w e largely refrain from working with ele- men ts.) The in vers e braiding is giv en by a similar form ula, with R replaced b y R − 1 21 ≡ τ H,H ◦ R − 1 . Besides R , also the inv erse R − 1 21 endo ws the category H -Mo d with the structures of a braided tensor category; the tw o bra idings are inequiv a len t unless R − 1 21 equals R , in which case the category is symmetric. Lik ewise o ne can act with R and with R − 1 21 from the righ t to o bt a in t wo diﬀerent braidings on the category of righ t H -mo dules. As a consequence , with resp ect to the chosen tensor pro duct on H -Bimo d there are tw o inequiv alent natural br a idings obtained b y either using R b oth on the left and on the righ t, or else using (sa y) R − 1 21 on the left and R on the righ t. F or our presen t purp oses (compare Lemma A.4(iii) ) the second of these p ossibilities turns out to b e the relev an t braiding c . Pictorially , describing the R- matrix a nd its in verse b y R = H H and R − 1 = H H (3.2) the braiding on H -Bimo d lo oks as follow s: c X,Y = X X Y Y R − 1 R τ X,Y (3.3) W e are no w in a p o sition to state Prop osition 3.1. The pr o duct m F of the F r ob enius algebr a F in H -Bim o d is c omm utative with r esp e ct to the br aiding (3.3): m F ◦ c F ,F = m F . (3.4) 13 Pr o of. W e ha ve m F ◦ c F ,F = H ∗ H ∗ H ∗ = H ∗ H ∗ H ∗ = H ∗ H ∗ H ∗ = H ∗ H ∗ H ∗ = H ∗ H ∗ H ∗ = m F . (3.5) Here in the second equalit y the deﬁnition (2.8) of the H -actions on F is inserted. The third equalit y holds b ecause the R- matrix satisﬁes ( s ⊗ id H ) ◦ R = R − 1 = ( id H ⊗ s − 1 ) ◦ R , (3.6) whic h implies ( s ⊗ s ) ◦ R − 1 = R − 1 as w ell as ( s − 1 ⊗ s − 1 ) ◦ R = R . The fourth equalit y fo llows b y the deﬁning prop ert y of R to in tertw ine the copro duct and opp osite copro duct of H . 4 Symmetry , sp ecialne s s and t wist By combining the dualities of V ect k with the an tip o de or its in verse , one obtains left and righ t dualities on the category H - Mo d of left mo dules o ver a ﬁnite-dimensional Hopf algebra H , and like wise fo r right H -mo dules. In the same w ay w e can deﬁne left and righ t dualities on H -Bimo d. Since the monoida l unit of H - Bimo d (with our choice o f tensor pro duct) is the ground ﬁeld k , we can actually take for the ev aluation and co ev a luation morphisms (and thus for the action of the functors on morphisms) just the ev alua tion and co ev aluation maps (2.11) of V ect k , and choose to deﬁne the actio n on ob j ects X = ( X , ρ, ρ ) ∈ H -Bimo d by X ∨ := ( X ∗ , ρ ∨ , ρ ∨ ) and ∨ X := ( X ∗ , ∨ ρ, ∨ ρ ) (4.1) 14 with ρ ∨ := H X ∗ ρ X ∗ ρ ∨ := X ∗ ρ X ∗ H ∨ ρ := H X ∗ ρ X ∗ ∨ ρ := ρ X ∗ X ∗ H (4.2) That the morphisms (4 .2) are (left resp ectiv ely righ t) H -a ctions fo llows from the fact that the an tip o de is an anti-algebra morphism, a nd that the ev a lua tions and co ev aluatio ns are bimo dule morphisms follo ws from the deﬁning prop erty m ◦ ( s ⊗ id H ) ◦ ∆ = η ◦ ε = m ◦ ( id H ⊗ s ) ◦ ∆ of the an tip o de. Note that with o ur deﬁnition of dualities 3 w e ha v e ∨ ( X ∨ ) = X = ( ∨ X ) ∨ (4.3) as equalities (no t just isomorphisms) of H -bimo dules. The c anonic al element (also called Dri n feld element ) u ∈ H of a quasitriangular Hopf algebra ( H , R ) with inv ertible antipo de is the elemen t u := m ◦ ( s ⊗ id H ) ◦ τ H,H ◦ R . (4.4) u is in ve r t ible and satisﬁes s 2 = ad u [Ka, Prop VI I I.4.1]. W e denote by t ∈ H the in v erse of the so-called sp ecial g roup-lik e elemen t, i.e. the pro duct t := u v − 1 ≡ m ◦ ( u ⊗ v − 1 ) (4.5) of the Drinfeld elemen t and the inv erse of the ribb on elemen t v . Since v is inv ertible a nd cen tral, w e ha v e ad t = s 2 and, as a consequence, m ◦ ( s ⊗ t ) = m ◦ ( t ⊗ s − 1 ) and m ◦ ( s − 1 ⊗ t − 1 ) = m ◦ ( t − 1 ⊗ s ) . (4.6) Also, since t is gro up- lik e w e ha ve ε ◦ t = 1 and s ◦ t = t − 1 = s − 1 ◦ t . (4.7) A sover eign structur e on a category with left and right dualities is a choic e of monoidal natural isomorphism π b et we en the left and righ t duality functors [Dr , Def. 2.7]. 4 The category H -Mo d of left mo dules ov er a ribb on Hopf algebra H is so v ereign iﬀ the square of the antipo de of H is inner [Bi, Dr]. Similarly , w e hav e 3 The left and right duals of any ob ject in a catego ry with dualities are unique up to distinguishe d isomor - phism. O ur choice do es not make use o f the fa ct that H is a ribb on Hopf algebra . Another realiza tion o f the dualities on H -Mo d (and analogo us ly on H -Bimo d), whic h inv olves the sp ecial group-like e le ment of H and hence do es use the ribb on structur e , is describ ed e.g. in [Vi, Lemma 4.2 ]. 4 Equiv alently [Y e1, Prop. 2.11] o ne may require the existence of monoidal natura l iso morphisms b etw een the (left or right) double dual functors and the identit y functor. The latter is ca lled a balanced structure (see e.g. Section 1.7 of [Da1]), or so metimes also a pivotal structure (see e.g . Section 3 o f [Sc ]). 15 Lemma 4.1. F or a ribb on Hopf algebr a H with invertible antip o d e , the family π X that is deﬁne d by π X := X ∗ X ∗ t t ∈ Hom k ( X ∗ , X ∗ ) (4.8) for X ∈ H -Bim o d (w i th X ∗ the ve ctor sp ac e dual to X ) furnishes a sover eign structur e on the c a te gory H -Bimo d of bim o d ules over H . Pr o of. W e mu st sho w that π X is an in v ertible bimo dule in tert winer from X ∨ to ∨ X (i.e. the dual bimo dules as deﬁned in (4.1)), tha t the family { π X } is natural, and that it is monoidal, i.e. π X ⊗ Y = π X ⊗ π Y . (i) That π X is a morphism in Hom H | H ( X ∨ , ∨ X ) is equiv alent to H t X X = H t X X and X X t H = X X t H (4.9) This in turn follows directly b y com bining (4.6) and the (left, resp ectiv ely righ t) represen tation prop erties. (ii) With the help o f the deﬁning prop erties o f the ev aluation and co ev aluation maps it is easily c hec ke d that π − 1 X := X ∗ X ∗ t − 1 t − 1 ∈ Hom k ( X ∗ , X ∗ ) (4.10) is a linear tw o-sided inv erse of π X . That π − 1 X is a bimo dule morphism is then automatic. (iii) That the fa mily { π X } o f isomorphisms furnishes a natural transformation from the rig ht to the left duality f unctor means tha t for an y morphism f ∈ Hom H | H ( X , Y ) one has ∨ f ◦ π Y = π X ◦ f ∨ (4.11) as morphisms in Hom H | H ( Y ∨ , ∨ X ). Now by sov ereignt y of V ect k w e know that ∨ f = f ∨ as linear maps from Y ∗ to X ∗ , a nd as a consequence ( 4 .11) is equiv alen t to f ◦ ϕ X = ϕ Y ◦ f (4.12) 16 as morphisms in Hom H | H ( X , Y ), where ϕ X is the left action o n X with t ∈ H comp osed with the righ t a ctio n on X with t . (4.12), in turn, is a direct conseq uence of the fact that f is a bimo dule mor phism. (iv) Th a t π X is monoidal follow s from the fact that t is group-lik e. Deﬁnition 4.2. (a) An invariant p airing o n an algebra A = ( A, m, η ) in a monoidal category ( C , ⊗ , 1 ) is a morphism κ ∈ Hom C ( A ⊗ A, 1 ) satisfying κ ◦ ( m ⊗ id A ) = κ ◦ ( id A ⊗ m ). b) A symmetric alg ebra ( A, κ ) in a sov ereign category C is a n a lgebra A in C together with an in v arian t pairing κ that is symmetric, i.e. satisﬁes = A κ A π − 1 A A A κ = A κ A π A C (4.13) Remark 4.3. (i) Unlike the pictures used so far (and most of the pictures b elo w), wh ich describe morphisms in V ect , ( 4 .13) refers to mor phisms in the category C rather tha n in V ect ; to emphasize this w e ha v e added the b o x C to the picture. Also note that the morphisms (4.13) inv olve the left and right dualities of C , but do not assume a braiding. Th us the natural setting f or the notion of symmetry of an a lgebra is the one of so ve reign categories C ; a bra iding on C is no t needed. (ii) An algebra with an in v arian t pairing κ is F rob enius iﬀ κ is non-degenerate, see e.g. [F St, Sect. 3 ]. (iii) The t w o equalities in (4.13) actually imply eac h other. In the case of the category H -Bimo d with sov ereign structure π as deﬁned in (4.8), the equalities (4.1 3) read A κ A t − 1 t − 1 = A A κ = A κ A t t (4.14) Theorem 4.4. F or any unimo dular ﬁnite-dimensio nal ribb on Hopf alge b r a H the p air ( F , κ F ) with F the c or e gular bimo dule ( w ith F r ob enius a lgebr a structur e as deﬁne d ab ove) and κ F := ε F ◦ m F = (∆ ◦ Λ) ∗ (4.15) is a symmetric F r ob enius algebr a in H -Bimo d. Pr o of. That the pairing κ F is inv arian t follo ws directly from the coa sso ciativit y of ∆. T o 17 establish that κ F is symmetric, consider the follo wing equalities: H H Λ = H H g Λ = H H t t Λ (4.16) The ﬁrst equality is Theorem 3( d) of [Ra2], and in v olves the right mo dular elemen t (also kno wn as distinguished gr o up-lik e elemen t) g of H , whic h b y deﬁnition satisﬁes g ◦ λ = ( id H ⊗ λ ) ◦ ∆. The second equalit y uses that g = t 2 (whic h holds b y Theorem 2(a) and Corollary 1 of [Ra1 ], sp ecialized to unimo dular H ) and s 2 = ad t . Using also the iden tit y ρ F ◦ ( t − 1 ⊗ id H ∗ ) =  m ◦ ( t ⊗ id H )  ∗ (whic h, in turn, uses (4.7)) , it follo ws that the equalit y of the left and right sides of (4.16 ) is nothing but the dualized vers ion o f the ﬁrst of the equalities (4.14) for the case A = F and κ = κ F . Next w e o bserv e: Lemma 4.5. The morphis m s (2.15) satisfy ε F ◦ η F = ε ◦ Λ and m F ◦ ∆ F = ( λ ◦ ε ) id H ∗ . (4.17) Pr o of. W e ha ve ε F ◦ η F = = and m F ◦ ∆ F = H ∗ H ∗ = H ∗ H ∗ = H ∗ H ∗ (4.18) Here the last equalit y uses the deﬁning pro p ert y of the an tip o de s of H . Deﬁnition 4.6. A F rob enius algebra ( A, m A , η A , ∆ A , ε A ) in a k -linear monoidal catego r y is called sp e cia l [FRS, EP] (o r strongly separable [M ¨ u]) iﬀ ε A ◦ η A = ξ id 1 and m A ◦ ∆ A = ζ id A with ξ , ζ ∈ k × . It is kno wn that a ﬁnite-dimensional Hopf a lgebra H is semisimple iﬀ the Masc hk e n um b er ε ◦ Λ ∈ k is non-zero, and it is cosemisimple iﬀ λ ◦ ε ∈ k is non-zero [LS]; also, in c ha r a cteristic zero cosemisimplicit y is implied by semisimplicit y [LR, Thm. 3.3]. Th us we ha v e Corollary 4.7. The F r ob enius alge br a F in H -Bimo d is sp e cial iﬀ H is semisimple. As a lr eady p oin ted out w e do not , how eve r , assume that H is semisimp le. W e no w note a consequenc e of the fact that F is comm uta t ive and symmetric, irresp ectiv e of whether H is semisimple or not. W e ﬁrst observ e: 18 Lemma 4.8. Th e br aide d monoidal c a te gory H -Bimo d of bimo dules over a ﬁn i te-d i m ensional ribb on Hopf k -algebr a H is b al a nc e d. Th e twist endom o rphisms ar e given by θ X = X X v v − 1 (4.19) with v the ribb on element of H . Pr o of. W e ha v e seen that H -Bimo d is braided, and according to Lemma 4.1 it is so v ereign. No w a braided monoidal category with a (left or right) duality is so ve reign iﬀ it is balanced, see e.g. Prop. 2.11 of [Y e1]. F or an y so vereign braided monoidal category C the t wist endomorphisms θ X can b e obtained b y com bining t he braiding, dualities and so ve reign structure according to θ X = X X c X,X π X C (4.20) With the explicit form (3.3) o f the br a iding and (4.8) of the sov ereign structure, this results in θ X = X X R R − 1 t t (4.21) Using t he relations t = v − 1 u a nd s 2 = ad t , the fact that v is cen t r a l and the relation (4.4) b et w een the canonical elemen t u and the R - matrix then giv es the fo rm ula (4.19). Remark 4.9. ( i) By using that v ∈ H is central and satisﬁes s ◦ v = v . it follo ws immediately from (4.19) that t he F rob enius alg ebra F has trivial twis t, θ F = id H ∗ . (4.22) (ii) In f act, a comm utativ e symmetric F rob enius algebra in any so vere ig n bra ided category has trivial twist. This was sho wn in Prop. 2.25(i) of [FFRS] f or the case that the catego ry is strictly sov ereign ( i.e. that the so ve r eign structure is trivial in the sense that π X = id X for all X ), and the pro of easily carries o v er to general so v ereign cat ego ries. Con v ersely , the fact that F is a sy mmetric algebra can be deriv ed b y com bining the trivialit y (4 .22) of t he t wist with comm utativity . (iii) That F is comm utative and symmetric implies [FFRS, Prop. 2.25(iii)] that it is co comm u- tativ e as w ell. 19 5 Mo d ular in v ariance Our fo cus so fa r has b een on a natural o b ject in the so vere ig n braided ﬁnite tensor category H -Bimo d, the symmetric F rob enius a lg ebra F . But in an y suc h category there exists another natural ob ject K , whic h has b een studied b y Lyubashenk o. K is a Hopf algebra, and it plays a crucial role in the construction o f mapping class group actions. The construction of these mapping class group actions relies on the presence of sev eral distinguished endomorphisms of K . The existence of these endomorphisms is a consequence of univ ersal prop erties c haracterizing the Hopf algebra ob ject K . More precise ly , apart from the an t ip o de s K of K , Lyubashenk o obtains in vertible endomorphisms S K , T K ∈ End C ( K ) t ha t ob ey the relations [Ly1, Thm 2.1.9] ( S K T K ) 3 = λ S 2 K and S 2 K = s − 1 K (5.1) with some scalar λ that depends on the category C in question. These endomorphisms are the central ingredien t fo r the construction of represen tations of mapping class groups o f punctured Riemann surfaces on morphism spaces of C . In particular, Lyubashenk o [Ly1, Sect. 4.3] constructed a pro jectiv e represen ta t io n of the mapping class group Γ 1; m of the m -punctured torus on morphism spaces of the form Hom C ( K , X 1 ⊗ X 2 ⊗ · · · ⊗ X m ), for ( X 1 , X 2 , ... , X m ) an y m -tuple of ob jects of C . Sp ecializing to the case of one puncture, m = 1, there is, for an y ob ject X of C , a pro jective Γ 1;1 -action Γ 1;1 × Hom C ( K , X ) → Hom C ( K , X ). The mapping class group Γ 1;1 is generated b y three generators S , T and D , where D is the Dehn t wist around the puncture. The represen tatio n of Γ 1;1 satisﬁes ( S, f ) 7→ f ◦ S − 1 K , ( T , f ) 7→ f ◦ T − 1 K and ( D , f ) 7→ θ X ◦ f (5.2) with θ the t wist of C . These general constructions a pply in pa r ticular to the ﬁnite tensor category H -Bimo d. The main goa l in this section is t o use the copro duct of the F rob enius algebra F to construct an elemen t in Hom H | H ( K , F ) that is in v ariant under the action o f Γ 1;1 . A similar construction allo ws o ne to deriv e an inv arian t elemen t in Hom H | H ( K ⊗ F , 1 ) from the pro duct of F . F or the motiv atio n to detect such eleme nts and for p ossible applications in full local conformal ﬁeld theory we refer to App endix B. 5.1 Distinguished endomorphisms of c o ends A ﬁnite tensor category [EO] is a k -linear ab elian rigid monoidal category with enough pro j ec- tiv es and with ﬁnitely man y simple ob jects up to isomorphism, with simple tensor unit, and with ev ery ob ject having a comp osition series of ﬁnite length. Both H -Mo d and H -Bimo d, for H a ﬁnite-dimensional unimo dular ribb on Hopf algebra, b elong to this class of categories, see [LM, Ly1]. Let C b e a sov ereign braided ﬁnite t ensor catego ry . As sho wn in [Ly1, Ke] (compare also [Vi] or, as a review, Sections 4.3 and 4.5 of [FSc]), there exis t s an ob ject K in C tha t carries a natura l structure of a Hopf algebra in C . Moreo ver, there is a t w o- sided in tegra l as w ell a s a Hopf pairing for this Hopf algebra K . Com bining the in t egr a l o f K and other structure of the category , one constructs [LM, Ly1, Ly2] distinguished morphisms S K and T K satisfying (5.1) in End C ( K ). 20 The Hopf algebra K can b e c haracterized as the c o end 5 K = Z X F ( X , X ) = Z X X ∨ ⊗ X (5.3) of the functor F that acts on ob jects as ( X , Y ) 7→ X ∨ ⊗ Y . As describ ed in some detail in App endix A.3, in the case C = H - Bimo d the ob ject K is the c o adjoint bimo dule H H ∗ ⊲⊳ . That is, the underlying v ector space is the tensor pro duct H ∗ ⊗ k H ∗ , and this space is endo w ed with a left H -a ction b y the coadjoint left action (A.26) on the ﬁrst factor, and with a right H -action b y the coa djoin t righ t action on the second factor. In the case o f a general braided ﬁnite tensor categor y C the morphisms S K and T K in End C ( K ) are deﬁned with the help of the braiding c and the t wist θ of C , respectiv ely [LM, Ly1, Ly2]. T K is giv en b y the dinatural family T K X ∨ ı K X X K = θ X ∨ X ∨ K ı K X X C (5.4) Here it is used that a morphism f with domain the co end K is uniquely determined b y the dinatural family { f ◦ ı K X } of morphisms. F or S one deﬁnes S K := ( ε K ⊗ id K ) ◦ Q K,K ◦ ( id K ⊗ Λ K ) , (5.5) where ε K and Λ K are the counit and t he t wo-sided in tegral of the Hopf algebra K , resp ectiv ely , while the morphism Q K,K ∈ End C ( K ⊗ K ) is determined through mono dromies c Y ∨ ,X ◦ c X,Y ∨ according to Q K,K ı K X X ∨ K X Y ∨ K Y ı K Y = X ∨ ı K X X K c c Y ∨ K ı K Y Y C (5.6) W e also note tha t the Hopf algebra K is endow ed with a Hopf pa iring ω K , giv en b y [Ly1] ω K = ( ε K ⊗ ε K ) ◦ Q K,K . (5.7) 5.2 The Drinfeld map Let us now sp ecialize the latter for mulas to the case o f our in terest, i.e. C = H - Bimo d. Then the co end is K = H H ∗ ⊲⊳ , with dinatural f amily ı K = i ⊲⊳ giv en b y (A.29), while the twist is giv en 5 The deﬁnition of the co end K of a functor G : C op × C → C , including the asso cia ted dinatural family ı K of morphisms ı K X ∈ Hom C ( F ( X , X ) , K ), will b e rec a lled at the beginning of Appendix A.1. 21 b y (4.19) and the braiding b y (3.3). F urther, the structure mor phisms of the categorical Hopf algebra H H ∗ ⊲⊳ can b e expressed through those of the algebraic Hopf algebra H ; in particular, the counit and integral are ε ⊲⊳ = η ∨ ⊗ η ∨ (see (A.32) ) and Λ ⊲⊳ = λ ∨ ⊗ λ ∨ (see (A.36) ) . The mon- o dromy morphism Q H H ∗ ⊲⊳ ,H H ∗ ⊲⊳ ∈ End H | H ( H H ∗ ⊲⊳ ⊗ H H ∗ ⊲⊳ ) that w as in tro duced in (5.6) then reads Q H H ∗ ⊲⊳ ,H H ∗ ⊲⊳ = H ∗ H ∗ H ∗ Q H ∗ s − 1 H ∗ Q − 1 s H ∗ H ∗ H ∗ (5.8) while the general fo rm ulas for T K and S K sp ecialize to the morphisms T ⊲⊳ = H ∗ v H ∗ H ∗ v − 1 H ∗ and S ⊲⊳ = H ∗ Q − 1 H ∗ λ H ∗ Q λ H ∗ (5.9) in End H | H ( H H ∗ ⊲⊳ ) . Note that the morphism S ⊲⊳ is comp osed of (v ariants o f ) the F rob enius map (2.13) and the D rinfeld map f Q := ( d H ⊗ id H ) ◦ ( id H ∗ ⊗ Q ) ∈ Ho m( H ∗ , H ) . (5.10) In o rder that S ⊲⊳ is inv ertible, whic h is necessary for ha ving pro jectiv e mapping class g r o up represen tations, it is necessary and suﬃcien t that the D rinfeld map f Q is in ve rt ible. Remark 5.1. By the results fo r general C , S ⊲⊳ is indeed a morphism in H -Bimo d. But this is also easily che ck ed directly: One just ha s to use that t he Drinfeld map intert wines the left coadjo in t action ρ ⊲ (see (A.26 ) ) of H on H ∗ and the left adjoint a ctio n ρ ad of H on itself [CW1, Prop. 2.5(5) ], i.e. that f Q ∈ Hom H ( H ∗ ⊲ , H ad ) , (5.11) together with the fact that the coin tegral λ satisﬁes (since H is unimo dular) [Ra2, Thm. 3] λ ◦ m = λ ◦ m ◦ τ H,H ◦ ( id H ⊗ s 2 ) . (5.12) Remark 5.2. The Dr inf eld map f Q of a ﬁnite-dimensional quasitriang ula r Hopf algebra H is in v ertible iﬀ H is factorizable. In the semisimple case, fa ctorizabilit y is the essen tial ingredi- en t f o r a Hopf a lg ebra to b e mo dular [NTV, Lemma 8.2]. Here, without an y a ssumption of semisimplicit y , w e see again a direct link b et w een in vertibilit y of S a nd factorizabilit y . 22 Remark 5.3. The coadjoint H -mo dule H ∗ ⊲ is actually the coend (5 .3) for the case that C is the category H -Mo d of left H -mo dules. In this case the endomorphism (5.5) is precisely the comp osition S ⊲ = Ψ ◦ f Q (5.13) of the Drinfeld and F rob enius maps (also called the quan t um F ourier tra nsform), see e.g. [LM, F GST]. F urther, the Drinfeld map f Q is related to the Hopf pairing ω H ∗ ⊲ from (5.7 ) for the co end H ∗ ⊲ in H -Mo d b y f Q = ( s ⊗ ω H ∗ ⊲ ) ◦ ( id H ⊗ τ H ∗ ,H ∗ ) ◦ ( b H ⊗ id H ∗ ) . (5.14) In particular, since the an tip o de is inv ertible, the Drinfeld map of H is in v ertible iﬀ the Hopf pairing of H ∗ ⊲ is non-degenerate. Remark 5.4. F or factorizable H , the D rinfeld map f Q maps any non- zero coin tegral λ o f H to a non-zero in tegra l Λ. Th us w e may (and do) c ho ose λ and Λ (uniquely , up to a common factor ± 1) suc h that b esides λ ◦ Λ = 1 w e also hav e f Q ( λ ) = Λ (5.15) (see [G W, Thm. 2.3.2 ] and [CW2, Rem. 2 .4 ]). T o gether with s ◦ Λ = Λ and the prop erty (5 .1 2) of the coin tegra l it then also follows that f Q − 1 ( λ ) = Λ as w ell as f Q − 1 ∈ Hom H ( H ∗ ⊲ , H ad ) , (5.16) where f Q − 1 is the morphism (5.10) with the mono dromy matrix Q replaced by its in verse Q − 1 . F urther, one ha s [CW2, Lemma 2.5] f Q ◦ Ψ ◦ f Q − 1 ◦ Ψ = id H , (5.17) whic h in turn by comparison with (5.13) sho ws that Ψ ◦ f Q − 1 = S − 1 ⊲ . F urther, the la t ter iden tity and (5.13) are equiv alent to the relatio ns H Q s λ Q − 1 H = H Λ H = H Q − 1 s − 1 λ Q H (5.18) Remark 5.5. According to the ﬁrst of the f orm ulas (5.9 ) w e ha ve T ⊲⊳ = T − 1 ⊲ ⊗ T ⊲ . Using (5 .12) and (5.13) it follows that S ⊲⊳ = S − 1 ⊲ ⊗ S ⊲ as w ell. As a consequence, the ﬁrst of the relations (5.1) is realized with λ = 1. Thus in the case of the category H -Bimo d o f our in t erest, with co end K , the pro jectiv e represen tation (5 .2 ) o f the mapping class gro up Γ 1;1 of the one-punctured torus is actually a gen uine Γ 1;1 -represen tation. 23 5.3 Action of Γ 1;1 on morp h ism spaces Consider now t he represen tation ρ K,Y of Γ 1;1 on the spaces Hom C ( K , Y ) just men tio ned. The group Γ 1;1 is generated b y three elemen ts S , T and D , where D is the Dehn t wist around the puncture, while S and T a re mo dular transformat ions that act o n the surface in the same wa y as in the absence of the puncture. The generator s are sub ject to the relatio ns ( S T ) 3 = S 2 (lik e for the mo dular group) and S 4 = D . Of particular in terest to us are invariants of the Γ 1;1 -action on the spaces Hom C ( K , Y ), i.e. morphisms g satisfying g ◦ ρ K,Y ( γ ) = g (5.19) for all γ ∈ Γ 1;1 . Morphisms in Hom C ( K , Y ) can in particular b e obtained b y deﬁning a linear map t Q from Hom C ( X , Y ⊗ X ) to Hom C ( K , Y ) as a univ ersal partial trace, to which w e refer a s the p artial mono dr o my tr ac e . Th us for f ∈ Hom C ( X , Y ⊗ X ) w e set t Q ( f ) := Q l K,X ε K K Y f X X π X C ∈ Hom C ( K , Y ) , (5.20) where Q l H ,Y ∈ End C ( K ⊗ Y ) is deﬁned b y Q l K,Y X ∨ K X Y Y := X ∨ K X c c Y Y C (5.21) Note that, by the natura lit y of the braiding, the morphisms (5.21) a r e natural in Y . Remark 5.6. (i) It follo ws from elemen ta r y prop erties o f the braiding that for any ob ject X of C the morphism ( ε K ⊗ id X ) ◦ Q l K,X endo ws X with the structure of a K -mo dule inte r nal to C . (ii) If C is a ( semisimple , strictly sov ereign) mo dular tensor category , then the inv ariance prop- ert y (5.1 9) for γ = S and g = t Q ( f ) is equiv alent to the deﬁnition of S - in v ariance of morphisms in Hom C ( Y ⊗ X , X ) that is given in [KoR, Def. 3.1(i)]. 24 Sp ecializing no w to C = H -Bimo d and X = Y b eing the F rob enius algebra F in H -Bimo d, w e can state one of the main results of this pap er: Theorem 5.7. The p artial mono dr omy tr ac e of the c opr o duct ∆ F is Γ 1;1 -invariant. W e will pro ve this stat ement b y establishing, in Lemma 5.8 a nd Lemma 5.9 b elow, separately in v ariance under the tw o gene r a tors T and S of Γ 1;1 . Note that this implies in particular in v ariance under the Dehn t wist D ; the latter can also b e directly deduced from the fact tha t D ⊲⊳ = θ F together with the result (4 .22) t ha t F has trivial t wist. Before in v estigating t he action of S and T , let us ﬁrst presen t the pa rtial mono drom y trace t Q (∆ F ) in a con v enien t form. T o this end w e ﬁrst note that, inv o king the explicit form (3 .3) of the braiding and (4.8) of the so v ereign structure o f H -Bimo d, w e hav e Q l K,F K F F = H ∗ H ∗ Q − 1 Q H ∗ H ∗ and hence t Q ( f ) = H ∗ H ∗ Q − 1 Q f t − 1 H ∗ t − 1 (5.22) for any f ∈ Hom H | H ( F , F ⊗ F ). Specializing (5.22) to the part ial mono dro my trace of the copro duct ∆ F , i.e. inserting ∆ F from (2.19), yields t Q (∆ F ) = H ∗ H ∗ Q − 1 Q λ t − 1 t − 1 H ∗ (5.23) This can b e rewritten as f o llo ws: t Q (∆ F ) = H ∗ H ∗ Q − 1 Q λ H ∗ = H ∗ H ∗ Q − 1 Q λ H ∗ = H ∗ H ∗ Q − 1 Q λ H ∗ (5.24) 25 Here in the ﬁrst step it is used that s 2 = ad t and that λ ◦ m ◦ ( g ⊗ id H ) = λ ◦ s ( which is a left coin tegral), while the second equalit y f ollo ws b y the fa ct that the antipo de is an anti-algebra morphism a nd b y asso ciativit y of m . W e can no w use the fact (see (5.16)) that t he morphism f Q − 1 in tertw ines the coadjoint and adjoin t actions; w e then hav e t Q (∆ F ) = H ∗ H ∗ Q − 1 Q λ H ∗ = H ∗ H ∗ Q − 1 Q λ H ∗ = H ∗ H ∗ Λ H ∗ (5.25) where the last equality uses the ﬁrst of the iden tities (5.18 ) together with the fa ct tha t the an tip o de is an a nti-coalgebra morphism and that s ◦ Λ = Λ. Lemma 5.8. The morphis m t Q (∆ F ) is T -invariant, i.e . satisﬁes t Q (∆ F ) ◦ T ⊲⊳ = t Q (∆ F ) . Pr o of. In v oking the expressions for T ⊲⊳ giv en in (5.9) and for t Q (∆ F ) given o n the right hand side of (5.25), and using t he cen tralit y of the ribb on elemen t v ∈ H , we hav e t Q (∆ F ) ◦ T ⊲⊳ = H ∗ H ∗ v v − 1 Λ H ∗ (5.26) Recalling now the iden tity (2.18), the cen tra l elemen ts v and v − 1 cancel eac h other, hence (5.26) equals t Q (∆ F ). Lemma 5.9. The morphis m t Q (∆ F ) is S -invariant, i. e. sa tisﬁes t Q (∆ F ) ◦ S ⊲⊳ = t Q (∆ F ) . Pr o of. W e will show that t Q (∆ F ) inv arian t under S − 1 . Applying deﬁnition (5.9), we can use the iden tities (5.18) to obtain t Q (∆ F ) ◦ S − 1 ⊲⊳ = H ∗ H ∗ Q Q − 1 Q − 1 Q λ λ λ H ∗ = H ∗ H ∗ Λ Λ λ H ∗ (5.27) 26 F urther w e note that owing t o t he iden tities ( 5 .12), s − 2 ◦ Λ = Λ and (2.14) we can write H ∗ H ∗ Λ λ H ∗ = H ∗ H ∗ Λ λ H ∗ = H ∗ H ∗ Λ λ H ∗ = H ∗ H ∗ Λ λ H ∗ = H ∗ H ∗ H ∗ (5.28) It follows that t Q (∆ F ) ◦ S − 1 ⊲⊳ = H ∗ H ∗ Λ H ∗ = H ∗ H ∗ Λ H ∗ (5.29) This coincides with the rig h t hand side of (5 .25) and thus with t Q (∆ F ). T o neatly summarize the results ab ov e w e state Deﬁnition 5.10. A coalgebra ( C , ∆ C , ε C ) in H -Bimo d is called mo dular invariant iﬀ the morphism t Q (∆ C ) ∈ Hom H | H ( K , C ) is Γ 1;1 -in v ariant. Th us what w e ha ve shown can b e rephrased as Corollary 5.11. The F r ob enius algebr a F ∈ H -B imo d intr o duc e d in (2.1 5) is m o d ular invariant. Pr o of. In v ariance of t Q (∆ C ) under the action of the generators T and S of Γ 1;1 has been sho wn in Lemma 5.8 and 5.9. Inv ariance under the action of the generator D of Γ 1;1 follo ws immediately from the fact that F has trivial t wist. Remark 5.12. ( i) The morphism t Q (∆ F ) is non-zero. Indeed one can sho w that ε F ◦ t Q (∆ F ) can b e expanded a s a bilinear form in the simple c hara cters of Lyubashenk o’s Hopf algebra H ∈ H - Mo d, with co eﬃcien ts giv en b y the Cartan matrix of H , i.e. b y the matrix that describes the comp osition series of indecomp o sable pro jectiv es, and these H - c haracters are non- zero. In conformal ﬁeld theory terms, this means that the Cartan matrix – whic h is a quan tity directly asso ciated to the cat ego ry – is the right substitute of the c harge conjugatio n mat r ix in the non-semisimple case. Ho w ev er, establishing this result requires metho ds diﬀeren t from those on whic h our f o cus is in this pap er. (ii) In the same wa y as the partial mo no drom y tra ce (5.20) asso ciates a morphism in Ho m C ( K , Y ) 27 to a morphism in Hom C ( X , Y ⊗ X ), one ma y in tro duce another partial mono drom y trace t ′ Q that maps morphisms in Hom C ( X ⊗ Y , X ) linearly to morphism in Hom C ( K ⊗ Y , 1 ), sa y as t ′ Q ( f ) := K Y Q l K,Y π − 1 X f X (5.30) It is not diﬃcult t o chec k that the morphism t ′ Q ( m F ) obta ined this w ay from the pro duct o f the F rob enius a lgebra F is modula r in v arian t in the sense that t ′ Q ( m F ) ◦ ρ K ⊗ F , 1 ( γ ) = t ′ Q ( m F ) for all γ ∈ Γ 1;1 . Indeed, t ′ Q ( m F ) is related to t Q (∆ F ) b y t ′ Q ( m F ) = H ∗ H ∗ H ∗ t Q (∆ F ) Ψ − 1 (5.31) with Ψ the F ro b enius ma p, and as a consequence (using that ( s − 2 ) ∗ ⊗ id H ∗ comm utes with the action of Γ 1;1 and that τ H ∗ ,H ∗ ◦ S ⊲⊳ = S − 1 ⊲⊳ and τ H ∗ ,H ∗ ◦ T ⊲⊳ = T − 1 ⊲⊳ ) mo dular inv a riance of t ′ Q ( m F ) is equiv alen t to mo dular in v a r ia nce of t Q (∆ F ). Accordingly , from the p ersp ectiv e of H -Bimo d alone we could as w ell ha v e referred to algebras rather than coa lg ebras in Def. 5 .1 0. Indeed, this is the optio n that was c hosen for the semisimple case in [KoR, Def. 3.1(ii)]. Our preference f o r coalgebras derive s fr om the fact that, as describ ed in App endix B, the morphism space Hom C ( K , F ) plays a more direct role than Hom C ( K ⊗ F , 1 ) in the motiv ating conte xt of mo dular functors and conformal ﬁeld theory . (iii) More generally , for any no n-negativ e in tegers m and n , the mapping class group Γ 1; m + n of the ( m + n )-punctured to r us acts on the morphism space Hom C ( K , F ⊗ m + n ) [Ly1, Sect. 4.3 ] and th us, using the canonical isomorphism of the F rob enius algebra F with its dual, also on Hom C ( K ⊗ F ⊗ m , F ⊗ n ). By suitably comp osing the morphisms (5.23) and (5.30) with pro duct and copro duct morphisms of F , one easily constructs a morphism in Hom C ( K ⊗ F ⊗ m , F ⊗ n ) that, o wing to commutativit y and co comm utativit y of F and t o the trivialit y of its t wist, is in v arian t under this action o f Γ 1; m + n . 28 6 The case of non-trivial Hopf algebra automorphisms A Hopf algeb r a automorphism of a Hopf algebra H is a linear map fr o m H to H that is b o t h an algebra a nd a coalgebra automorphism and comm utes with t he an t ip o de. F or H a ribb on Hopf algebra with R-matrix R and ribb on elemen t v , a n automorphism ω of H is said to b e a ribb on Hopf algebr a automorphis m iﬀ ( ω ⊗ ω )( R ) = R and ω ( v ) = v . F or an y H -bimodule ( X , ρ, ρ ) and an y pair o f algebra automorphisms ω , ω ′ of H there is a corr esp o nding ( ω , ω ′ )- twiste d bimo dule ω X ω ′ = ( X , ρ ◦ ( ω ⊗ id X ) , ρ ◦ ( id X ⊗ ω ′ )). If ω and ω ′ are Hopf algebra a utomorphisms, then the t wisting is compatible with the monoidal structure of H -Bimo d, and if they a re ev en ribb on Hopf algebra automorphisms, then it is compatible with the ribb on structure of H -Bimo d. In this section w e observ e that t o any ﬁnite-dimensional factorizable ribb on Hopf algebra H and an y ribb on Hopf algebra automorphism of H there is again asso ciated a F rob enius algebra in H -Bimo d, whic h moreov er shares all the prop erties, in particular mo dular inv ariance, of the F rob enius algebra F that w e obtained in the previous sections. The arg uments needed to establish this result ar e simple mo diﬁcations of those used previously . Accordingly w e will b e quite brief. Prop osition 6.1. (i) F or H a ﬁ n ite-dimensional factorizable ribb on Hopf alge br a ove r k an d ω a Hopf algebr a automorphism of H , the b i m o d ule F ω := id H ( F ) ω c a rrie s the s tructur e of a F r ob enius algebr a. The structur e morphisms of F ω as a F r ob enius a l g e br a ar e given by the formulas (2.15) (thus as line ar maps they ar e the same as for F ≡ F id H ). (ii) F ω is c omm utative and symmetric, and it i s sp e cial i ﬀ H is semisimp le. (iii) If ω is a ribb on Hopf alg ebr a automorphism, then F ω is m o d ular inva riant. Pr o of. The pro ofs of all statemen ts are completely parallel to those in t he case ω = id H . The only diﬀerence is that the v arious morphisms one deals with, a lb eit coinciding as linear maps with those encoun tered befo r e, are now morphisms b etw een diﬀeren t H -bimo dules than previ- ously . That they do intert wine the relev an t bimo dule structures fo llows by com bining the simple facts that (since ω is compatible with the ribb on structure of H - Bimo d) ( F ⊗ F ) ω = F ω ⊗ F ω as a bimo dule and that a linear map f ∈ Hom( X , Y ) for X , Y ∈ H -Bimo d lies in the subspace Hom H | H ( X , Y ) iﬀ it lies in the subspace Hom H | H ( X ω , Y ω ). F urthermore, again the F rob enius algebra F ω is canonically asso ciated with H -Bimo d as an a bstract category . Ind eed, analogously as in Prop osition A.3, one sees that F ω can b e constructed as a co end, namely the o ne of the functor G H ; ω ⊗ k : H -Mo d op × H -Mo d → H -Bimo d that acts on morphisms as f × g 7→ f ∨ ⊗ k g and on ob jects by mapping ( X , ρ X ) × ( Y , ρ Y ) to  X ∗ ⊗ k Y , [ ρ X ∨ ◦ ( ω − 1 ⊗ id X ∗ )] ⊗ id Y , id X ∗ ⊗ ( ρ Y ◦ τ Y , H ◦ ( id Y ⊗ s − 1 ))  (6.1) (or, in other words , G H ; ω ⊗ k = (? ω − 1 × Id ) ◦ G H ⊗ k with the f unctor G H ⊗ k (whose co end is F ) g iv en b y (A.2)): Prop osition 6.2. The H -bimo dule F ω to gether with the dinatur al famil y of morphisms ı F ω X := ( ω − 1 ) ∗ ◦ ı F X , (6.2) with ı F X as deﬁne d in (A.5), is the c o end of the functor G H ; ω ⊗ k . 29 Pr o of. Again the pro of is parallel to the one for the case ω = id H , the diﬀerence b eing that the automorphisms ω ± 1 need to b e inserted at a ppro priate places. F or instance, the equalities H ρ X ∨ ρ X X ∗ X H ∗ ω − 1 ω − 1 ∗ = H X ∗ X H ∗ ω − 1 ω − 1 = H X ∗ X H ∗ ω − 1 ∗ = H X ∗ ρ X X H ∗ ρ F ω − 1 ∗ (6.3) and X ∗ X H H ∗ ω − 1 ∗ = X ∗ X H H ∗ ω − 1 = X ∗ X H H ∗ ω − 1 = X ∗ X H ρ F H ∗ ω − 1 ∗ ω (6.4) whic h generalize the relations (A.8) and (A.9), resp ectiv ely , demonstrate that the linear maps ı F ω X ∈ Hom( X ∗ ⊗ k X , H ∗ ) are indeed bimo dule morphisms in Hom H | H ( G H ; ω ⊗ k ( X , X ) , F ω ). Remark 6.3. (i) When discussing tw ists of F w e can restrict to the case that only the, sa y , righ t mo dule structure is twis ted, b ecause the bimo dule ω H ω ′ is isomorphic to id H H ω − 1 ◦ ω ′ . (ii) It follow s fr o m the automorphism prop erty of ω that together with Λ also ω (Λ) is a non-zero t wo-side d in tegral of H . As a conseque nce, just lik e in the case ω = id H considered in R emark 2.11, the counit ε F of F ω is uniquely determined up to a non- zero scalar. 30 A Co end construc t ions A.1 The coregular bimo dule as a c o end A di n atur al tr ans f o rmation F ⇒ B fr o m a functor F : C op × C → D , to an ob ject B ∈ D is a family of morphisms ϕ = { ϕ X : F ( X , X ) → B } X ∈C suc h tha t the diagram F ( Y , X ) F ( id Y ,f ) / / F ( f , id X )   F ( Y , Y ) ϕ Y   F ( X , X ) ϕ X / / B (A.1) comm utes for all f ∈ Hom( X , Y ). F or instance, the family { d X } of ev aluation morphisms of a rigid monoidal category C forms a dinatural transformation f rom the f unctor t hat acts a s X × Y 7→ X ∨ ⊗ Y t o the monoidal unit 1 ∈ C . Dinatural tr a nsformations from a giv en functor F to an ob ject o f D form a category , with the morphisms from ( F ⇒ B , ϕ ) to ( F ⇒ B ′ , ϕ ′ ) being g iven b y morphisms f ∈ Hom D ( B , B ′ ) satisfying f ◦ ϕ X = ϕ ′ X for a ll X ∈ C . A c o end ( A, ι ) fo r the functor F is an initial o b ject in this category . If the co end o f F exists, then it is unique up to unique isomorphism; one denotes it b y R X F ( X , X ). A morphism with domain R X F ( X , X ) and co domain Y is equiv alen t to a family { f X } X ∈C of morphisms from F ( X , X ) to Y suc h that ( Y , f ) is a dinatural tra nsformation. F or H a ﬁnite-dimensional Hopf algebra ov er k , endo w the categories H -Mo d and H - Bimo d of left H -mo dules and o f H -bimo dules, respectiv ely , with the tensor pro ducts (2 .6) and (2.5) and with the dualities describ ed at the b eginning of Section 4. Consider the tensor pro duct (bi)functor G H ⊗ k : H -Mo d op × H -Mo d → H -Bimo d (A.2) that acts on ob jects as ( X , ρ X ) × ( Y , ρ Y ) G H ⊗ k 7− →  X ∗ ⊗ k Y , ρ X ∨ ⊗ id Y , id X ∗ ⊗ ( ρ Y ◦ τ Y , H ◦ ( id Y ⊗ s − 1 ))  (A.3) and on morphisms as f × g 7→ f ∨ ⊗ k g . Pictorially , the action on ob jects is H X X ρ X × H Y Y ρ Y G H ⊗ k 7− → H X ∨ X ∨ Y Y H s − 1 ≡ s H X ∗ ρ X X ∗ Y Y ρ Y H s − 1 (A.4) Remark A.1. The categor y H -Mo d op × H -Mo d is naturally endo wed with a tensor pro duct, acting on o b jects as ( X × Y ) × ( X ′ × Y ′ ) 7→ ( X ⊗ H -Mo d X ′ ) × ( Y ′ ⊗ H -Mo d Y ). With resp ect to this tensor pro duct and the tensor pro duct (2.5) on H -Bimo d, G H ⊗ k together with the asso cia- tivit y constrain ts fr o m V ect k is a monoidal functor. 31 In this appendix w e sho w that the coregular H -bimo dule F intro duced in Def. 2.8 is the co end of t he functor G H ⊗ k . W e ﬁrst presen t the appropriat e dinatural fa mily . Lemma A.2. The family ( ı F X ) of mo rp hisms ı F X := ( d X ⊗ id H ∗ ) ◦ [ id X ∗ ⊗ ( ρ X ◦ τ X,H ) ⊗ id H ∗ ] ◦ ( id X ∗ ⊗ id X ⊗ b H ) (A.5) in H -Bimo d, pictorial ly given by X ∗ ı F X H ∗ X = X ∗ X ρ X H ∗ (A.6) is dinatur al for the functor G H ⊗ k , i.e. ı F Y ◦ G H ⊗ k ( id Y , f ) = ı F x ◦ G H ⊗ k ( f , id X ) (A.7) for any f ∈ Hom H ( X , Y ) . Pr o of. (i) F irst note that the maps (A.5) ar e a priori just linear maps in Hom k ( X ∗ ⊗ k X , H ∗ ). Ho we ver, when H ∗ is endo we d with the H -bimo dule structure (2.8) and X ∗ ⊗ k X with the one implied b y (A.2), w e ha v e the c hain o f equalities H ρ X ∨ ρ X X ∗ X H ∗ = H X ∗ ρ X X H ∗ = H X ∗ X H ∗ = H X ∗ X H ∗ = H X ∗ ρ X X H ∗ ρ F (A.8) sho wing that (A.6 ) in tertw ines the left action of H , a nd X ∗ X H H ∗ = X ∗ X H H ∗ = X ∗ X H H ∗ = X ∗ X H ρ F H ∗ (A.9) sho wing that it also in tertwine s the right a ctio n. 32 (ii) The dinaturalness prop ert y a moun ts to the equalit y o f the left and r ig h t ha nd sides of X ∗ Y ∗ f ∗ X H ∗ = Y ∗ X f H ∗ = Y ∗ X f Y H ∗ (A.10) for an y mo dule morphism f from X to Y . Now the ﬁrst equality in (A.10) ho lds b y deﬁnition of f ∗ , a nd the second equality holds b ecause f is a mo dule mor phism. Prop osition A.3. The H -bimo dule F to gether with the dina tur al famil y ( ı F X ) give n by (A.5) is the c o end of the functor G H ⊗ k , ( F , ı F ) = Z X G H ⊗ k ( X , X ) . (A.11) Pr o of. W e ha ve to sho w that ( F , ı F ) is a n initial ob ject in the category of dinatural transfor- mations from G H ⊗ k to a constan t . (i) Let j Z b e a dinatural t ransformation from G H ⊗ k to Z ∈ H -Bimo d. Giv en an y X ∈ H -Mo d and a n y x ◦ ∈ Hom k ( k , X ) (i.e. elemen t of X ), applying the dinaturalness prop ert y of j Z to the morphism f x ◦ := ρ X ◦ ( id H ⊗ x ◦ ) ∈ Hom H ( H , X ) (with H rega rded as an H -mo dule via the regular left actio n) yields j Z X ◦ ( id X ∗ ⊗ x ◦ ) = j Z H ◦ ( ı F X ⊗ η ) ◦ ( id X ∗ ⊗ x ◦ ). Namely , w e hav e 000 000 111 111 X ∗ j Z X Z H x ◦ ≡ X ∗ j Z X Z H f x ◦ = f ∗ x ◦ X ∗ j Z H Z H ≡ 00 00 00 11 11 11 X ∗ x ◦ j Z H Z H (A.12) and th us, after comp osition with id X ∗ ⊗ η , j Z X ◦ ( id X ∗ ⊗ x 0 ) = 000 000 000 111 111 111 X ∗ j Z X Z = 00 00 11 11 X ∗ j Z H Z = j Z H ◦ ( ı F X ⊗ η ) ◦ ( id X ∗ ⊗ x ◦ ) (A.13) with ı F X from (A.5). Since x ◦ ∈ Hom k ( k , X ) is arbitrary , w e actually hav e j Z X = j Z H ◦ ( ı F X ⊗ η ) (A.14) 33 for an y bimo dule Z and dinatural transformatio n j Z from G H ⊗ k to Z . (ii) Now consider the linear map κ Z := j Z H ◦ ( id H ∗ ⊗ η ) (A.15) from H ∗ to Z . This is in fact a bimo dule morphism from F to Z : Compatibility with the left H -a ctio n follows directly from t he fact that j Z H is a morphism of bimo dules, and th us in particular of left mo dules, while compatibilit y with the rig ht H -action is seen a s follow s: ρ F H ∗ j Z H Z h = H ∗ Z h j Z H = H ∗ Z h j Z H = H ∗ Z h j Z H = H ∗ Z ρ Z h j Z H (A.16) Here the elemen t h ∈ Hom k ( k , H ) is a rbitrary; the second equalit y inv okes the dinaturalness of j Z for the map m ◦ ( id H ⊗ ( s − 1 ◦ h )) ∈ End H ( H ). (iii) In terms of the morphism κ Z , (A.14) amounts to j Z X = κ Z ◦ ı F X . (A.17) This establishes existence of the morphism fro m F to Z that is required fo r the univ ersal prop erty of the co end. (iv) It remains to sho w that κ Z is uniquely determined. This just follo ws by sp ecializing (A.17) to the case X = H a nd observing that ı F H has a right-in v erse. The latter prop ert y ho lds b ecause of ı F H ◦ ( id H ∗ ⊗ η ) = ( d H ⊗ id H ∗ ) ◦ ( id H ∗ ⊗ b H ) = id H ∗ . A.2 Some equiv alences of braided monoidal categories W e note the following equiv a lences, where as usual H op is H with opp o site pro duct m ◦ τ H,H (and with the same copro duct), and H coop is H with opp osite copro duct τ H,H ◦ ∆ ( a nd with the same pro duct). Lemma A.4. (i) F or any Hopf algebr a H ther e ar e e quivalen c e s H -Bimo d ≃ ( H ⊗ k H )-Mo d ≃ ( H ⊗ k H op )-Mo d (A.18) as ab e lian c ate gories. (ii) The e quivalenc es (A.18) extend to e quivalenc es H -Bimo d ≃ ( H ⊗ k H coop )-Mo d ≃ ( H ⊗ k H op )-Mo d (A.19) as monoida l c ate gories, with r esp e ct to the tenso r pr o ducts (2.6) on H -Mo d and (2.5) on H -Bimo d. The c ons tr aint mo rp h isms for the tenso r functor structur es of the e quivalenc e func- tors ar e al l identities. 34 (iii) If the Hopf algebr a H is quasitriangular with R-matrix R , then the e quivalenc es (A.19) extend to e quivalen c es H -Bimo d ≃ ( H ⊗ k H coop )-Mo d ≃ ( H ⊗ k H op )-Mo d (A.20) as br ai d e d monoida l c ate go ries, wher e H -Bimo d is endo we d with the br aidin g (3.3) and H is H with R-m atrix R − 1 21 . (Also, H op is en dowe d with the natur al quasitriangular structur e inherite d fr om H , i.e. has R-matrix R 21 .) Pr o of. (i) W e deriv e eac h of the equiv alences in a somewhat more general context. F or any tw o Hopf algebras H and H ′ there is an equiv alence H - H ′ -Bimo d ≃ ( H ⊗ k H ′ )-Mo d as ab elian categories. The equiv alence is furnished by the t w o functors whic h on morphisms are the iden tity and whic h map ob jects according to H H ′ M M ρ H ⊗ H ′ 7→ H M M H ′ s ′− 1 ρ H ⊗ H ′ ρ H ⊗ H ′ and H ρ H M M ρ H ′ H ′ 7→ H H ′ M M s ′ (A.21) resp ectiv ely . Similarly , an equiv alence H - H ′ -Bimo d ≃ ( H ⊗ k H ′ op )-Mo d as ab elian categories is furnished b y functors that diﬀer from those in ( A.2 1 ) by just omitting the (inv erse) an tip o de ( compar e e.g. [FRS, Prop. 4.6]). (ii) F or the ﬁrst equiv alence, compatibilit y with the tensor pro duct follows for t he second functor in (A.21) a s H M M N N H ′ 7→ H H ′ M M N N = H H ′ M M N N = H H ′ M M N N (A.22) and analogously fo r the ﬁrst functor, as well as for the second equiv alence. (iii) The Hopf a lgebras H ⊗ k H coop and H ⊗ k H op ha ve natural quasitriangular structures, with R-matrices giv en by ( id H ⊗ c H,H ⊗ id H ) ◦ ( R − 1 21 ⊗ R − 1 ). By direct calculation one ch ec ks that the tw o functors given in (A.21) (with H ′ = H ), resp ectiv ely the o nes with the o ccurences of the an tip o de remo ved, not only furnish a n equiv a lence b etw een H -Bimo d and ( H ⊗ k H coop )-Mo d, resp ectiv ely ( H ⊗ k H op )-Mo d, as ab elian monoidal categories, but map the braidings of these categories to each other as w ell. Also note that the R-matrix furnis hes an equiv alence b etw een H coop -Mo d and H -Mo d as monoidal categories, so tha t in the equiv alences (A.20) w e could as w ell use H instead of H coop . 35 Remark A.5. In view of Lemma A.4 , Prop. A.3 is implied by Theorem 7.4 .13 of [KL]. The signiﬁcance of the co end (A.11) and of the equiv alences in Lemma A.4 actually tran- scends the framew ork of the (bi)mo dule categories considered in this pap er. Namely , one can consider the situation that H - Mo d is replaced b y a more general ribb on category C , while the role of H -Bimo d is take n ov er b y the Deligne pro duct of C with itself. Recall [De, Sect. 5] that the Deligne tensor pro duct of t wo k -linear ab elian categories C and D tha t are lo cally ﬁnite, i.e. all morphism spaces of whic h are ﬁnite-dimensional and all o b j ects of whic h hav e ﬁnite length, is a category C ⊠ D together with a bifunctor ⊠ : C × D → C ⊠ D that is righ t exact and k - linear in b oth v a r iables and has the follo wing univ ersal prop ert y: for any bif unctor G from C × D to a k -linear ab elian category E being righ t exact and k -linear in b oth v ariables t here exists a unique righ t exact k -linear functor G ✷ : C ⊠ D → E suc h that G ∼ = G ✷ ◦ ⊠ . In short, bifunctors from C × D b ecome functors from C ⊠ D . The category C ⊠ D is again k -linear ab elian and lo cally ﬁnite. By the univ ersal prop ert y of the Deligne pro duct, there is a unique functor G H ✷ : H -Mo d ⊠ H -Mo d − → H - Bimo d (A.23) suc h that the bifunctor (A.2) can b e written as the comp osition G H ⊗ k = G H ✷ ◦ (? ∨ ⊠ Id ), with the functor ? ∨ ⊠ Id = ⊠ ◦ (? ∨ × Id ) acting as X × Y 7→ X ∨ ⊠ Y and f × g 7→ f ∨ ⊠ g . On ob jects of C ⊠ D that a re o f the form U ⊠ V with U ∈ C and V ∈ D , the functor G H ✷ acts as ( X , ρ X ) ⊠ ( Y , ρ Y ) G H ✷ 7− →  X ⊗ Y , ρ X ⊗ id Y , id X ⊗ ( ρ Y ◦ τ Y , H ◦ ( id Y ⊗ s − 1 ))  . (A.24) No w b y combining Prop. 5.3 of [De] with the ﬁrst equiv alence in (A.18 ), one sees (com- pare also e.g. [F r, Ex. 7.10]) that the functor G H ✷ is an equiv alence of ab elian categories. F ur- ther, H -Mo d ⊠ H -Mo d has a natural monoida l structure [De, Prop. 5.17] as w ell as a braiding (whic h o n ob jects of the f orm U ⊠ V acts as ( c H -Mo d U ′ ,U ) − 1 ⊗ k c H -Mo d V , V ′ ). With r espect to these the equiv alence (A.23) can b e endow ed with the structure of an equiv alence of braided mono ida l categories. Observ ations analogous to those made here for the category C H = H -Mo d in fact apply to an y lo cally ﬁnite k -linear ab elian ribb on categor y C . Hereby the F rob enius algebra F in H -Bimo d can b e understoo d as a particular case of the co end F C := Z X X ∨ ⊠ X (A.25) of the functor ? ∨ ⊠ Id : C op × C → C ⊠ C (where C is C with opp osite braiding ) , whic h exists for an y suc h category C . This co end F C has already b een considered in [Ke] and [K L, Sect 5.1.3]. It is natural to exp ect t ha t also in this general setting the co end F C still car r ies a na tural F rob enius algebra structure. How ev er, so far w e only kno w that F C is naturally a unital asso ciative algebra in C ⊠ C . F or C = H -Mo d, the category C ⊠ C is also equiv alen t, as a ribb on category , to the cen ter of C , and thus to the catego ry of Y etter-Drinfeld mo dules o ver H . Hence instead of with H - bimo dules w e could equiv alently w ork with Y etter-Drinfeld mo dules o v er H . In particular, the F rob enius alg ebra F can b e recognized as the so-called [FFRS, Da2] full cen ter of the tensor unit of H -Mo d; in the Y etter-Drinfeld setting, t his is describ ed in Example 5.5 of [D a3]. 36 A.3 The co end H H ∗ ⊲⊳ in H -Bimo d The coadjoint left and right actions ρ ⊲ ∈ Hom( H ⊗ H ∗ , H ∗ ) and ρ ⊳ ∈ Hom( H ∗ ⊗ H , H ∗ ) of H on its dual H ∗ are b y deﬁnition the morphisms ρ ⊲ = H H ∗ s H ∗ and ρ ⊳ = H ∗ s H ∗ H s − 1 (A.26) W e call the H - bimo dule that consists of the v ector space H ∗ ⊗ k H ∗ , endo w ed with the coadjoint left H -action on the ﬁrst tensor factor and with the coa djoin t rig ht H -a ctio n on the second factor, the c o adjoint bi m o d ule and denote it by H H ∗ ⊲⊳ . That is, H H ∗ ⊲⊳ = ( H ∗ ⊗ k H ∗ , ρ ⊲ ⊗ id H ∗ , id H ∗ ⊗ ρ ⊳ ) . (A.27) W e will now show that this bimo dule arises as the co end of the functor ⊗ ◦ (? ∨ × Id ) : H -Bimo d op × H -Bimo d → H -Bimo d , (A.28) where ⊗ and ? ∨ are the tensor pro duct (2.5) and right duality (4.1) of H -Bimo d. A crucial input is the bra ided monoidal equiv a lence described in Lemma A.4(iii). Prop osition A.6. L et H b e a ﬁnite-dimensional ri b b on Hopf k -alge b r a. T h en the H -bimo dule H H ∗ ⊲⊳ to gether with the family i ⊲⊳ of morphisms H ∗ ⊗ H ∗ X ∨ i ⊲⊳ X X := X ∗ ρ X X ρ X H ∗ H ∗ (A.29) fr om X ∨ ⊗ X to H H ∗ ⊲⊳ , for X = ( X , ρ X , ρ X ) ∈ H -Bimo d, is the c o e n d fo r the functor (A.28): ( H H ∗ ⊲⊳ , i ⊲⊳ ) = Z X X ∨ ⊗ X . (A.30) Pr o of. The statemen t follo ws fro m the results of [Ly1, Sect. 1.2] and [Vi, Se ct. 4.5] for the co end of the f unctor ⊗ ◦ ( ? ∨ × Id ) from H ′ -Mo d op × H ′ -Mo d to H ′ -Mo d, with the Ho pf algebra H ′ = H ⊗ H op , b y transp orting them via the equiv alence (A.20) to H - Bimo d. W e omit t he details, but ﬁnd it instructiv e to compare a few asp ects of a direct pro of to the 37 corresp onding pa rts of the pro o f of Lemma A.2 a nd of Prop osition A.3: First, dinaturalness follo ws b y an argumen t completely parallel to the one used in (A.10) to sho w dinaturalness of the family (A.5). Second, the r ole o f the morphism f x ◦ (that is, left action of H on an elemen t x ◦ of X ) t hat app ears in fo r mula ( A.12) is tak en o ve r b y the map H X x ◦ H (A.31) This map is a bimo dule morphism fr om H ⊗ k H – regarded as an H - bimo dule ( H ⊗ k H ) reg via the regular left and rig ht actions on the second and ﬁrst factor, resp ectiv ely – to X . Analo- gously a s in (A.13) o ne sho ws that f o r an y dinatural transforma t io n j Z from the functor (A.28) to Z ∈ H -Bimo d one has j Z X ◦ ( id X ∗ ⊗ x ◦ ) = κ Z ◦ i ⊲⊳ X ◦ ( id X ∗ ⊗ x ◦ ), with the map κ Z deﬁned b y κ Z := j Z ( H ⊗ k H ) reg ◦ ( id H ∗ ⊗ id H ∗ ⊗ η ⊗ η ). And aga in κ Z is a bimo dule morphism, so that the existence part of the univ ersal pro p ert y of the co end is established. Uniqueness follows b y sp ecializing to the case X = ( H ⊗ k H ) reg and observing that i ⊲⊳ ( H ⊗ k H ) reg is an epimorphism (as is e.g. seen from i ⊲⊳ ( H ⊗ k H ) reg ◦ ( id H ∗ ⊗ id H ∗ ⊗ η ⊗ η ) = id H ∗ ⊗ id H ∗ ). Corollary A .7. T he H -b imo dule H H ∗ ⊲⊳ c a rrie s the struct ur e of a Hop f alge br a, with structur e morphisms given as fol l o ws. The unit, c ounit and c opr o duct ar e η ⊲⊳ = ε ∨ ⊗ ε ∨ , ε ⊲⊳ = η ∨ ⊗ η ∨ , ∆ ⊲⊳ = ( id H ∗ ⊗ τ H ∗ ,H ∗ ⊗ id H ∗ ) ◦  ( m op ) ∨ ⊗ m ∨  , (A.32) the pr o duct is m ⊲⊳ = H ∗ H ∗ R − 1 H ∗ H ∗ H ∗ R H ∗ = H ∗ H ∗ R − 1 H ∗ H ∗ H ∗ R H ∗ (A.33) 38 and the antip o de is s ⊲⊳ = H ∗ H ∗ R − 1 H ∗ H ∗ R (A.34) Pr o of. W e just ha ve to sp ecialize the general results of [Ly2], whic h a pply to the co end of the functor ⊗ ◦ (? ∨ ⊗ Id ) : C op × C → C in an y k -linear ab elian ribb on category C , to the case C = H - Bimo d. The calculations are straig h tforward, and except for the multiplic a tion a nd the an tip o de they a r e ve ry short. Let us just men tio n that the ﬁrst equality in (A.33) fo llo ws from the general results (see [Ly2, Prop. 2,3], as w ell as [Vi, Sect. 1.6] or [FSc, Sect. 4.3]) together with (4.2 ) and t he deﬁning relation (2.1 ) of the R -matrix. The second equalit y in (A.33) follows with the help of standard manipulations fr o m the fact that the R -matrix in tertwine s the copro duct and the opp osite copro duct. Prop osition A.8. (i ) I f Λ is a two-side d inte gr al of H , then λ ⊲⊳ := Λ ∨ ⊗ Λ ∨ (A.35) is two-side d c ointe gr al o f the Hopf algebr a ( H H ∗ ⊲⊳ , m ⊲⊳ , η ⊲⊳ , ∆ ⊲⊳ , ε ⊲⊳ , s ⊲⊳ ) . (ii) If λ is a right c ointe gr al of H , then Λ ⊲⊳ := λ ∨ ⊗ λ ∨ (A.36) is a two-side d inte gr al of ( H H ∗ ⊲⊳ , m ⊲⊳ , η ⊲⊳ , ∆ ⊲⊳ , ε ⊲⊳ , s ⊲⊳ ) . Pr o of. (i) Inserting the deﬁnitions one has ( λ ⊲⊳ ⊗ id H ∗ ⊗ H ∗ ) ◦ ∆ ⊲⊳ =  m ◦ (Λ ⊗ id H ∗ )  ∗ ⊗  m ◦ ( id H ∗ ⊗ Λ)  ∗ and ( id H ∗ ⊗ H ∗ ⊗ λ ⊲⊳ ) ◦ ∆ ⊲⊳ =  m ◦ ( id H ∗ ⊗ Λ)  ∗ ⊗  m ◦ (Λ ⊗ id H ∗ )  ∗ . (A.37) Since Λ is a tw o-sided integral of H , b o th of these expressions a re equal to η ⊲⊳ ◦ λ ⊲⊳ . (ii) That Λ ⊲⊳ is a left inte g ral readily follo ws fro m the ﬁrst expression for the pro duct in (A.33 ) together with the fact that λ is a righ t coin tegral and that it satisﬁes (5.1 2). That Λ ⊲⊳ is also a righ t cointegral fo llows in the same w ay b y using instead the second expression in (A.33 ) for the pro duct. 39 B Motiv ation from co n formal ﬁe l d the o ry A ma jor motiv ation fo r t he mathematical results of this pap er comes from structures t hat originate in f ull lo cal tw o- dimensional conformal ﬁeld theory . In this app endix w e brieﬂy describe some of these structures. In represen tation theoretic approache s to conformal ﬁe ld theory the starting p oin t is a c hiral symmetry algebra together with its category C of represen tations. F or a ny mathematical structure that f o rmalizes the ph ysical concept of chiral symmetry algebra, the category C can b e endow ed with a lot of additional structure. In particular, in ma ny cases it leads to a so- called mo dular functor. A mo dular f unctor actually consists of a collection of functors. Namely , to an y compact Riemann surface Σ g ,n of gen us g and with a ﬁnite n umber n of mar ked p oints it assigns a functor F Σ g,n : C ⊠ n → V ect (B.1) from C ⊠ n to the category V ect of ﬁnite-dimensional complex vector spaces. This collection of functors is required to ob ey a system of compatibilit y conditions, whic h in particular expresses factorization constrain ts and accommo dates actions of mapping class gr o ups of surfaces. Th us, selecting for a genus - g surface Σ g ,n with n mark ed p oin t s any n -tuple ( V 1 , V 2 , ... , V n ) of ob jects o f the category C , we obtain a ﬁnite-dimensional complex v ector space F Σ g,n ( V 1 , V 2 , ... , V n ) whic h carries an action of the mapping class gro up of Σ g ,n . In chiral conformal ﬁeld theory , this space pla ys the ro le of the space of confo rmal blo ck s with c hiral insertion of t yp e V i at the i th mark ed p oin t of Σ g ,n . In the particular case that the category C is ﬁnitely semisimple, t he structure of a mo dular functor is reasonably w ell understo o d. Sp eciﬁcally , precise conditions are kno wn under whic h the r epresen ta tion category of a v ertex algebra V is a mo dular tensor category . In this case the Reshetikhin-T uraev construction allow s one t o obtain a mo dular functor j ust on the basis of C as an abstract category . In a remark able dev elopment, L yubashenk o and others (see [KL] and references cited there) hav e extended man y asp ects of this story to a larg er class of monoidal categories that ar e not necessarily semisimple an y longer. In particular, g iv en an abstract monoidal category with adequate additional pro p erties, one can still construct represen tations of mapping class gro ups. Represen tation categories that are not semis imple are of considerable ph ysical in terest; they a rise in particular in v arious sys tems of statistical mec hanics. The corresponding mo dels of conformal ﬁeld theory ha ve b een termed “lo garithmic” conforma l ﬁeld theories. A complete c haracterization of this class of mo dels ha s not b een ac hiev ed ye t , but a necessary requiremen t ensuring tractability is that the catego ry C , while b eing no n-semisimple, still p o ssess es certain ﬁniteness prop erties, e.g. eac h ob j ect should ha ve a comp osition series of ﬁnite length. In the presen t pap er w e consider an ev en mor e restricted, but non-empt y , sub class, namely the one for whic h the monoidal c a tegory C is equiv a lent to the represen tation category of a ﬁnite-dimensional factorizable ribb on Hopf algebra. Finite-dimensional Hopf algebras H KL ha ve indeed b een asso ciated, via a Ka zhdan-Lusztig correspo ndence, to certain classes of log- arithmic conformal ﬁeld theories. These Hopf algebras H KL do not hav e an R -matrix, alb eit they do hav e a mono drom y matrix that is ev en factorizable [FGST] (so that in particular the partial mono drom y traces whic h w e in t ro duced in section 5 can still b e deﬁned). Accordingly our results do not p erfectly matc h the presen tly a v aila ble conformal ﬁeld theoretic prop osals. On the other hand, it is apparen t tha t the Hopf algebras H KL are not quite the appro priate 40 algebraic structures: their represen ta tion categories, alb eit b eing equiv alen t to the represen ta - tion categories o f the relev ant v ertex algebras as ab elian categories, are not equiv alent to them as monoidal categories. 6 The Riemann surface o f interes t to us is Σ 1 , 1 , a one-punctured torus. This surface is dis- tinguished by the fact [Y e2] t ha t it carries a natural Hopf algebra structure in the category of three-cob ordisms. F or general reasons, the functor F Σ 1 , 1 is represen table: F Σ 1 , 1 ∼ = Hom C ( K C , − ) . (B.2) In this w ay w e obtain fo r the category C a distinguished ob ject, and this ob ject is actually a Hopf alg ebra in C . The construction in [Ly1] turns the logic around: it starts with a Hopf algebra ob ject K C ∈ C canonically asso ciated to the braided category and constructs the functor as F Σ 1 , 1 ( V ) = Hom C ( K C , V ) for any ob ject V ∈ C . F rom the p oin t of view of t wo-dimens io nal conformal ﬁeld theory , the one-punctured torus is the surface relev a n t fo r part ition f unctions. W e are in terested in this pap er in a candidate for the par t it ion function of the space of bulk ﬁelds and th us in one-p oint functions of bulk ﬁelds on the torus. The space H bulk of bulk ﬁelds carries t he structure of a bimo dule ov er the c hir a l symmetry algebra V . In the case that the cat ego ry C is semisimple, a particularly simple solution is giv en b y the bulk state space L i S ∨ i ⊗ C S i , where the (ﬁnite) summation is o ve r all isomorphism classes [ S i ] o f simple V -mo dules. The corresp onding partition f unction is the so-called char ge c on jugation mo dular inv arian t. It has b een conjectured [QS, GR] that this t yp e of bulk state space exists in the non-semisimple case as w ell, and that as a left V -mo dule it decomp oses as H bulk ∼ = M i P ∨ i ⊗ C S i , (B.3) with P i the pro jectiv e co ver of the simple V -mo dule S i . According to the principle of holomorphic factorizatio n, a correlation function for a con- formal real surface is an eleme nt in the space of conformal blo c ks associated to the oriented double of the surface. Th us a one-p oint function on t he torus is a sp eciﬁc elemen t in the space of conformal blo c ks asso ciated to the do uble o f the torus (as a real surface), that is, o f the discon- nected sum of t w o copies of Σ 1 , 1 with opp osite orien tation. F or a ny selection of a pair ( V 1 , V 2 ) of ob jects of C at the t w o p o in ts on the double cov er that lie o ve r the o ne insertion p oint on the torus, t his space of conformal blo ck s is the tensor pro duct Hom C ( K C , V 1 ) ∗ ⊗ C Hom C ( K C , V 2 ). More compactly , this space can b e written a s a morphism space of another bra ided tensor cat- egory D := C ⊠ C , whic h has its own canonical Hopf algebra ob ject K D . As we ha ve noted in Section A.2, if C is the category H -Mo d of left mo dules o ve r a ﬁnite-dimensional factor izable ribb on Hopf algebra H , then D can b e iden tiﬁed with the category of bimo dules o ver H , with a tensor pro duct deriv ed fro m the copro duct on H . Compatibilit y of (B.3 ) with short exact seque nces implies that the c hara cter of the pro jectiv e co v er P i is a linear com bination of simple ch aracters, with co eﬃcien t s giv en by the en tries o f the Cartan ma t r ix of the categor y . Th us in the charge conjugation case the Cartan matrix pro vides the co eﬃcien ts in the bilinear combination of c haracters that, o wing to holomorphic 6 Also, constructing alg ebras with the help of the Kazhdan-Lusztig corr esp ondence in volves some arbitrar i- ness. It has been suggested [ST] that one should b etter work with Hopf algebr as in a category of Y etter-Drinfeld mo dules built from bra ided v e c tor s paces, rather than Hopf algebr as in V ect . 41 factorization, describes the bulk partit io n function. Indeed, as men tioned in Remark 5.12(iii), the same structure is seen when expres sing the morphism ε F ◦ t Q (∆ F ), whic h (as follo ws f rom Remark 5.6(i)) is nothing but the character of F for the alg ebra K D , as a bilinear combination of c haracters for t he algebra K C . Correlation functions in conformal ﬁeld theory should b e inv a rian t under the relev an t map- ping class group. F or the one-p oin t corr elatio n function on the torus w e are thu s inte rested in ﬁnding an ob ject F ∈ D corresp onding to the space o f bulk ﬁelds as w ell as a v ector Z F ∈ Hom D ( K D , F ) (B.4) that is in v ar ia n t under the action of the mapping class group Γ 1;1 of the one-punctured tor us. Moreo v er, comparison with the semisimple situation, in whic h C is a mo dular tensor category , indicates that the ob j ect F should p ossess a structure of a commutativ e symmetric F rob enius algebra in D . The partition function, give n b y ε F ◦ Z F , is then inv arian t under the mo dular group SL(2 , Z ). This is precisely what the presen t paper achie ves for the case C ≃ H -Mo d: given a ribb on Hopf alg ebra automorphism ω o f H , w e obtain a commutativ e symmetric F rob enius algebra F ω in the category H -Bimo d. As an ob ject, F ω is the t wisted coregular bimo dule id H ( F ) ω , so that e.g. its decomp osition as a left H - mo dule precisely repro duces the decomp osition (B.3) ab o v e. (The conjecture (B.3 ) has only b een made for the case corresp onding to trivial automorphism ω = id H , though.) Also note that according to Remark 6.3(ii) the counit o f F ω is unique up to a non-zero scalar; in the conformal ﬁeld theory context this amo unts to uniqueness o f the v acuum state. The partial mono drom y tr a ce (5.23) of the copro duct ∆ : F ω → F ω ⊗ F ω furnishes a Γ 1;1 -in v ariant morphism Z ω ∈ Hom( K D , F ω ). The morphism ε F ω ◦ Z ω , asso ciated to H and ω , is a natural candidate for a mo dular in v ariant partition function on t he torus. A cknowledgments: JF is la rgely supp orted by VR under pro ject no. 621 - 2009-3 993. CSc is partially supp or t ed by the Collab orat ive Researc h Cen tre 676 “P articles, String s and the Early Univ erse - the Structure of Matter and Space-Time” and b y t he DFG Priorit y Programme 1388 “Represen tation Theory”. W e tha nk Jens Fjelstad for discussions. JF is grateful to Ham burg univ ersit y , and in particular to CSc and Astrid D¨ orh¨ ofer, for their hospitalit y during the time when this study was initia ted. 42 References [Bi] J. Bic hon, Cosover eign Hopf algebr as , J. Pur e App l. Alg. 157 (2001) 121–133 [math.QA /9902030 ] [Bu] S. Burciu, A class of Drinfeld doubles that ar e ribb on algebr as , J. Algebra 320 (2008) 2053– 2078 [CW1] M. Cohen and S. W estreic h, Some interr elations b etwe en Hopf algebr as and their duals , J . 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