Hopf algebras and Frobenius algebras in finite tensor categories

ZMP-HH/10-8 Hamburger Beitr¨ age zur Mathematik Nr. 366 Marc h 2010 HOPF ALGEBRAS AND FR OBENIUS ALGEBRAS IN FINITE TENSOR CA TEGORIES Christoph Sc h w eigert a and J ¨ urgen F uc hs b a Or ganisationseinheit Mathematik, Universit¨ at Hambur g Ber eich A lgebr a und Zahlenthe orie Bundesstr aße 55, D – 20 146 Hambur g b T e or etisk fysik, Karlstads Universi tet Universitetsgatan 21, S – 651 88 Karlstad Abstract W e discuss alge braic and representat ion theoretic stru ctures in br aided tensor catego ries C wh ic h ob ey certain fi niteness conditions. Muc h interesti ng structure of suc h a category is enco ded in a Hopf algebra H in C . In p articular, the Hopf algebra H giv es r ise to rep- resen tations of the mo du lar group SL( 2 , Z ) on v arious morphism spaces. W e also explain ho w ev ery symmetric sp ecial F rob enius algebra in a semisimple mo dular cat egory pro vides additional structure related to th ese represen tations. 1 Braided finite te nsor catego ries Algebra and represen tation theory in semisimple ribb on catego ries has b een an activ e field o v er the las t decade, ha ving applications to quan tum gro ups, lo w-dimensional top ology and quan tum field theory . More recen tly , partly in connection with progress in the understanding of log arithmic conformal field theories, there has b een increased interest in tensor catego r ies that are not semisimple an y longer, but still ob ey certain finiteness conditions [EO]. Owing to the w ork of v arious groups (for some recen t results s ee e.g. [GT, NT]), examples of suc h categories are b y now r ather explic itly understo o d, at least as ab elian catego ries. In t his section w e describe a class of categories that has receiv ed part icular atten tion. This will allow us to define the structure of a semisimple mo dular tensor catego r y . T o extend the notion of mo dular tensor category to the non-semisimple case requires further categorical constructions in v olving Hopf algebras and co ends; these will b e introduced in sec tion 2. These constructions also pro vide represen tations of the mo dular group SL(2 , Z ) on certain morphism spaces. In section 3 w e sho w that symmetric sp ecial F rob enius algebras in semisimple mo dular tensor categories giv e rise to structures related to suc h SL(2 , Z )- represen tations. Let k b e an algebraically closed field of c haracteristic zero and V ec t fin ( k ) the categor y of finite-dimensional k -v ector spaces. Definition 1.1. A finite category C is a n ab elia n c ate gory enriche d over V ect fin ( k ) with the fol lowing additional pr op erties: 1. Ev e ry obje ct h as finite length. 2. Ev e ry obje ct X ∈ C ha s a pr oje ctive c over P ( X ) ∈ C . 3. T he set I of isom o rphism classes of simple o b je cts is finite. It can be sho wn that an abelian category is a finite category if and only if it is equiv alent to the category of (left, say) mo dules ov er a finite-dimen sional k -algebra. W e w ill b e concerned with finite categories that ha v e additional structure. First, they ar e tensor categories, i.e., for our purposes, so v ereign monoidal categories: Definition 1.2. A tensor category o v er a field k is a k -line ar ab elian m onoidal c ate gory C with simple tensor unit 1 and with a left and a right d uality in the sens e o f [K a , Def. XIV.2.1] , such that the c ate gory is sover e i g n, i.e. the two functors ? ∨ , ∨ ? : C → C opp that ar e induc e d by the left and right dualities c oi n cide. Thus for an y obje ct V ∈ C ther e exists an obje ct V ∨ = ∨ V ∈ C to gether w ith morphisms b V : 1 → V ⊗ V ∨ and d V : V ∨ ⊗ V → 1 (right duality) and ˜ b V : 1 → V ∨ ⊗ V and ˜ d V : V ⊗ V ∨ → 1 (left duality), ob eying the r elations (id V ⊗ d V ) ◦ ( b V ⊗ id V ) = id V and ( d V ⊗ id V ∨ ) ◦ (id V ∨ ⊗ b V ) = id V ∨ and analo gous r elations for the right d uality, an d the duality functors not only c oinc ide on obje cts, but also on morphisms, i. e . ( d V ⊗ id U ∨ ) ◦ (id V ∨ ⊗ f ⊗ id U ∨ ) ◦ (id V ∨ ⊗ b U ) = (id U ∨ ⊗ ˜ d V ) ◦ (id U ∨ ⊗ f ⊗ id V ∨ ) ◦ ( ˜ b U ⊗ id V ∨ ) for al l morphisms f : U → V . T o give an example, the category of finite-dimensional left mo dules ov er an y finite-dimen- sional complex Hopf algebra H is a finite tensor category . As a direct consequence of t he definition, the tensor pro duct functor ⊗ is exact in b oth argumen ts. W e will imp ose on the dualities the additional requiremen t that left and righ t dualit y lead to the same cyclic trace tr : End( U ) → End ( 1 ), and th us to a dimension dim( U ) = tr(id U ). The categories o f our in terest ha ve in addition a braiding: 2 Definition 1.3. A braiding on a tensor c ate gory C is a natur al isomorphism c : ⊗ → ⊗ opp that is c omp atible with the tensor pr o duct, i.e. satisfie s c U ⊗ V ,W = ( c U,W ⊗ id V ) ◦ (id U ⊗ c V ,W ) and c U,V ⊗ W = ( id V ⊗ c U,W ) ◦ ( c U,V ⊗ id W ) . W e c ho ose a set { U i } i ∈ I of represen ta tiv es fo r the isomorphism classes of simple ob jects a nd tak e the tensor unit to be the repres en tativ e of its isomorphism class, writing 1 = U 0 . W e are now ready to for mulate the notion of a mo dular tensor category . Our definition will, ho w ev er, still be preliminary , as it has the disadv an tage of being sensible only for semisimple categories. Definition 1.4. A semisimple mo dular tensor category is a semis i m ple finite br aide d tensor c ate gory such that the matrix ( S ij ) i,j ∈ I with entries S ij := tr( c U j ,U i ◦ c U i ,U j ) is non- d e gener ate. Tw o r emarks are in order: Remarks 1.5. 1. The r epr e sentation c ate g o ries of sever al alg ebr aic structur es give exam p l e s of s e misimple mo dular tensor c ate g o ries: (a) L eft mo dules over c onne cte d factorizab le ribb on we ak Hopf algebr as with Haar inte gr al over an a lgebr aic al ly close d field [NTV] . (b) L o c al se ctors of a finite µ -i n dex net of von Neumann algebr as on R , if the ne t is str ongly add itive and split [KLM ] . (c) R epr esentations of selfdual C 2 -c ofinite vertex algebr as with an ad ditional finiteness c ondition on the homo gene ous c omp onents and which have semisimple r epr ese n tation c ate gories [Hu ] . 2. By the r esults of R eshetikhin a nd T ur aev [R T, T ] , every C -line ar semi s i m ple mo dular tensor c ate go ry C pr ov i d es a thr e e-dimension al top olo gic al field the ory, i.e. a tensor f unc tor tft C : cob ord C 3 , 2 → V ect fin ( C ) . Her e cob ord C 3 , 2 is a c ate go ry of thr e e-dimen sional c ob or dism s wi th emb e dde d rib b on gr aphs that ar e de c or ate d by obje cts a n d morphisms of C . Ther e ar e also vario us r esults f o r the c ase of non- s emisimple mo d ular c ate gorie s . We r efer to [He, L 1 , V] for the c onstruction of thr e e-m anifold invaria n ts, to [L1] for the c onstruction of r epr esentations of mapping class gr oups, and to [KL] for an attempt to unify these c ons tructions in terms of a top o lo gic al quantum field the ory define d on a do uble c ate gory of manifolds with c orners. 3 2 Hopf algebras, co en ds and mo dul ar te n sor categori es Our go a l is to study some algebraic and represen tation theoretic structures in tensor categories of the type introduced a b o v e. T o simplify the exp osition, w e supp ose that we ha ve replaced the tensor category C b y an equiv alen t strict tensor category . Definition 2.1. A (unital, asso ciative) algebra in a (strict) tensor c ate gory C is a triple c onsisting of an obje ct A ∈ C , a m ultiplic ation m orphism m ∈ Hom( A ⊗ A, A ) and a unit morph ism η ∈ Hom( 1 , A ) , subje ct to the r elations m ◦ ( m ⊗ id A ) = m ◦ (id A ⊗ m ) and m ◦ ( η ⊗ id A ) = id A = m ◦ (id A ⊗ η ) . which expr e s s asso cia tivity and unitality. A nalo gously, a coalgebra in C is a triple c onsisting o f an ob je ct C , a c omultiplic ation ∆ : C → C ⊗ C and a c ounit ε : C → 1 o b eying c o ass o ciativity and c ounit c onditions. Similarly one generalizes other basic notions of algebra to the categorical setting and in- tro duces mo dules, bimo dules, como dules, etc. (F or a more complete exp osition we refer to [FRS1].) T o pro ceed w e observ e that the m ultiplication of an algebra A endows b oth A itself and A ⊗ A with the structure of an A - bimo dule. F urther, if the category C is braided, then the ob ject A ⊗ A can b e endow ed with the structure of a unital asso ciative algebra b y taking the morphisms ( m ⊗ m ) ◦ (id A ⊗ c A,A ⊗ id A ) as the pro duct and η ⊗ η as the counit. Definition 2.2. L et C b e a tensor c ate gory and A ∈ C an obje ct whi c h is en d owe d with b oth the structur e ( A, m, η ) of a unital asso ciative algebr a and the structur e ( A, ∆ , ε ) of a c ounital c o asso ciative c o algeb r a. 1. ( A, m, η , ∆ , ε ) is c al le d a F rob enius algebra iff ∆ : A → A ⊗ A is a morphism of bimo dules. 2. ( A, m, η , ∆ , ε ) is c al le d a bialgebra iff ∆ : A → A ⊗ A is a morphism of unital algebr as. 3. A bialgebr a with an antip o de S : A → A (with pr op erties analo g o us to the classic al c ase) is c al le d a Hopf algebra . T o construct concrete examples of suc h structures, w e recall a few notions from category theory . Definition 2.3. L et C and D b e c ate gories and F : C opp × C → D b e a functor. 1. F or B an o bje ct of D , a dinatural tra nsformation ϕ : F ⇒ B is a family o f morphisms ϕ X : F ( X, X ) → B for every obje ct X ∈ C such that the diagr am F ( Y , X ) F (id Y ,f ) / / F ( f, id X )   F ( Y , Y ) ϕ Y   F ( X , X ) ϕ X / / B c ommutes for al l morphisms X f → Y in C . 4 2. A co end for the functor F is a dinatur al tr a nsformation ι : F ⇒ A with the universal pr op erty that an y din atur al tr ansform ation ϕ : F ⇒ B uniquely factorizes: F ( Y , X ) F (id Y ,f ) / / F ( f, id X )   F ( Y , Y ) ι Y   ϕ Y   F ( X , X ) ϕ X 2 2 ι X / / A " " F F F F F B If the co end exis ts, it is unique up to unique isomorphism. It is denoted b y R X F ( X , X ). The univ ersal prop erty implies that a morphism with domain R X F ( X , X ) can be sp ecified by a dinatural f a mily of morphisms X ∨ ⊗ X → B for eac h ob ject X ∈ C . W e a re no w ready to fo rm ulat e the follo wing result. Theorem 2.4. [L2] In a finite br aide d tensor c ate gory C , the c o end H := Z X X ∨ ⊗ X of the functor F : C opp × C → C ( U, V ) 7→ U ∨ ⊗ V exists, and it has a natur al struct ur e of a Hopf algebr a in C . Pro of: F or a pro of we refer e.g. to [V]. Here we only indicate ho w the structural morphisms of the Hopf algebra are cons tructed. Owing to the univ ersal prop erty , the counit ε H : H → 1 can b e sp ecified b y the dinatural family ε H ◦ ι X = d X : X ∨ ⊗ X → 1 of morphisms . Similarly , the copro duct is giv en b y the dinatural fa mily ∆ H ◦ ι H = ( ι X ⊗ ι X ) ◦ (id X ∨ ⊗ b X ⊗ id X ) : X ∨ ⊗ X → H ⊗ H . It should b e appreciated that the braiding do es not enter in the coalgebra struc ture of H . It do es en ter in the pro duct, though. W e refrain from writing out the pro duct as a f or- m ula. Ins tead, w e use the graphical formalism [JS, FRS1] to display all structural morphisms ( m H , ∆ H , η H , ǫ H , S H ) of the Hopf algebra H . More precisely , w e displa y dinatural families of 5 morphisms so tha t the iden tities apply to all X, Y ∈ C : X ∨ X H m H Y ∨ Y ι X ι Y = γ X,Y id Y | X X ∨ X H Y ∨ Y ( Y ⊗ X ) ∨ Y ⊗ X H H X ∨ X ∆ H = H H X ∨ X H η H = H ε H X ∨ X = X ∨ X S H H = H X ∨ ∨ X ∨ (Here γ X,Y is the canonical iden t ificatio n of X ∨ ⊗ Y ∨ with ( Y ⊗ X ) ∨ , and id X | Y is the one of id X ⊗ id Y with id X ⊗ Y .)  An explicit description of the Hopf alg ebra H ∈ C is av ailable in the follo wing sp ecific situ- ations: Examples 2.5. 1. F or C = H -mo d the c ate gory of left mo dules ove r a finite-dimension a l ribb on Hopf algebr a H , the c o end H = R X X ∨ ⊗ X is the dual sp ac e H ∗ = Hom k ( H , k ) endowe d with the c o adjoint r epr esentation. T he structur e morphism for the c o end for a mo dule M ∈ H -mo d is ι M : M ∨ ⊗ M → H ∗ ˜ m ⊗ m 7→ ( h 7→ h ˜ m, h.m i ) . F or mor e details se e [V, Sect. 4.5] . 2. I f the finite tensor c ate g ory C is semisimple, then the Hopf al g e br a de c omp o s e s as an obje ct as H = L i ∈ I U ∨ i ⊗ U i , se e [V, Sect. 3.2] . The Hopf algebra in question has additional structure: it comes with an integral and with a Hopf pairing . Definition 2.6. A left in tegral of a b ialgebr a ( H, m, η , ∆ , ε ) in C is a non-zer o morphism µ l ∈ Hom( 1 , H ) satis- fying m ◦ (id H ⊗ µ l ) = µ l ◦ ε . A right cointe gral of H is a non-zer o morphism λ r ∈ Hom( H , 1 ) s atisfying ( λ ⊗ id H ) ◦ ∆ = η ◦ λ . Right i n te gr als µ r and left c ointe gr als λ l ar e de fine d analo gously. 6 The Hopf algebra H in a n y finite braided tensor category has left and right in tegrals, as can b e sho wn [L2] b y a generalization of t he classic al argument of Sw eedler tha t an in tegral exists for a n y finite-dimen sional Hopf algebra. If C is semisimple, then the in tegral o f H can b e give n explicitly [Ke, Sec t. 2.5]: µ l = µ r = M i ∈ I dim( U i ) b U i . Remarks 2.7. 1. I f the left and right inte gr als of H c oincide, then the inte gr al c an b e use d as a Kirby el- ement and pr ovid e s invariants of thr e e-manifo l d s [V] . If the c ate gory C is the c ate gory of r epr esentations of a finite-dimensiona l Hopf alge b r a, this is the Hennings-Lyub ashenko [L1] invariant. 2. T he c ate gory C is semisimple if and on l y if the morphism ε ◦ µ ∈ Hom( 1 , 1 ) do es not vanish, i.e. iff the c o nstant D 2 of pr op ortion a lity in ε ◦ µ = D 2 id 1 is non-zer o. (This gener aliz e s Maschke’s the or em .) This c onstant, in turn, which in the semisimple c ase (with µ l = µ r normalize d as ab ove ) has the value D 2 = P i ∈ I (dim U i ) 2 , cru- cial ly enters the normalizations in the R e shetikhin-T ur aev c onstruction of top o lo gic al field the ories (se e e.g. chapter II of [T] ) . Invariants b ase d on nonsemisimple c ate gories, like the Hennings invariant, vanish o n many thr e e-manif o lds. This c an b e tr ac e d b a c k to the vanishing o f ε ◦ µ [C KS] . 3. Any Hopf alg ebr a H in C with invertible antip o de that has a left inte gr al µ and a right c oin- te gr al λ with λ ◦ µ 6 = 0 is natur al ly also a F r ob eni us algebr a, with the same alge b r a structur e. Definition 2.8. A Hopf pairing of a Hopf algebr a H in C is a m o rphism ω H : H ⊗ H → 1 such that ω H ◦ ( m ⊗ id H ) = ( ω H ⊗ ω H ) ◦ (id H ⊗ c H,H ⊗ id H ) ◦ (id H ⊗ id H ⊗ ∆) , ω H ◦ (id H ⊗ m ) = ( ω H ⊗ ω H ) ◦ (id H ⊗ c − 1 H,H ⊗ id H ) ◦ (∆ ⊗ id H ⊗ id H ) and ω H ◦ ( η ⊗ id H ) = ε = ω H ◦ (id H ⊗ η ) . As one easily c hec ks, a non-degenerate Hopf pairing giv es an isomorphism H → H ∨ of Hopf algebras. The dinatural fa mily of morphisms ( d X ⊗ d Y ) ◦ [id X ∨ ⊗ ( c Y ∨ ,X ◦ c X,Y ∨ ) ⊗ id Y ] induces a bilinear pair ing ω H : H ⊗ H → 1 on the co end H = R X X ∨ ⊗ X of a finite braided tensor category . It endo ws [L1] the Hopf algebra H with a symmetric Hopf pairing. W e are no w finally in a p osition to g iv e a conceptual de finition of a modular finite tensor category without r equiring it to be semisimple: 7 Definition 2.9. [KL, Def. 5.2.7] A mo dular finite tensor category is a br aide d finite tenso r c ate gory for which the Hopf p airing ω H is non-de g e n er ate. Example 2.10. The c ate gory H -m o d of left mo d ules over a finite-dimensio nal factorizable ribb o n Hopf algebr a H is a mo dular finite tensor c ate gory [LM, L1] . One can show [L2, Thm. 6.11] that if C is mo dular in t he sense of definition 2 .9, then the left in tegral and the r ig h t inte gral of H coincide. As the terminology suggests, there is a relation with the mo dular group SL(2 , Z ). T o see this, w e will no w obtain eleme n ts S H , T H ∈ End( H ) that satisfy the relations for generators of SL(2 , Z ). Recall the notion o f the cen ter Z ( C ) of a category a s the algebra of natural endotransforma- tions of the iden tity endofunctor o f C [Ma]. Giv en suc h a natural transformation ( φ X ) X ∈C with φ X ∈ End( X ), one chec ks that ( ι X ◦ (id X ∨ ⊗ φ X )) X ∈C is a dinatural family , so that the univ ersal prop ert y of the co end giv es us a unique endomorphism φ H of H suc h that the diagra m X ∨ ⊗ X id ⊗ φ X / / ι X   X ∨ ⊗ X ι X   H φ H / / _ _ _ _ _ _ _ _ _ H comm ut es, leading to an inj ective linear map Z ( C ) → End( H ). Since H has in particular t he structure of a coalgebra and 1 the structure of a n algebra, the v ector space Hom( H , 1 ) ha s a natural structure of a k - algebra. Concatenating with the counit ε H giv es a map Z ( C ) − → End( H ) ( ε H ) ∗ − − − → Hom( H , 1 ) , whic h can b e shown [Ke, Lemma 4] to b e an isomorphism of k -alg ebras. The v ector space o n the righ t ha nd side is dual t o the ve ctor space Ho m( 1 , H ), of whic h one can think as the appropriate substitute fo r the space of class functions. Hence Hom( 1 , H ) would b e a na t ural starting p o in t for constructing a ve ctor space assigned to the torus T 2 b y a topolo g ical field theory based on C . If the catego r y C is a ribbon category , w e ha v e the ribb on elemen t ν ∈ Z ( C ). W e set T H := ν H ∈ End( H ) . Pictorially , ν H X ∨ X H = X ∨ X H 8 Another morphism Σ : H ⊗ H → H is obtained from the follo wing family of morphisms whic h is dinatural both in X and in Y : Σ X ∨ X Y ∨ Y H := X ∨ X Y ∨ Y H Comp osing this morphism to H with a left or righ t inte gral µ : 1 → H one a rriv es a t an endo- morphism S H := Σ ◦ (id H ⊗ µ ) ∈ End( H ) . F or ξ ∈ k × , denote b y k ξ SL(2 , Z ) the tw isted group algebra of SL(2 , Z ) with relatio ns S 4 = 1 and ( S T ) 3 = ξ S 2 . The previous construction and the follo wing result a re due to Lyub ashenk o. Theorem 2.11. [L2, Sect. 6] L et C b e m o dular. Then the two-si d e d inte gr al of H c an b e norm a lize d in such a way that the endomorphism s S H and T H of H pr ovide a morphism of algebr as k ξ SL(2 , Z ) − → End( H ) for some ξ ∈ k × . Since for ev ery U ∈ C the morphism space Hom( U, H ) is, by push-forw ar d, a left mo dule o v er the algebra End ( H ), w e obtain this wa y pro jectiv e represe n tations of SL(2 , Z ) on a ll v ector spaces Hom( U, H ). T o se t the stage f or the results in the next section, w e consider the map Ob j( C ) → Hom( 1 , H ) U 7→ χ U with χ U : 1 b U − → U ∨ ⊗ U ι U − → H . It factorizes to a morphism o f rings K 0 ( C ) → Hom( 1 , H ) = tft C ( T 2 ) . If the category C is semisimple, t hen Hom( 1 , H ) ∼ = L i ∈ I Hom( 1 , U ∨ i ⊗ U i ), so that { χ U i } i ∈ I constitutes a basis o f the v ector space Hom( 1 , H ). If C is not sem isimple, these elemen ts are still linearly independen t, but they do not form a basis an y more. Pseudo-c hara cters [Mi, GT] ha v e b een proposed as a (non-cano nical) comple men t o f this linearly independen t set. 3 F rob enius algebras and b r aid ed induct i o n In this section w e show that symmetric sp ecial F rob enius a lg ebras (i.e. F rob enius algebras with t w o further prop erties, to be defined b elo w) in a mo dular tensor catego ry allo w one to sp ecify in teresting structure related to the SL(2 , Z )-represen ta tion that w e ha v e just explained. 9 Giv en an algebra A in a braided (strict) tensor category , we consider the t w o tensor functors α ± A : C → A - bimo d U 7→ α ± A ( U ) whic h assign to an ob ject U ∈ C t he bimo dule ( A ⊗ U, ρ l , ρ r ) for whic h the left action is giv en b y m ultiplication and the righ t action by multiplication composed with a braiding, ρ l = m ⊗ id U ∈ Hom( A ⊗ A ⊗ U, A ⊗ U ) and ρ + r = ( m ⊗ id U ) ◦ (id A ⊗ c U,A ) and ρ − r = ( m ⊗ id U ) ◦ (id A ⊗ c − 1 A,U ) . W e call these functors br aide d induction functors. They hav e been intro duced, under the name α -induction, in op erato r algebra theory [LR, X, BE]. F or more details in a category-theoretic framew ork w e refer to [O, Sect. 5.1]. W e pause to recall that [VZ] an Azumaya alge b r a A is an algebra for whic h the t w o functors α ± A are equiv alences of tensor categories. This s hould b e compared to the textb o ok definition of an Azuma ya algebra in the t ensor category of mo dules ov er a comm utative k -algebra A , requiring in particular the morphism ψ A : A ⊗ A opp → End( A ) a ⊗ a ′ 7→ ( x 7→ a · x · a ′ ) to be an isomorphism of algebras. Indeed, in this situation for a n Az umay a algebra A one ha s the follo wing c hain of eq uiv alences: A -bimo d ∼ − → A ⊗ A opp -mo d ψ A − − → End( A )-mo d Morita − − − − − → V ect ( k ) . W e now intro duce the prop erties of an alg ebra A to b e symmetric and special. Definition 3.1. L et C b e a tensor c ate gory. 1. F or C enriche d over the c ate gory of k -ve ctor sp a c es, a sp ecial algebra in C is an obje c t A of C that is endowe d with an algebr a struct ur e ( A, m, η ) and a c o algebr a structur e ( A, ∆ , ε ) such that ε ◦ η = β 1 id 1 and m ◦ ∆ = β A id A with invertible elements β 1 , β A ∈ k × . 2. A symme tric algebra i n C is an algebr a ( A, m, η ) to g e ther with a morphism ε ∈ Hom( A, 1 ) such that the two morphisms Φ 1 := [( ε ◦ m ) ⊗ id A ∨ ] ◦ (id A ⊗ b A ) ∈ Hom( A, A ∨ ) and (1) Φ 2 := [id A ∨ ⊗ ( ε ◦ m )] ◦ ( ˜ b A ⊗ id A ) ∈ Hom( A, A ∨ ) (2) ar e id entic al. Sp ecial algebras are in part icular separable, and as a consequence their catego r ies of mo dules and bimo dules are semisimple. A class of examples of sp ecial F rob enius alg ebras is supplied by the F rob enius algebra structure o n a Hopf algebra H in C , pro vided H is semisimple. 10 W e no w consider the case of a semis imple mo dular tensor category C and introduce f o r a ny algebra A in C the square matrix ( Z ij ) i,j ∈ I with en tries Z ij ( A ) := dim k Hom A | A ( α − A ( U i ) , α + A ( U ∨ j )) , where Ho m A | A stands for homomorphisms of bimo dules. Iden tifying A -bimo d with the tensor category of mo dule endofunctors of A -mo d, one sees that the non-negativ e in tegers Z ij ( A ) only dep end on the Morita class of A . In this setting, and in case that the algebra A is sym metric and special, w e can mak e the follo wing statemen ts. Theorem 3.2. [FRS1, Thm. 5.1(i)] F or C a semisimple m o dular tensor c ate gory and A a sp e cial symmetric F r ob enius algebr a in C , the morphism X i,j ∈ I Z ij ( A ) χ i ⊗ χ j ∈ Hom( 1 , H ) ⊗ k Hom( 1 , H ) (3) is invari a n t under the diagonal a c tion of SL (2 , Z ) . Remarks 3.3. 1. In c onfo rmal field the ory, the ex p r e s s ion (3) has the interpr etation of a p artition function for bulk field s. 2. F o r s emisimple tensor c ate gories b ase d on the sl (2) affine Lie algebr a, an A-D-E p a ttern app e ars [K O] . W e finally summarize a few other results that hold under the assumption that C is a semisim- ple mo dular tensor cat ego ry and A a sym metric sp ecial F rob enius algebra in C . T o formulate them, w e need the follo wing ingredien ts: Th e fusion algeb r a R C := K 0 ( C ) ⊗ Z k is a separable comm utative algebra with a natural basis { [ U i ] } i ∈ I giv en by the isomorphism classes of simple ob jects. The mat rix S introduced in definition 1.4 pro vides a natural bijection from the set of isomorphism class es of irreducible repres en tations of R C to I . Theorem 3.4. [FRS1, Thm. 5.18] F or any sp e cial symmetric F r ob en i us algebr a A the ve ctor sp ac e K 0 ( A -mo d ) ⊗ Z k is an R C - mo dule. The multiset Exp ( A -mo d ) that c ontains the irr e ducible R C -r epr esentations , with their multiplicities in this R C -mo dule, c an b e expr esse d in terms of the matrix Z ( A ) : Exp( A -mo d ) = Exp( Z ( A )) := { i ∈ I with multiplicity Z ii ( A ) } . The observ ation that the vec tor space K 0 ( A -mo d) ⊗ Z k ha s a natural basis provided b y the classes of simple A -mo dules giv es Corollary 3.5. The numb er of isomorphism classes of simple A -mo dules e q uals tr( Z ( A )) . The category A -bimo d of A -bimo dules has the structure of a tensor category . F r o m the fact that A is a symmetric sp ecial F robenius algebra, it follo ws [FS] that A -bimod inherits left and righ t dualities from C . Hence the tensor pro duct on A -bimo d is exact and thus K 0 ( A -bimo d) is a ring. The corresponding k -algebra can again b e describ ed in t erms of the matrix Z ( A ): 11 Theorem 3.6. [O, FRS2] Ther e is an isomorphism K 0 ( A -bimo d ) ⊗ Z k ∼ = M i,j ∈ I Mat Z ij ( A ) ( k ) , of k -algebr as, w ith Mat n ( k ) denoting the algebr a of k -value d n × n -matric es . Corollary 3.7. The numb er of isomorphism classes of simple A -bimo dules e quals tr( Z Z t ) . Theorem 3.8. [FFRS, Prop. 4.7] A ny A -bimo dule is a sub quotient of a bimo dule of the form α + A ( U ) ⊗ A α − A ( V ) for some p air of obje cts U, V ∈ C . 4 Outlo ok W e conclude this brief review with a few commen ts. First, all the results ab out algebra and represen tation theory in braided tensor cat ego ries that we ha v e pres en ted ab o v e a r e motiv ated b y a construction of correlation functions of a rational confor mal field theory as elemen ts o f v ector spaces whic h are assigned b y a t o p ological field theory to a tw o-manifold. F or details of this construction w e refer to [SFR] and the litera t ur e giv en there. In the conformal field theory con text the matrix Z describ es the partition function of bulk fields. The three-dimensional topolo gy inv olv ed in the RC FT construction provides in particular a motiv a t ion for us ing the differen t braidings whic h lead to the functors α + A and α − A as w ell as in the definition of Z ( A ). T o extend the results obtained in connection with rational conformal field theory to non- semisimple finite bra ided tensor catego ries remains a ma j o r c hallenge. 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