Tensor categories: A selective guided tour

T ensor categories: A selectiv e guided tour Mic hael M ¨ uger Institute for Mathematics , Astroph ysics and P article Ph ysic s Radb oud Univ ersit y Nijmegen The Netherlands No v em ber 26, 2024 Abstract These are the, somewha t p olished a nd updated, lecture notes for a three hour course on tensor categorie s, given a t the C IRM, Marseille, in April 200 8. The co verage in these notes is relatively non-technical, fo cusing on the essen tial ideas. They a re mea n t to be acces sible for beginner s, but it is ho ped that a lso some of the exp erts will ﬁnd something interesting in them. Once the basic deﬁnitions are given, the focus is mainly on categories that are linear ov er a ﬁeld k and hav e ﬁnite dimensional hom- spaces. Connections with quantum gr oups and lo w dimensional top ology are pointed out, but these notes hav e no pr etension to co ver the latter sub jects to an y depth. Essen tially , these notes s hould be considered a s annota tions to the extensiv e bibliography . W e also rec o mmend the rec e n t review [43], which cov ers less gro und in a deeper wa y . 1 T ensor categories These informal notes are a n o utg rowth of the three hours o f lectures that I g av e at the Centre Int ernationa l de Renco n tres Ma thematiques, Mar seille, in April 2008. The orig inal version of text was pro jected to the screen and therefor e kept maximally concis e. F or this publication, I hav e corrected the languag e where needed, but no s erious attempt has b een made to make these notes conform with the highes t standar ds of exp osition. I still believe that publishing them in this form has a purpo se, even if only pr oviding so me po in ters to the litera ture. 1.1 Strict tensor categories W e b egin with strict tenso r categor ies, despite their limited immediate applicability . • W e assume that the reader has a w orking knowledge of ca tegories, functors a nd natur al trans - formations. Cf . the s tandard reference [180]. Instead o f s ∈ Hom( X , Y ) we will o ccasio nally write s : X → Y . • W e a re int erested in “categorie s with multiplication”. (This was the title o f a pap er [24] b y B´ enab ou 19 63, cf. also Mac Lane [178] fro m the sa me year). This term was so on replaced by ‘monoidal categories’ or ‘tensor categories’. (W e use thes e synon ymously .) It is m ysterious to this author wh y the explicit formalization of tensor ca tegories to ok tw en t y years to arrive after tha t o f ca tegories, in particula r since monoidal catego ries app ear in protean form, e.g ., in T annak a’s work [25 5]. • A strict tensor category (strict monoidal ca tegory) is a triple ( C , ⊗ , 1 ), where C is a cate- gory , 1 a distinguished ob ject a nd ⊗ : C × C → C is a functor, satisfying ( X ⊗ Y ) ⊗ Z = X ⊗ ( Y ⊗ Z ) and X ⊗ 1 = X = 1 ⊗ X ∀ X , Y , Z. 1 If ( C , ⊗ , 1 ) , ( C ′ , ⊗ ′ , 1 ′ ) are strict tensor categ ories, a s trict tenso r functor C → C ′ is a functor F : C → C ′ such that F ( X ⊗ Y ) = F ( X ) ⊗ ′ F ( Y ) , F ( 1 ) = 1 ′ . If F , F ′ : C → C ′ are strict tenso r functors, a natural tra nsformation α : F → F ′ is mo noidal if and o nly if α 1 = id 1 ′ and α X ⊗ Y = α X ⊗ α Y ∀ X , Y ∈ C . (Both sides live in Hom( F ( X ⊗ Y ) , F ′ ( X ⊗ Y )) = Hom( F ( X ) ⊗ ′ F ( Y ) , F ′ ( X ) ⊗ ′ F ′ ( Y )).) • W ARNING: The coherence theorems, to be discus sed in a bit mor e detail in Subsection 1.2, will imply that, in a sense, strict tensor categories are s uﬃcie nt for all purposes . Ho wev er, even when dealing with strict tenso r catego r ies, o ne needs no n-strict tensor functors! • B a sic examples: – Let C b e a n y categ ory and let End C b e the catego ry of functors C → C a nd their natural transformatio ns. Then End C is a strict ⊗ -categor y , with compositio n of functors as tensor pro duct. It is a lso denoted as the ‘center’ Z 0 ( C ). (The subscript is needed since v ar ious other centers will be encountered.) – T o every gro up G , we asso ciate the dis crete tensor category C ( G ): Ob j C ( G ) = G, Hom( g , h ) =  { id g } g = h ∅ g 6 = h , g ⊗ h = g h. – The symmetric category S : Ob j S = Z + , Hom( n, m ) =  S n n = m ∅ n 6 = m , n ⊗ m = n + m with tensor pro duct of morphisms given b y the obvious map S n × S m → S n + m . Remark: 1. S is the fr ee symmetric tensor ca tegory on o ne monoidal ge ne r ator. 2. S is equiv a lent to the categ ory of ﬁnite sets a nd bijectiv e maps. 2. This constructio n w orks with an y family ( G i ) of gr oups with an asso cia tiv e comp osition G i × G j → G i + j . – Let A b e a unital as s o c iative algebra with unit ov er some ﬁeld. W e deﬁne End A to hav e as ob jects the unital algebra homomor phisms ρ : A → A . The morphisms are deﬁned by Hom( ρ, σ ) = { x ∈ A | xρ ( y ) = σ ( y ) x ∀ y ∈ A} with s ◦ t = s t and s ⊗ t = sρ ( t ) = ρ ′ ( t ) s for s ∈ Hom( ρ, ρ ′ ) , t ∈ Hom( σ , σ ′ ). This con- struction has impo r tant applications in in subfactor theory [169] and (algebra ic) quantum ﬁeld theory [68, 90]. Y amagami [284] prov ed that ev ery co untably gener ated C ∗ -tensor category with conjuga tes (cf. b elow) em b eds fully into End A for some von Neumann- algebra A = A ( C ). (See the ﬁnal section for a co njecture co ncerning a n algebr a that should work for all such ca tegories.) – The T emp erley-Lieb categories T L ( τ ). (Cf. e.g. [107].) Let k be a ﬁeld and τ ∈ k ∗ . W e deﬁne Ob j T L ( τ ) = Z + , n ⊗ m = n + m, as for the free s ymmetric catego ry S . But now Hom( n, m ) = s pan k { Isotopy c lasses of ( n, m )- TL diagrams } . Here, an ( n, m )- dia gram is a planar diag ram where n p oints o n a line and m p oints on a parallel line are connected b y lines without crossings. The following example of a (7,5)-TL diag ram will explain this suﬃciently: 2 The tensor pr o duct of morphisms is given b y ho rizontal juxtap osition, whereas comp o- sition of morphisms is deﬁned by vertical juxtap osition, follow ed b y remov al all newly formed close d circles and multiplication by a factor τ for ea ch circle. (This makes sens e since the category is k -linear.) Remark: 1. The T emp erley-Lieb algebr as TL( n, τ ) = E nd T L ( τ ) ( n ) ﬁrst app ear ed in the theory of exactly soluble lattice mo de ls of sta tistical mechanics. They , as well a s T L ( τ ) are close ly related to the Jones p olyno mial [127] and the quantum group S L q (2). Cf. [262, Chapter XII]. 2. The T emp erley-Lie b algebras, as well as the categories T L ( τ ) can b e deﬁned purely algebraic ally in terms of genera to rs a nd relations. – In dealing with (strict) tenso r categ ories, it is often conv enient to adopt a graphical notation for morphisms: s : X → Y ⇔ Y ✎ ✍ ☞ ✌ s X If s : X → Y , t : Y → Z , u : Z → W then we write t ◦ s : X → Z ⇔ Z ✎ ✍ ☞ ✌ t ✎ ✍ ☞ ✌ s X s ⊗ u : X ⊗ Z → Y ⊗ W ⇔ Y W ✎ ✍ ☞ ✌ s ✎ ✍ ☞ ✌ u X Z The usefulness of this notation be c omes apparen t when ther e are morphisms with ‘diﬀeren t nu mbers of in- a nd outputs’: Le t, e.g., a : X → S ⊗ T , b : 1 → U ⊗ Z , c : S → 1 , d : T ⊗ U → V , e : Z ⊗ Y → W and consider the comp osite mo rphism c ⊗ d ⊗ e ◦ a ⊗ b ⊗ id Y : X ⊗ Y → V ⊗ W . (1.1) This formula is a lmost unin telligible. (In or der to econo mize on brackets, we follow the ma jorit y of author s and dec lare ⊗ to bind strong er than ◦ , i.e. a ◦ b ⊗ c ≡ a ◦ ( b ⊗ c ). Notice that inserting brack ets in (1.1) do es nothing to render the formula noticeably more int elligible.) It is not even clea r whether it represents a morphism in the category . This is 3 immediately obvious from the diagra m: V W c d e S T    U Z ✁ ✁ ✁ a b X Y Often, there is mor e than one way to translate a diagra m into a formula, e.g. Z Z ′ ✎ ✍ ☞ ✌ t ✎ ✍ ☞ ✌ t ′ ✎ ✍ ☞ ✌ t ✎ ✍ ☞ ✌ t ′ X X ′ can be read as t ⊗ t ′ ◦ s ⊗ s ′ or a s ( t ◦ s ) ⊗ ( t ′ ⊗ s ′ ). But b y the interc hange law (which is just the functoriality of ⊗ ), these tw o morphisms coincide. F or proofs of consistency of the formalism, cf. [129, 94] or [137]. 1.2 Non-strict tensor categories • F or almost all situatio ns where tensor catego ries arise, s tr ict tensor categ ories are not general enough, the main reas o ns being: – Requir ing equa lit y of ob jects as in ( X ⊗ Y ) ⊗ Z = X ⊗ ( Y ⊗ Z ) is highly unnatur al from a catego rical point of view. – Ma ny would-be tensor catego ries are not strict; in pa r ticular this is the case for V ect k , as well as for representation c ategories of g roups (irresp ective of the class of groups and representations under consider ation). • The obvious minimal modiﬁcation, namely to require only existence o f isomorphisms ( X ⊗ Y ) ⊗ Z ∼ = X ⊗ ( Y ⊗ Z ) for all X , Y , Z and 1 ⊗ X ∼ = X ∼ = X ⊗ 1 for all X , turns out to b e to o weak to b e useful. • The cor r ect deﬁnition of not-necessar ily-strict tensor categories w as giv en in [24]: It is a sextuplet ( C , ⊗ , 1 , α, λ, ρ ), where C is a ca tegory , ⊗ : C × C → C a functor, 1 an ob ject, and α : ⊗ ◦ ( ⊗ × id) → ⊗ ◦ (id × ⊗ ), λ : 1 ⊗ − → id, ρ : − ⊗ 1 → id are natural iso morphisms (i.e., for all X , Y , Z w e hav e is omorphisms α X,Y ,Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) and λ X : 1 ⊗ X → X , ρ X : X ⊗ 1 → X ) such that all mor phisms b etw een the s ame pair of ob jects that can be built from α, λ, ρ coincide. (Examples of what this means are given by the commutativit y of the following tw o diagr ams.) • Ther e are tw o versions of the coherence theor em for tensor categor ies: V ersion I (Mac L a ne [17 8, 180]): All morphisms built fro m α, λ, ρ are unique provided α satisﬁes the p entagon identit y , i.e. c o mm utativity o f (( X ⊗ Y ) ⊗ Z ) ⊗ T α X,Y ,Z ⊗ id T ✲ ( X ⊗ ( Y ⊗ Z )) ⊗ T α X,Y ⊗ Z, T ✲ X ⊗ (( Y ⊗ Z ) ⊗ T ) ( X ⊗ Y ) ⊗ ( Z ⊗ T ) α X ⊗ Y ,Z,T ❄ α X,Y ,Z ⊗ T ✲ X ⊗ ( Y ⊗ ( Z ⊗ T )) id X ⊗ α Y ,Z,T ❄ 4 and λ, ρ s atisfy the unit identit y ( X ⊗ 1 ) ⊗ Y α X, 1 ,Y ✲ X ⊗ ( 1 ⊗ Y ) X ⊗ Y ρ X ⊗ id Y ❄ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ X ⊗ Y id X ⊗ λ Y ❄ F or mo dern exp ositions of the co herence theo r em s ee [180, 137]. (Notice that the orig inal deﬁnition of non-strict tensor categor ies given in [1 78] was modiﬁed in s light ly [14 6, 147].) • E xamples of no n-strict tensor ca teg ories: – Let C b e a category with pro ducts and terminal ob ject T . Deﬁne X ⊗ Y = X Q Y (for each pair X , Y c ho ose a pro duct, non-uniquely) and 1 = T . Then ( C , ⊗ , 1 ) is non- strict tensor category . (Exis tence of asso ciator and unit isomorphisms follows from the universal prop erties of pro duct and terminal ob ject). An analogous co ns truction works with copr o duct and initial ob ject. – V ect k with α U,V ,W deﬁned on simple tensors by ( u ⊗ v ) ⊗ w 7→ u ⊗ ( v ⊗ w ). Note: T his trivially satisﬁes the p entagon identit y , but the other choice ( u ⊗ v ) ⊗ w 7→ − u ⊗ ( v ⊗ w ) do es not! – Let G b e a gro up, A an ab elian group (written mult iplicatively) and ω ∈ Z 3 ( G, A ), i.e. ω ( h, k , l ) ω ( g , hk , l ) ω ( g , h, k ) = ω ( g h, k , l ) ω ( g, h, kl ) ∀ g , h, k , l ∈ G. Deﬁne C ( G, ω ) by Ob j C = G, Hom( g , h ) =  A g = h ∅ g 6 = h , g ⊗ h = g h. with ass o c iator α = ω , cf. [245]. If k is a ﬁeld, A = k ∗ , one ha s a k -linear version wher e Hom( g , h ) =  k g = h { 0 } g 6 = h . I denote this by C k ( G, ω ), but also V ect G ω app ears in the literature. The imp ortance of this exa mple lies in its showing relations b etw een catego ries and cohomolog y , whic h are reinforced by ‘higher category theory ’, c f. e.g. [14]. But also the concrete example is r elev ant for the cla ssiﬁcation of fusion catego ries, at least the la rge class of ‘group theoretical categories ’. (Cf. Ostrik et al. [22 3, 84].) See Section 3. – A ca tegorical group is a tensor catego ry that is a gr oup o id (a ll morphisms ar e invertible) and where ev ery ob ject has a tensor-inv erse, i.e. fo r ev ery X there is a n ob ject X such that X ⊗ X ∼ = 1 . The categor ies C ( G, ω ) are just the skeletal categorica l g roups. • Now we can give the gener al deﬁnition of a tenso r functor (betw een non-strict tensor ca tegories or non-str ic t tensor functors b et ween strict tens o r categor ies): A tensor functor b etw een tensor categor ies ( C , ⊗ , 1 , α, λ, ρ ) , ( C ′ , ⊗ ′ , 1 ′ , α ′ , λ ′ , ρ ′ ) consis ts of a functor F : C → C ′ , an isomorphism e F : F ( 1 ) → 1 ′ and a family of natural isomor phisms d F X,Y : F ( X ) ⊗ F ( Y ) → F ( X ⊗ Y ) satisfying comm utativity of ( F ( X ) ⊗ ′ F ( Y )) ⊗ ′ F ( Z ) d X,Y ⊗ id ✲ F ( X ⊗ Y ) ⊗ ′ F ( Z ) d X ⊗ Y ,Z ✲ F (( X ⊗ Y ) ⊗ Z ) F ( X ) ⊗ ′ ( F ( Y ) ⊗ ′ F ( Z )) α ′ F ( X ) ,F ( Y ) ,F ( Z ) ❄ id ⊗ d Y ,Z ✲ F ( X ) ⊗ ′ F ( Y ⊗ Z ) d X,Y ⊗ Z ✲ F ( X ⊗ ( Y ⊗ Z )) F ( α X,Y ,Z ) ❄ 5 (notice that this is a 2-co cy cle condition, in pa rticular when α ≡ id) and F ( X ) ⊗ F ( 1 ) id ⊗ e F ✲ F ( X ) ⊗ 1 ′ F ( X ⊗ 1 ) d F X, 1 ❄ F ( ρ X ) ✲ F ( X ) ρ ′ F ( X ) ❄ (and simila r for λ X ) Remark: Oc casionally , functor s a s deﬁned above a re called strong tensor functors in order to distinguish them from the lax v ariant, where the d F X,Y and e F are not required to be isomorphisms. (In this case it also ma kes sense to co nsider d F , e F with source and ta rget exchanged.) • Le t ( C , ⊗ , 1 , α, λ, ρ ) , ( C ′ , ⊗ ′ , 1 ′ , α ′ , λ ′ , ρ ′ ) b e tensor categ o ries and ( F, d, e ) , ( F ′ , d ′ , e ′ ) : C → C ′ tensor functors. Then a natur al transfor mation α : F → F ′ is mono idal if F ( X ) ⊗ ′ F ( Y ) d X,Y ✲ F ( X ⊗ Y ) F ′ ( X ) ⊗ ′ F ′ ( Y ) α X ⊗ α Y ❄ d ′ X,Y ✲ F ′ ( X ⊗ Y ) α X ⊗ Y ❄ F or strict tenso r functors, we hav e d ≡ id ≡ d ′ , and w e obtain the ea rlier condition. • A tensor functor F : ( C , ⊗ , 1 , α, λ, ρ ) → ( C ′ , ⊗ ′ , 1 ′ , α ′ , λ ′ , ρ ′ ) is called an equiv alence if there exist a tensor functor G : C ′ → C a nd natur al monoidal isomorphisms α : G ◦ F → id C and β : F ◦ G → id C ′ . F or the existence o f such a G it is necessary and suﬃcien t that F b e full, faithful and es s en tially surjective (a nd of cour se mono idal), cf. [23 8]. (W e follow the practice of not worrying to o muc h ab out size issues and as s uming a suﬃciently strong version of the axiom of choice fo r class es. O n this matter, cf. the diﬀerent discussions of foundational issues given in the tw o editions of [180].) • Given a gr oup G and ω , ω ′ ∈ Z 3 ( G, A ), the iden tity functor is part of a monoidal equiv alence C ( G, ω ) → C ( G, ω ′ ) if and only if [ ω ] = [ ω ′ ] in H 3 ( G, A ). Cf. e.g. [54, Chapter 2]. Since categoric al groups form a 2-categor y C G , they are best clas s iﬁed b y providing a 2-e q uiv alence betw een C G and a 2-categor y H 3 deﬁned in terms o f co homology g roups H 3 ( G, A ). The details are to o in volv ed to give here; cf. [128]. (Unfortunately , the theo r y of categ orical groups is marred b y the fact that importa n t w orks [24 5, 128] were never formally published. F or a comprehensive r ecent trea tmen t cf. [1 2].) • V ersio n I I of the Coherence theor em (equiv alen t to V ersion I): Every tensor ca tegory is monoidally equiv alent to a strict o ne. [180, 137]. As mentioned earlier , this allows us to pretend that all tenso r ca tegories are strict. (But w e ca nnot res trict ourselves to strict tensor functors!) • O ne may ask what the strictiﬁca tion of C ( G, ω ) lo oks like. The answ er is somewhat compli- cated, cf. [128]: It inv olves the free g roup o n the set underlying G . (This shows that s ometimes it is a ctually more conv enien t to work with non- s trict categor ies!) • As shown in [241], many non-s tr ict tensor categor ies can b e turned in to equiv alent strict o nes by changing only the tensor functor ⊗ , but le aving the underlying c ate gory unchange d . • W e reca ll the “Eckmann-Hilton a rgument”: If a set has t wo monoid structures ⋆ 1 , ⋆ 2 satisfying ( a ⋆ 2 b ) ⋆ 1 ( c ⋆ 2 d ) = ( a ⋆ 1 c ) ⋆ 2 ( b ⋆ 1 d ) with the same unit, the t wo pro ducts coincide and are commutativ e. If C is a tensor catego ry and we co nsider End 1 with ⋆ 1 = ◦ , ⋆ 2 = ⊗ we ﬁnd that 6 End 1 is commutativ e, cf. [148]. In the Ab- ( k -linea r) case, deﬁned in Subsection 1.6, E nd 1 is a commutativ e unital ring ( k -alg e bra). (Another classical application o f the Eckmann-Hilton argument is the ab eliannes s o f the higher homo top y groups π n ( X ) , n ≥ 2 and of π 1 ( M ) for a top ological monoid M .) 1.3 Generalization: 2-categories and bicategories • T ensor ca tegories hav e a very natura l a nd useful generaliza tion. W e b egin with ‘2-catego ries’, which generalize strict tens o r ca tegories: A 2-categor y E consists of a s et (cla s s) of ob jects and, for every X , Y ∈ Ob j E , a category HOM( X , Y ). The ob jects (morphisms) in HOM( X, Y ) are called 1- mo rphisms (2- morphisms) o f E . F or the detaile d axio ms we refer to the references given b elow. In particular , w e hav e functors ◦ : HO M( A , B ) × HOM( B , C ) → HOM( A , C ), and ◦ is asso cia tiv e (on the nose). • The pr ototypical example of a 2-catego ry is the 2- category C AT . Its ob jects are the small categorie s, its 1-morphisms ar e functors a nd the 2-mo rphisms are natur al transfor mations. • W e notice that if E is a 2- category and X ∈ Ob j E , then END( X ) = HO M( X, X ) is a str ict tensor category . This leads to the non-strict version of 2- c ategories ca lled bicategories: W e replace the a sso ciativity of the compo sition ◦ o f 1-mor phisms by the existence of inv ertible 2-morphisms ( X ◦ Y ) ◦ Z → X ◦ ( Y ◦ Z ) satisfying axioms g eneralizing those of a tensor category . Now, if E is a bicategory and X ∈ Ob j E , then END( X ) = HOM( X , X ) is a (no n- strict) tensor category . Bicatego ries are a very imp ortant generaliza tion of tensor ca teg ories, and we’ll meet them again. Also the relation b et ween bicategories and tensor categories is prototypical for ‘higher ca tegory theory’. References: [15 0] for 2-ca teg ories and [26] for bicategories , as w ell as the very recent r eview by Lack [162]. 1.4 Categoriﬁcation of monoids T ensor ca tegories (or monoidal categorie s ) can b e co nsidered as the categor iﬁcation of the notion of a monoid. This has in teresting co nsequences: • Mo noids in monoidal catego ries: Let ( C , ⊗ , 1 ) b e a str ict ⊗ -categor y . A monoid in C (B´ ena bo u [25]) is a triple ( A, m, η ) with A ∈ C , m : A ⊗ A → A, η : 1 → A satisfying m ◦ m ⊗ id A = m ◦ id A ⊗ m, m ◦ η ⊗ id A = id A = m ◦ id A ⊗ η . (In the non-strict case, insert an asso cia tor at the obvious place .) A mono id in Ab (V ect k ) is a ring ( k -algebra ). Therefor e, in the recent litera ture monoids are often ca lled ‘alge bras’. Monoids in monoidal categories are a prototypical example of the ‘micro cosm pr inciple’ of Baez and Dola n [1 1] aﬃrming that “c ertain a lg ebraic structur es can b e deﬁned in any categor y equipp e d with a categor iﬁed v ersion of the sa me structure”. • If C is an y categor y , monoids in the tensor ca teg ory End C are k nown as ‘mo nads’. As suc h they ar e older than tensor catego r ies! Cf. [180]. • If ( A, m, η ) is a mono id in the strict tensor ca teg ory C , a left A-mo dule is a pa ir ( X, µ ), where X ∈ C and µ : A ⊗ X → X s a tisﬁes µ ◦ m ⊗ id X = µ ◦ id A ⊗ µ, µ ◦ η ⊗ id X = id X . T ogether with the obvious notion of A-mo dule morphism Hom A − Mo d (( X, µ ) , ( X ′ , µ ′ )) = { s ∈ Hom C ( X, X ′ ) | s ◦ µ = µ ′ ◦ id A ⊗ s } , A -mo dules form a catego ry . Rig ht A-mo dules a nd A − A bimo dules ar e deﬁned analog ously . The free A-mo dule of r ank 1 is just ( A, m ). 7 • If C is a belia n, then A − Mo d C is ab elian under weak assumptions, cf. [6]. (The latter are satisﬁed when A has duals, as e.g. when it is a s trongly separable F robe nius a lg ebra [98]. All this could also b e deduced from [76].) • E very monoid ( A, m, η ) in C gives rise to a monoid Γ A = Hom( 1 , A ) in the categ ory S E T of sets. W e ca ll it the el emen ts of A . (Γ A is r elated to the endomo r phisms of the unit ob ject in the tensor categories of A − A -bimo dules and A - mo dules (in the braided ca se), when the latter exist.) • Le t C b e a belia n and ( A, m, η ) an alg ebra in C . An i deal in A is an A-mo dule ( X, µ ) tog ether with a monic morphism ( X , µ ) ֒ → ( A, m ). Much a s in o rdinary algebra, o ne ca n deﬁne a quotient algebr a A/I . F urthermore, every ideal is contained in a maxima l ideal, a nd an ideal I ⊂ A in a comm utative monoid is maximal if a nd only if the ring Γ A/I is a ﬁeld. (F or the last claim, cf. [197].) • Co algebras and their como dules are deﬁned analogously . In a tensor ca tegory equipp ed with a symmetry or br aiding c (cf. b elow), it makes sense to say that an (co)algebra is (co) commut ativ e . F or an algebr a ( A, m, η ) this mea ns that m ◦ c A,A = m . • (B) Just as monoids can act o n sets, tensor categories ca n act on categor ies: Let C b e a tensor ca tegory . A left C -m o dule category is a pair ( M , F ) where M is a category and F : C → E nd M is a tensor functor . (Here, End M is a s in o ur ﬁrst example of a tensor categ ory .) This is equiv alen t to having a functor F ′ : C × M → M and natural isomorphisms β X,Y ,A : F ′ ( X ⊗ Y , A ) → F ( X , F ( Y , A )) satisfying a pentagon-type c o herence law, unit co ns traints, etc. Now one can deﬁne indecomp osable module categor ie s, etc. (Ostrik [222]) • Ther e is a close connectio n betw een module catego ries a nd categ o ries of mo dules: If ( A, m, η ) is an a lgebra in C , then there is an natural right C - module structure on the category A − Mo d C of left A-mo dules: F ′ : A − Mo d C × C , ( M , µ ) × X 7→ ( M ⊗ X , µ ⊗ id X ) . (In the case wher e ( M , µ ) is the free rank-one mo dule ( A, m ), this gives the free A -mo dules F ′ (( A, m ) , X ) = ( A ⊗ X , m ⊗ id X ).) F or a fusion ca tegory (cf. b elow), one can s how that every semisimple indecomposa ble left C -module category a rises in this w a y from an algebra in C , cf. [2 22]. 1.5 Dualit y in tensor categor ies I • If G is a gr oup and π a repres en tation on a ﬁnite dimensio nal vector space V , we deﬁne the ‘dual’ or ‘conjuga te’ r epresentation π on the dual vector space V ∗ by h π ( g ) φ, x i = h φ, π ( g ) x i . Denoting by π 0 the triv ial representation, one ﬁnds Hom Rep G ( π ⊗ π , π 0 ) ∼ = Hom Rep G ( π , π ), implying π ⊗ π ≻ π 0 . If π is irreducible, then so is π and the multiplicit y of π 0 in π ⊗ π is one by Sch ur’s lemma. Since the ab ove discussion is quite sp eciﬁc to the group situation, it clear ly ne e ds to b e generalized. • Le t ( C , ⊗ , 1 ) b e a strict tenso r categ ory and X , Y ∈ C . W e say that Y is a left dual of X if there are morphisms e : Y ⊗ X → 1 a nd d : 1 → X ⊗ Y satisfying id X ⊗ e ◦ d ⊗ id X = id X , e ⊗ id Y ◦ id Y ⊗ d = id Y , or, represe n ting e : Y ⊗ X → 1 and d : 1 → X ⊗ Y by ☛ ✟ and ✡ ✠ , resp ectively , X ☛ ✟ e Y d ✡ ✠ X = X X Y e ☛ ✟ X ✡ ✠ d Y = Y Y 8 ( e stands for ‘ev aluation’ and d for ‘dua l’.). In this situation, X is called a right dual of Y . Example: C = V ect ﬁn k , X ∈ C . Let Y = X ∗ , the dual vector spa ce. Then e : Y ⊗ X → 1 is the usual pairing. With the canonical isomor phism f : X ∗ ⊗ X ∼ = − → End X , we hav e d = f − 1 (id X ). W e state so me facts: 1. Whether an ob ject X a dmits a left or r ight dual is not for us to choo se. It is a pro per ty of the tenso r catego ry . 2. If Y , Y ′ are left (or rig h t) duals o f X then Y ∼ = Y ′ . 3. If ∨ A, ∨ B are left duals o f A, B , resp ectively , then ∨ B ⊗ ∨ A is a left dual for A ⊗ B , a nd similarly for r ight duals. 4. If X has a left dua l Y and a right dual Z , we may or may no t ha v e Y ∼ = Z ! (Again, that is a prop erty of X .) While duals, if they exist, ar e unique up to iso morphisms, it is often conv enient to mak e choices. One therefore deﬁnes a left duali t y of a str ict tensor categ ory ( C , ⊗ , 1 ) to b e a map that as signs to each ob ject X a left dual ∨ X a nd mor phisms e X : ∨ X ⊗ X → 1 a nd d X : 1 → X ⊗ ∨ X satisfying the ab ove identities. Given a left dua lit y and a morphis m, s : X → Y we de ﬁne ∨ s = e Y ⊗ id ∨ X ◦ id ∨ Y ⊗ s ⊗ id ∨ X ◦ id ∨ Y ⊗ d X = ∨ X e Y ☛ ✟ ✎ ✍ ☞ ✌ s ✡ ✠ d X ∨ Y Then ( X 7→ ∨ X , s 7→ ∨ s ) is a contrav ariant functor. (W e cannot recover the e’s a nd d’s from the functor!) It can be equipped with a natural (anti-)monoidal isomorphism ∨ ( A ⊗ B ) → ∨ B ⊗ ∨ A, ∨ 1 → 1 . Often, the dualit y functor co mes with a given an ti-monoidal structur e, e.g. in the case of piv otal ca tegories, cf. Section 3. • A chosen right duali t y X 7→ ( X ∨ , e ′ X : X ⊗ X ∨ → 1 , d ′ X : 1 → X ∨ ⊗ X ) a lso give rise to a contra v a riant ant i-monoida l functor X 7→ X ∨ . • Ca tegories equipp e d with a left (right) dua lit y are called left (rig h t) rigid (or autonomous ). Categorie s with le ft and right duality are c a lled rigid (or autonomous). • E xamples: V ect ﬁn k , Rep G ar e rigid. • No tice that ∨∨ X ∼ = X holds if and only if ∨ X ∼ = X ∨ , for which there is no genera l r eason. • If every ob ject X ∈ C admits a left dual ∨ X and a right dual X ∨ , and b oth ar e isomorphic , we say that C ha s t w o-sided duals a nd write X . W e will only cons ider such categor ies, but we will need strong er axioms. • Le t C b e a ∗ -catego r y (cf. below) with left dualit y . If ( ∨ X , e X , d X ) is a left dual of X ∈ C then ( X ∨ = ∨ X , d ∗ X , e ∗ X ) is a rig h t dual. Thus duals in ∗ -ca tegories ar e a utomatically t w o-sided. F or this reason, duals in ∗ -catego ry a re often a xiomatized in a symmetric fashion by s aying that a conjugate , cf. [70, 1 7 2], of an ob ject X is a triple ( X , r, r ), wher e r : 1 → X ⊗ X, r : 1 → X ⊗ X satis fy id X ⊗ r ∗ ◦ r ⊗ id X = id X , id X ⊗ r ∗ ◦ r ⊗ id X = id X . It is c le ar that then ( X , r ∗ , r ) is a left dual and ( X , r ∗ , r ) a right dual. • Unfor tunately , there is an a lmost Bab ylonian inﬂation of slight ly diﬀerent notions concern- ing duals, in par ticular when braidings are in volv ed: A categ ory can b e rig id, autonomo us, sov ereign, pivotal, spherical, ribb on, tortile, balanced, closed, categor y with conjugates, etc. T o make things worse, these terms a re not a lw ays use d in the same w ay! 9 • B e fore w e contin ue the discussion of dualit y in tenso r catego r ies, we will disc uss symmetries. F or symmetric tensor catego ries, the discussion o f dualit y is somewhat simpler than in the general cas e . Pro ceeding like this seems justiﬁed s inc e symmetric (tensor ) catego ries a lready app eared in the second pa per ([178] 1 9 63) on tensor categorie s. 1.6 Additiv e, linear and ∗ -struct ure • The discussion so far is quite general, but often one e nc o un ters ca tegories with more structure. • W e beg in with ‘Ab-categories ’ (= categories ‘enr iched ov er ab elian groups’): F or such a cate- gory , each Ho m( X, Y ) is an ab elian gro up, and ◦ is bi-additive, cf. [180, Se c tion I.8]. Example: The catego r y Ab of ab elian gr oups. In ⊗ - c a tegories, also ⊗ must b e bi-additive on the mor - phisms. F unctors of Ab-tensor ca tegories requir ed to b e a dditiv e on hom-sets. • If X , Y , Z ar e ob jects in a n Ab-ca tegory , Z is called a direct sum o f X and y if ther e a re morphisms X u → Z u ′ → X , Y v → Z v ′ → Y s atisfying u ◦ u ′ + v ◦ v ′ = id Z , u ′ ◦ u = id X , v ′ ◦ v = id Y . An additiv e ca tegory is an Ab-catego ry having direct sums for all pairs of o b jects and a zero ob ject. • An ab elian category is an additive c ategory whe r e ev ery morphism has a kernel and a cokernel and every monic (epic) is a kernel (cokernel). W e do not hav e the space to go further int o this and must refer to the litera ture, e.g. [180]. • A catego ry is said to ha ve splitting idempotents (o r is ‘K aroubian’) if p = p ◦ p ∈ End X implies the existence of an ob ject Y and o f morphisms u : Y → X, u ′ : X → Y such tha t u ′ ◦ u = id Y and u ◦ u ′ = p . An additive category with splitting idemp otents is called pseudo- ab e lian . Every a belia n categor y is pseudo-ab elian. • In an ab elian category with dua ls, the functors − ⊗ X a nd X ⊗ − are automatically exact, cf. [64, Prop osition 1.16]. B ut without rigidity this is far from tr ue. • A semis imple ca tegory is a n ab elian ca tegory where every shor t exa ct sequence splits. An alternative, a nd more p edestria n, w ay to deﬁne semisimple catego ries is a s pseudo -ab elian categorie s admitting a family of simple ob jects X i , i ∈ I such that every X ∈ C is a ﬁnite direct sum of X i ’s. Standard examples: The category Rep G of ﬁnite dimensional representations of a compact group G , the categ o ry H − Mo d of ﬁnite dimensional left modules for a ﬁnite dimensional semisimple Hopf algebra H . • In k -linear categor ies, each Hom( X, Y ) is k -vector space (often r equired ﬁnite dimensional), and ◦ (and ⊗ in the mono idal case) is bilinear. F u nctors m ust b e k -linear. Exa mple: V ect k . • P seudo-ab elian categor ies that ar e k -linea r with ﬁnite-dimensional ho m- sets are called Krul l - Sc hmidt ca tegories. (This is slightly w eaker than semisimplicity .) • A fusi on category is a s emisimple k -linear catego ry with ﬁnite dimensiona l ho m-sets, ﬁnitely many isomor phism cla sses of simple ob jects a nd End 1 = k . W e also require that C has 2 -sided duals. • A ﬁnite tensor category (Etingof, Ostrik [85]) is a k -linear tens or c a tegory with End 1 = k that is equiv a len t (as a category ) to the categ ory of modules ov er a ﬁnite dimensional k-algebr a. (There is a more in trinsic c hara cterization.) Notice that semisimplicit y is no t assumed. • Dr o pping the condition End 1 = k id k , one ar rives at a multi-fusion category (Etingo f, Nikshyc h, Ostrik [84]). • Despite the recent work on generalizatio ns , mo st o f these lectures will b e concerned with semisimple k -linear categor ies sa tisfying End 1 = k id 1 , including inﬁnite ones! (But see the remarks at the end of this s e c tion.) 10 • If C is a semisimple tensor category , one can choo se representers { X i , i ∈ I } of the simple isomorphism class es and deﬁne N k i,j ∈ Z + by X i ⊗ X j ∼ = M k ∈ I N k i,j X k . There is a distinguished element 0 ∈ I s uc h that X 0 ∼ = I , thus N k i, 0 = N k 0 ,i = δ i,k . By asso ciativity o f ⊗ (up to isomo r phism) X n N n i,j N l n,k = X m N l i,m N m j,k ∀ i, j, k , l ∈ I . If C has t wo-sided duals, ther e is an inv olution i 7→ ı such that X i ∼ = X ı . One has N 0 i,j = δ i,  . The quadruple ( I , { N k i,j } , 0 , i 7→ ı ) is called the fus ion ring or fusion h yp ergroup o f C . • The ab ov e do e s not work when C is no t semisimple. But: In a n y ab elian tens or ca teg ory , one ca n co nsider the Grothendieck ring R ( C ), the free abelian group g enerated b y the isomorphism class es [ X ] of ob jects in C , with a re la tion [ X ] + [ Z ] = [ Y ] for every sho rt exact sequence 0 → X → Y → Z → 0 and [ X ] · [ Y ] = [ X ⊗ Y ]. In the semisimple case, the Grothendieck ring has { [ X i ] , i ∈ I } as Z -basis and [ X i ] · [ X j ] = P k N k i,j [ X k ]. O bviously , an is o morphism of h yper groups gives rise to a ring isomorphism of Grothendieck rings, but the conv erse is not o b vious. While the author is not aw are o f counterexamples, in or der to rule out this annoying even tualit y , some a utho rs w ork with isomorphisms of the Grothendieck semi r ing or the ordered Grothendieck ring, cf e .g. [112]. Back to hypergr oups: • The h ype r group contains impo rtant infor mation ab out a tens o r categ ory , but it misses that enco ded in the asso ciativity constraint. In fact, the hype r group of Rep G for a ﬁnite group G contains exactly the sa me information as the c haracter table o f G , a nd it is well known that there ar e non-is o morphic ﬁnite groups with isomo rphic character tables. (The simplest example is given b y the dihedral g roup D 8 = Z 4 ⋊ Z 2 and the quaternion g r oup Q , cf. any elemen tary textbo ok, e.g. [123].) Since D 8 and Q have the same n um b er of ir reducible representations, the categor ies Rep D 8 and Rep Q are e quiv alent (as c a tegories). They are not equiv alen t as symmetric tensor categor ies, since this would imply D 8 ∼ = Q by the dualit y theorems of Doplicher and Rober ts [7 0] or Deligne [58] (which we will dis cuss in Section 3). In fact, D 8 and Q ar e alr eady ineq uiv alent as tensor ca tegories (i.e. they ar e not iso catego rical in the sense discussed below). Cf. [25 4], where fusion categories with the fusion h yp ergro up of D 8 are cla ssiﬁed (among other things). • O n the pos itive side: (1) If a ﬁnite gro up G has the s ame fusion hypergro up (or c hara c ter table) as a ﬁnite simple g roup G ′ , then G ∼ = G ′ , cf. [51]. (The pro of uses the cla ssiﬁcation of ﬁnite simple gro ups.) (2) Compact groups that a re a belia n or co nnected are determined by their fusion rings (by Pontrjagin duality , resp ectively b y a r esult of McMullen [188] and Handelman [112]. The latter is ﬁrst prov en for simple compact Lie groups and then one deduces the general result via the structure theor em for connec ted compact gr o ups.) • If a ll o b jects in a semisimple categ ory C are inv ertible, the fusion hypergr oup b ecomes a group. Such fusion categories are called p ointed and are just the linear v ersions o f the ca tegorical groups encountered earlier . This situation is very spec ial, but: • T o each h yp ergro up { I , N , 0 , i 7→ ı } one ca n asso cia te a group G ( I ) as follows: Let ∼ be the smallest equiv alence rela tion on I such that i ∼ j whenever ∃ m, n ∈ I : i ≺ mn ≻ j (i.e. N i n,m 6 = 0 6 = N j n,m ) . Now let G ( I ) = I / ∼ and deﬁne [ i ] · [ j ] = [ k ] for a ny k ≺ ij, [ i ] − 1 = [ ı ] , e = [0] . 11 Then G ( I ) is a group, and it ha s the universal prop erty that every map p : I → K , K a group, satisfying p ( k ) = p ( i ) p ( j ) when k ≺ ij factors throug h the map I → G ( I ) , i 7→ [ i ]. In analogy to the ab elianization of a non-a belia n gr oup, G ( I ) should p erhaps b e called the groupiﬁcation of the hypergr oup I . But it was called the univ ersal grading group by Gelaki/Nikshyc h [102], to whic h this is due in the ab ov e generality , since e very group- grading on the o b jects o f a fusion categ ory having fusio n hypergr oup I facto rs through the map I → G ( I ). • In the symmetric case (where I and G ( I ) are ab elian, but ev erything else as ab ov e) this groupiﬁcatio n is due to Baumg¨ artel/Lled´ o [21], who sp oke of the ‘chain g roup’. They stated the conjecture that if K is a compac t gro up, then the (discrete) univ ersal grading group G (Rep K ) o f Rep K is the Pon trjagin dual of the (compact) center Z ( K ). Thus: The c ent er of a c omp act gr oup K c an b e re c over e d fr om t he fusion ring of K , even if K itself in gener al cannot! This conjecture was prov en in [195], but the who le cir c le of ideas is already implicit in [18 8]. Example: The r epresentations o f K = S U (2 ) are lab elled by Z + with i ⊗ j = | i − j | ⊕ · · · ⊕ i + j − 2 ⊕ i + j. F ro m this one easily sees that there are tw o ∼ - equiv alence cla sses, c o nsisting of the even and o dd int eger s . This is compatible with Z ( S U (2)) = Z / 2 Z . Cf. [21]. • Ther e is ano ther a pplication of G ( C ): If C is k -linea r semisimple then group of natural monoidal isomor phisms of id C is g iv en by Aut ⊗ (id C ) ∼ = Hom( G ( C ) , k ∗ ). • Given a fusion categor y C (where we hav e t wo-sided duals X ), Gelak i/Nikshyc h [102] deﬁne the full sub catego r y C ad ⊂ C to be the generated by the ob jects X ⊗ X where X runs through the simple ob jects. Notice that C ad is just the full sub categor y of ob jects of universal g r ading zero. Example: If G is a compact group then (Rep G ) ad = Rep( G/ Z ( G )). A fusion ca tegory C is called nilp o ten t [1 02] when its upp er ce n tral se r ies C ⊃ C ad ⊃ ( C ad ) ad ⊃ · · · leads to the trivial category after ﬁnitely many steps. Example: If G is a ﬁnite group then Rep G is nilpotent if and only if G is nilpo ten t. • W e call a square n × n - matrix A indecomp osable if there is no pro p er subset S ⊂ { 1 , . . . , n } such that A maps the c o o rdinate subspace span { e s | s ∈ S } in to itself. L e t A be an inde- comp osable square matrix A with non-neg ative entries and eigenv alues λ i . Then the theorem of Perron and F rob enius states that there is a unique non-negative eig en v alue λ , the Perron- F ro benius eigenv alue, such that λ = max i | λ i | . F urthermor e, the a sso ciated eig enspace is one-dimensional and c o n tains a vector with all comp onents non-neg ative. Now, given a ﬁnite hypergro up ( I , { N k i,j } , 0 , i 7→ ı ) and i ∈ I , deﬁne N i ∈ Mat( | I | × | I | ) by ( N i ) j k = N k i,j . Due to the existence of duals, this matrix is indecompo sable. Now the P erron-F rob enius dim en- sion d F P ( i ) of i ∈ I is deﬁned as the Perron-F rob enius eigenv alue of N i . Cf. e.g. [96, Section 3.2]. Then: d F P ( i ) d F P ( j ) = X k N k i,j d F P ( k ) . Also the hypergr oup I has a Perron-F rob enius dimension: F P − dim( I ) = P i d F P ( i ) 2 . This also deﬁnes the PF-dimensio n of a fusio n catego ry , cf. [84] • O cnean u rigidity : Up to equiv alence there are only ﬁnitely many fusion categor ies with given fusion hypergr oup. The gener al statement was announced by Blanchard/A. W ass e rmann, and a pr o o f is given in [84], using the defor mation co homology theory of Davydo v [53] and Y etter [290]. 12 • O cneanu rigidity was preceded and motiv ated by s e v eral related res ults o n Hopf algebras : Stefan [249] proved that the num ber of isomorphism clas ses o f semisimple and co- semisimple Hopf a lgebras of given ﬁnite dimension is ﬁnite. F or Hopf ∗ - algebras , Blanchard [30] even prov ed a b ound o n the n umber of iso -classes in terms of the dimensio n. Ther e a lso is an upper b ound on the num ber of is o -classes of semisimple Hopf algebr as with given num b er o f irreducible r epresentations, cf. E tingof ’s a ppendix to [22 4]. • Ther e is an eno r mous literature o n hypergr oups. Much of this concerns harmonic ana lysis on the latter and is not too relev an t to tensor categories. But the no tion of amenability of hyperg roups do es hav e such a pplications, cf. e.g. [119]. F or a review of some asp ects of hypergro ups, in particular the discrete ones relev ant here, cf. [27 8]. • A consider able fraction of the liter ature o n tensor catego ries is devoted to categories that ar e k -linear ov er a ﬁeld k with ﬁnite dimensional Hom-spaces. Clear ly this a rather restrictive condition. It is therefore very remar k able tha t k - linearity ca n actually be deduced under suitable assumptions , cf. [16 1]. • ∗ -categories : A ‘ ∗ -o per ation’ o n a C -linear category C is a contrav a riant functor ∗ : C → C which acts trivially on the ob jects, is antilinear, inv olutiv e ( s ∗∗ = s ) a nd mo noidal ( s ⊗ t ) ∗ = s ∗ ⊗ t ∗ (when C is monoida l). A ∗ -op eration is ca lled po sitive if s ∗ ◦ s = 0 implies s = 0. Categorie s with (positive) ∗ -op eration are a ls o called he r mitian (unitar y). W e will use ‘ ∗ - category ’ as a syno n ym for ‘unitary ca tegory’.) Ex ample: The catego ry of Hilb ert spaces HI LB with bo unded linear maps and ∗ given by the adjoint. • It is ea sy to prov e that a ﬁnite dimensional C -algebra with p ositive ∗ - ope r ation is semisimple. Therefore, a unitary catego ry with ﬁnite dimensio nal hom-sets ha s semisimple endomorphism algebras . If it has direct sums and splitting idempo ten ts then it is semisimple. • B a nach-, C ∗ - and v on Neumann catego ries: A Banach ca tegory [135 ] is a C -linear a dditiv e category , where each Hom( X , Y ) is a Ba nach spac e, and the norms satisfy k s ◦ t k ≤ k s k k t k , k s ∗ ◦ s k = k s k 2 . (They were introduced b y Karo ubi with a view to applica tions in K- theory , cf. [135].) A Banach ∗ -categ ory is a B anach category with a po sitive ∗ -op era tion. A C ∗ -categor y is a Banach ∗ -categor y satisfying k s ∗ ◦ s k = k s k for any mo rphism s (not only endomorphisms). In a C ∗ -categor y , ea c h End( X ) is a C ∗ -algebra . Just as an additive category is a ‘r ing with several ob jects’, a C ∗ -categor y is a “ C ∗ -algebra with sev eral o b jects” . V o n Neumann ca tegories are deﬁned similarly , cf. [105]. They tur ne d out to have applications to L 2 -cohomolo gy (cf. F arb er [88]), repr esentation theory of quantum groups (W oronowicz [280]), subfacto rs [172], etc. Remark: A ∗ -catego ry with ﬁnite dimensional hom-spaces and End 1 = C automatically is a C ∗ -categor y in a uniq ue wa y . (Cf. [190].) • If C is a C ∗ -tensor categ ory , E nd 1 is a commutativ e C ∗ -algebra , thus ∼ = C ( S ) for so me compact Hausdorﬀ spa ce S . Under cer tain technical conditions, the spa ces Hom( X, Y ) can be considere d as v ector bundles ov er S , or at least as (semi)con tinuous ﬁelds of vector spaces. (W or k by Zito [2 91] and V asselli [271].) In the case where End 1 is ﬁnite dimensional, this bo ils down to a direct s um decomp osition of C = ⊕ i C i , where each C i is a tensor catego ry with End C i ( 1 C i ) = C . (In this c o nnection, cf. Ba ez’ comments a Doplicher-Rob erts type theorem for ﬁnite group oids [9].) 2 Symmetric tensor c ategorie s • Ma n y of the obvious ex a mples of tensor catego ries enc o un tered in Section 1, like the categories S E T , V ect k , representation categ ories of g r oups and Car tesian ca tegories (tenso r pr o duct ⊗ given by the ca tegorical pro duct), have an additional piece of s tructure, to which this section is dedicated. 13 • A symm etry o n a tensor categ o ry ( C , ⊗ , 1 , α, ρ, λ ) is a na tur al isomorphism c : ⊗ → ⊗ ◦ σ , where σ : C × C → C × C is the ﬂip a utomorphism of C × C , suc h that c 2 = id. (I.e., for an y t wo o b jects X , Y there is an isomor phism c X,Y : X ⊗ Y → Y ⊗ X , natural w.r .t. X , Y suc h that c Y ,X ◦ c X,Y = id X ⊗ Y .), wher e “all pr op erly built diagr ams co mm ute”, i.e. the c ategory is coher en t. A symmetric tensor category (STC) is a tensor categ ory equipp ed with a symmetry . W e represent the symmetry g raphically by c X,Y = Y X ❅ ❅ ❅    X Y • As for tensor categories, there are tw o v ersions of the Coherence Theorem. V ersio n I (Mac Lane [178]): Let ( C , ⊗ , 1 , α, ρ, λ ) b e a tensor catego ry . Then a natural is o morphism c : ⊗ → ⊗ ◦ σ satisfying c 2 = id is a symmetry if and o nly if ( X ⊗ Y ) ⊗ Z c X,Y ⊗ id Z ✲ ( Y ⊗ X ) ⊗ Z α Y ,X ,Z ✲ Y ⊗ ( X ⊗ Z ) id Y ⊗ c X,Z ✲ Y ⊗ ( Z ⊗ X ) X ⊗ ( Y ⊗ Z ) α X,Y ,Z ❄ c X,Y ⊗ Z ✲ ( Y ⊗ Z ) ⊗ X α Y ,Z,X ✻ commutes. (In the strict case, this reduces to c X,Y ⊗ Z = id Y ⊗ c X,Z ◦ c X,Y ⊗ id Z .) A symmetric tensor functor is a tenso r functor F s uch that F ( c X,Y ) = c ′ F ( X ) ,F ( Y ) . Notice that a natural trans fo rmation b etw een s ymmetric tensor functors is just a monoidal natura l transformatio n, i.e. there is no new condition. • Now we can state version I I of the Co herence theorem: Every symmetric tens o r categ ory is equiv a len t (by a symmetric tensor functor) to a strict o ne . • E xamples of s y mmetric tensor c a tegories: – The categor y S deﬁned ea rlier, when c n,m : n + m → n + m is ta ken to b e the ele men t of S n + m deﬁned by (1 , . . . , n + m ) 7→ ( n + 1 , . . . , n + m, 1 , . . . n ). It is the free symmetric monoidal catego r y generated by one ob ject. – Non- strict symmetr ic categorica l g roups w ere c la ssiﬁed by Sinh [245]. W e p ostp one our discussion to Section 4, where we will als o c o nsider the braided cas e . – V ect k , represe ntation categ ories of g roups: W e have the canonical symmetry c X,Y : X ⊗ Y → Y ⊗ X, x ⊗ y 7→ y ⊗ x . – The tensor categ ories obtained using pro ducts o r copro ducts ar e symmetric. • Le t C be a str ict STC, X ∈ C and n ∈ N . Then there is a unique homomorphism Π X n : S n → Aut X ⊗ n such that σ i 7→ id X ⊗ ( i − 1) ⊗ c X,X ⊗ id X ⊗ ( n − i − 1) . Pro of: This is immediate by the deﬁnition o f STCs and the presentation S n = { σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i when | i − j | > 1 , σ 2 i = 1 } of the s ymmetric gr oups. These homomorphisms in fact com bine to a symmetric tensor functor F : S → C s uch that F ( n ) = X ⊗ n . (This is why S is called the free sy mmetric tensor c ategory on one genera to r.) 14 • In the ⊗ -categ ory C = V ect ﬁn k , Hom( V , W ) is itself an o b ject of C , giving rise to an internal hom-functor: C op × C → C , X × Y 7→ [ X, Y ] = Hom( X, Y ) satisfying some axio ms . In the older literature, a symmetric tensor ca tegory with such an in ternal-hom functor is called a closed ca teg ory . There are co her ence theorems for closed categ ories. [149, 148]. Since in V ect ﬁn k we hav e Hom( V , W ) ∼ = V ∗ ⊗ W , it is suﬃcien t – and more tra nsparent – to axiomatize duals V 7→ V ∗ , as is customary in the mor e rec e n t literature. W e w on’t mention ‘closed’ categ ories again. (Which do esn’t mea n that they hav e no uses!) • W e have seen that, ev en if a tens o r category has left and r ight duals ∨ X , X ∨ for every ob ject, they don’t need to be isomorphic. But if C is s ymmetric and X 7→ ( ∨ X , e X , d X ) is a left duality , then deﬁning X ∨ = ∨ X , e ′ X = e X ◦ c X, ∨ X , d ′ X = c X, ∨ X ◦ d X , one easily chec ks that X 7→ ( X ∨ , e ′ X , d ′ X ) deﬁnes a right dua lit y . W e can thus ta k e ∨ X = X ∨ and denote this mor e symmetrically by X . • Le t C be s ymmetric with g iv en left duals and with right duals as just deﬁned, and let X ∈ C . Deﬁne the (left) trace T r X : End X → E nd 1 by T r X ( s ) = e X ◦ id X ⊗ s ◦ d ′ X = ✛ ✘ e X X ✎ ✍ ☞ ✌ s ✚ ✙ d ′ X                 = ✛ ✘ e X X ✎ ✍ ☞ ✌ s ❅ ❅ ❅    ✚ ✙ d X                 Without muc h eﬀort, one can prove the trace prop erty T r X ( ab ) = T r X ( ba ) and multiplicativit y under ⊗ : T r X ⊗ Y ( a ⊗ b ) = T r X ( a )T r Y ( b ). Finally , T r X equals the right trac e deﬁned using e ′ X , d X . F or mo re on tra ces in tensor categor ies cf. e.g. [1 34, 185]. • Using the ab ov e, we deﬁne the categorical dimension of an o b ject X by d ( X ) = T r X (id X ) ∈ End 1 . If End 1 = k id 1 , we can use this identiﬁcation to obta in d ( X ) ∈ k . With this dimension and the usual symmetry and dualit y on V ect ﬁn k , one v eriﬁes d ( V ) = dim k V · 1 k . How ev er, in the category SV ect k of super vector spaces (whic h coincides with the repr esen- tation category Rep k Z 2 , but has the symmetry modiﬁed by the Koszul rule) it giv es the sup e r-dimension, which can b e negative, while o ne might pr e fer the total dimension. Suc h situations can be taken care of (without changing the sy mmetry) by intro ducing twists. • If ( C , ⊗ , 1 ) is s tr ict symmetric, we deﬁne a twist to be natura l family { Θ X ∈ End X , X ∈ C } of isomor phisms satisfying Θ X ⊗ Y = Θ X ⊗ Θ Y , Θ 1 = id 1 (2.1) i.e., Θ is a monoidal natural isomor phis m of the functor id C . If C has a left duality , w e also require ∨ (Θ X ) = Θ ∨ X . The second co ndition implies Θ 2 X = id. Notice that Θ X = id X ∀ X is a leg al choice. This will not remain tr ue in braided tensor c a tegories! Example: If G is a compact gr oup and C = Rep G , then the t wists Θ s atisfying o nly (2.1) are in bijection with Z ( G ). The second condition reduces this to central element s o f or der t wo. (Cf. e.g. [1 97].) 15 • Given a strict symmetric tensor category with left duality and a t wist, we can deﬁne a rig h t duality by X ∨ = ∨ X , writing X = ∨ X = X ∨ , but now e ′ X = e X ◦ c X, X ◦ Θ X ⊗ id X , d ′ X = id X ⊗ Θ X ◦ c X,X ◦ d X , (2.2) still deﬁning a right duality and the maps T r X : End X → E nd 1 still a re trace s . • Co n versely , the twist can b e recov ered from X 7→ ( X , e X , d X , e ′ X , d ′ X ) by Θ X = (T r X ⊗ id)( c X,X ) = X e X ☛ ✟ X ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ d ′ X ✡ ✠ X Thu s: Given a symmetric tensor ca tegory with ﬁxed left dualit y , every twist gives ris e to a right duality , a nd every rig h t duality tha t is ‘compatible’ with the left dua lit y gives a t wist. (The trivial t wist Θ ≡ id co rresp onds to the o riginal deﬁnition of r ight duality . The latter do es not w ork in prope r braided ca tegories!) This co mpatibilit y makes sense even for catego ries without symmetr y (or bra iding) and will be discussed later ( ❀ piv otal ca tegories). • The symmetric categories with Θ ≡ id a r e now called ev en . • The category SV ect k of s uper vector spac e s with Θ deﬁned in ter ms o f the Z 2 -grading no w satisﬁes dim( V ) ≥ 0 for all V . • The standar d examples for STCs are V ect k , S V ect k , Rep G and the representation categories of s uper g roups. In fact, rigid STCs are not far from b eing representation catego ries of (su- per )groups. Ho wev er, they not a lw ays are, cf. [103] for examples of no n-T a nna kian symmetric categorie s.) • A category C is called concrete if its ob jects ar e sets and Hom C ( X, Y ) ⊂ Hom Sets ( X, Y ). A k -linear categor y is ca lled co ncrete if the ob jects are ﬁn.dim. vector spaces over k and Hom C ( X, Y ) ⊂ Ho m V ec t k ( X, Y ). How ever, a be tter w a y of thinking of a c o ncrete catego ry is as a (abstr act) categ o ry C equipped with a ﬁb er functor , i.e. a faithful functor E : C → S ets , resp ectively E : C → V ect k . The la tter is required to be monoidal when C is monoidal. • E xample: G a group. Then C := Rep k G should be c onsidered as an abstra ct k -linear ⊗ - category together with a fa ithful ⊗ - functor E : C → V ect k . • The po in t o f this that a ca tegory C may hav e inequiv alent ﬁb er functors!! • B ut: If k is a lgebraically closed of c hara cteristic zero, C is rig id symm et ric k -linear with End 1 = k and F , F ′ are symmetric ﬁber functors then F ∼ = F ′ (as ⊗ -functors). (Saav edra Riv ano [238, 64]). • The ﬁrst non-trivial application of (symmetric ) tensor categor ies probably were the r econ- struction theor ems of T annak a [255] (193 9!) and Saavedra Riv a no [2 38, 64]. Let k b e alg ebraically clo s ed. Let C b e rig id sy mmetr ic k - linear with End 1 = k a nd E : C → V ect k faithful tensor functor. (T annak a did this for k = C , C a ∗ - category and E ∗ - preserving .) Let G = Aut ⊗ E be the group of natural monoida l [unitary] a utomorphisms o f E . Deﬁne a functor F : C → Rep G [unitary r epresentations] by F ( X ) = ( E ( X ) , π X ) , π X ( g ) = g X ( g ∈ G ) . Then G is pr o-algebr aic [compact] and F is an equiv alence of sy mmetr ic tensor [ ∗ -]c a tegories. Pro of: The idea is the following (Grothendieck, Sa av edra Riv ano [238], cf. Bich on [27]): Let E 1 , E 2 : C → V ect k be ﬁb er functors. Deﬁne a unital k -alg ebra A 0 ( E 1 , E 2 ) by A 0 ( E 1 , E 2 ) = M X ∈ C Hom V ec t ( E 2 ( X ) , E 1 ( X )) , 16 spanned by elements [ X, s ] , X ∈ C , s ∈ Hom( E 2 ( X ) , E 1 ( X )), with [ X, s ] · [ Y , t ] = [ X ⊗ Y , u ], where u is the co mpos ite E 2 ( X ⊗ Y ) ( d 2 X,Y ) − 1 ✲ E 2 ( X ) ⊗ E 2 ( Y ) s ⊗ t ✲ E 1 ( X ) ⊗ E 1 ( Y ) d 1 X,Y ✲ E 1 ( X ⊗ Y ) . This is a unital asso ciative algebra, and A ( E 1 , E 2 ) is deﬁned a s the quotient by the idea l generated by the elemen ts [ X , a ◦ E 2 ( s )] − [ Y , E 1 ( s ) ◦ a ], where s ∈ Ho m C ( X, Y ) , a ∈ Hom V ec t ( E 2 ( Y ) , E 1 ( X )). • Rema rk: Let E 1 , E 2 : C → V ect k be ﬁb er functors as a bove. Then the map X × Y 7→ Hom V ec t k ( E 2 ( X ) , E 1 ( Y )) extends to a functor F : C op × C → V ect k . Now the alg ebra A ( E 1 , E 2 ) is just the co end R X F ( X, X ) of F , a universal ob ject. Co ends are a categ orical, non-linear version o f traces, but we refra in fro m going into them since it takes some time to appre ciate the co ncept. (Cf. [180].) • Now one pr oves [27, 197]: – If E 1 , E 2 are symmetr ic tensor functor s then A ( E 1 , E 2 ) is c o mm utative. – If C is ∗ -category and E 1 , E 2 are *-preser ving then A ( E 1 , E 2 ) is a ∗ -algebra and has a C ∗ -completion. – If C is ﬁnitely generated (i.e. there exists a mono ida l generator Z ∈ C such that every X ∈ C is direct summand of s ome Z ⊗ N ) then A ( E 1 , E 2 ) is ﬁnitely genera ted. – Ther e is a bijection b etw een na tur al monoidal (unitary) isomor phis ms α : E 1 → E 2 and ( ∗ -)characters on A ( E 1 , E 2 ). Thu s: If E 1 , E 2 are symmetric and either C is ﬁnitely generated or a ∗ -catego ry , the a lgebra A ( E 1 , E 2 ) a dmits ch ara cters (by the Nullstellensatz or b y Gelfand’s theory), thus E 1 ∼ = E 2 . One also ﬁnds that G = Aut ⊗ E ∼ = ( ∗ -)Char( A ( E , E )) a nd A ( E ) = F un( G ) (representativ e resp ectively con tinuous functions). This is use d to prov e that F : C → Rep G is an equiv alence. • Rema rks: 1. While it has become customary to speak of T a nnakian catego ries, the w ork of Kre ˘ ın, cf. [158], [118, Section 30], sho uld also b e men tioned since it can b e considered as a pr e- cursor of the la ter genera lizations to no n- symmetric ca tegories, in particular in W oronowicz’s approach. 2. The uniqueness of the symmetric ﬁb er funct or E implies that G is unique up to iso mo r- phism. 3. F o r the ab ov e construction, we need to have a ﬁb er functor. Around 1989 , Doplic her and Rob erts [70], and indep e ndently Deligne [58] construct such a functor under weak assumptions on C . See b elow. 4. The uniquene s s pro of fails if either o f E 1 , E 2 is not symmetric (or C is not symmetric). Given a group G , there is a tautological ﬁb er functor E . The fact that there may b e (non- symmetric) ﬁb er functors that are not naturally isomorphic to E reﬂects the fact that there can be groups G ′ such that Rep G ≃ Rep G ′ as tensor ca tegories, but not as symmetric tensor categorie s! This phenomenon was indepe ndently discov ered by Etingof/Gelaki [8 0], who ca lled such G, G ′ iso categorical and pr o duced e x amples of is oc ategorica l but non-isomorphic ﬁnite groups, by Davydo v [5 5] and b y Izumi and Ko saki [122]. The treatment in [8 0] relies o n the fact that if G, G ′ are iso categor ical then C G ′ ∼ = C J for some Drinfeld t wist J . A more categoric al approa c h, a llowing also an extens io n to compact groups, will b e g iv en in [202]. A group G is called categor ically rig id if every G ′ iso categor ical to G is actually isomorphic to G . (Co mpa ct gr oups that are abelia n or connected are ca tegorically r igid in a strong se nse since they are determined already by their fusion hypergr oups.) 17 • Co nsider the free rigid symmetr ic tensor ∗ -catego ry C with End 1 = C genera ted by one ob ject X of dimens io n d . If d ∈ N then C is equiv a len t to Rep U ( d ) or Rep O ( d ) or Rep S p ( d ), dep end- ing on whether X is non-selfdual o r or thogonal o r symplectic. The pro of [9] is straightforward once one has the Do plicher-Rob erts theorem. • The free r igid symmetr ic categor ies just mentioned can be construc ted in a top ological wa y , in a fashion very similar to the construction of the T emp erley-Lieb catego ries TL( τ ). The ma in diﬀerence is that one allows the lines in the pictures deﬁning the morphisms to cross. (But they still live in a plane.) Now one quotient s o ut the negligible mor phisms and completes w.r.t. direct sums and splitting idemp otents. (In the non- self dual case, the ob jects are words ov er the a lphabet { + , − } and the lines in the morphisms are directed.) All this is noted in passing by Deligne in a pap er [59] dedicated to the exce ptio nal gro ups! Notice that when d 6∈ N , these ca tegories are exa mples of rigid symmetric c ategories that a re not T annakia n. • The above results alr eady establish stro ng co nnec tio ns b etw een tensor catego ries and repre- sentation theory , but there is muc h more to say . 3 Bac k to general tensor c ategories • In a general tensor category , left and right duals need no t co incide. This can alre a dy b e seen for the left mo dule c a tegory H − Mo d of a Hopf algebra H . This category ha s left and right duals, related to S and S − 1 . ( S m ust b e invertible, but can b e a per io dic !) They coincide when S 2 ( x ) = uxu − 1 with u ∈ H . • W e only consider tensor categor ie s that hav e isomorphic left and right dua ls, i.e. t wo-sided duals, which we de no te X . • If C is k -line a r with E nd 1 = k id and End X = k id ( X is simple/ir r educible), one can canonically deﬁne the squa red dimensions d 2 ( X ) ∈ k b y d 2 ( X ) = ( e X ◦ d ′ X ) · ( e ′ X ◦ d X ) ∈ End 1 . (Since X is simple, the morphis ms d, d ′ , e, e ′ are unique up to sca lars, and well-deﬁnedness of d 2 follows from the e q uations inv olving ( d, e ) , ( d ′ , e ′ ) bilinear ly .) Cf. [191]. • If C is a fusion categor y , we deﬁne its dimensi on by dim C = P i d 2 ( X i ). • If H is a ﬁnite dimensional semisimple and co-semisimple Hopf algebra then dim H − Mod = dim k H . (A ﬁnite dimensional Hopf alg ebra is co-semis imple if and o nly if the dual Ho pf algebra b H is se mis imple.) • E ven if C is semisimple, it is not cle a r whether one ca n cho ose r o ots d ( X ) o f the ab ove num bers d 2 ( X ) in such a wa y that d is additive and multiplicativ e! • In pivotal categ o ries this ca n b e done. A str ict pivotal categor y [93, 94] is a strict left rig id category with a monoidal structure on the functor X 7→ ∨ X and a monoidal equiv ale nc e of the functors id C and X 7→ ∨∨ X . As a cons e quence, one can deﬁne a rig h t duality satisfying X ∨ = ∨ X . • In a str ict pivotal categories we can deﬁne left and r ight trace s for every endomorphism: T r L X ( s ) = ✛ ✘ e X X ✎ ✍ ☞ ✌ s ✚ ✙ d ′ X T r R X ( s ) = ✛ ✘ e ′ X ✎ ✍ ☞ ✌ s X ✚ ✙ d X (3.1) Notice: In g eneral T r L X ( s ) 6 = T r R X ( s ). 18 • W e no w deﬁne dimensions by d ( X ) = T r L X (id X ) ∈ E nd 1 . One then automatically has d ( X ) = T r R X (id X ), whic h can diﬀer from d ( X ). But for simple X we hav e d ( X ) d ( X ) = d 2 ( X ) with d 2 ( X ) a s ab ov e. • In a pivotal catego ry , we can use the trace to deﬁne pair ings Hom( X , Y ) × Hom( Y , X ) → End 1 b y ( s, t ) 7→ T r L X ( t ◦ s ). In the semisimple k -linear case with End 1 , these pairings are non-degenera te for all X , Y . Cf. e.g. [104]. In general, a mor phism s : X → Y is ca lle d negligi ble if T r( t ◦ s ) = 0 for all t : Y → X . W e call an Ab-category non-de gener ate if only the zero mor phisms are negligible. The negligible morphisms for m a monoidal ideal, i.e. comp osing or tensor ing a neglig ible morphism with any morphism yields a neg ligible morphism. It follows that o ne can quotient out the negligible morphisms in a stra ightf orward wa y , obtaining a non-de g enerate catego ry . A non-degenerate ab elian category is se mis imple [61], but a count erexa mple given ther e shows that no n-degeneracy plus ps eudo-ab elianness do not imply semisimplicity! • A spherical ca teg ory [20] is a pivotal category where the left a nd right traces coincide. Equiv alen tly , it is a strict autonomous category (i.e. a tensor category eq uipped with a left and a r ight duality) for which the r esulting functors X 7→ X ∨ and X 7→ ∨ X coincide. Sphericity implies d ( X ) = d ( X ), and if C is semisimple, the c o n verse implication holds. • The T emp erley- L ieb categor ie s T L ( τ ) are spherical. • A ﬁnite dimensional Hopf algebr a that is involutiv e, i.e. satisﬁes S 2 = id , giv es rise to a spherical ca tegory . (It is known that every semisimple a nd co-semisimple Hopf algebra is inv olutiv e.) Mo re generally , ‘spheric al Hopf algebras’, deﬁned as satisfying S 2 ( x ) = w xw − 1 , where w ∈ H is invertible with ∆( w ) = w ⊗ w and T r( θ w ) = T r( θ w − 1 ) fo r an y ﬁnitely generated pro jective le ft H -mo dule V , give r ise to spherical catego ries [20]. • In a ∗ -ca tegory with conjuga tes, trac es of endomo rphisms, in particular dimensio ns of ob jects, can b e deﬁned uniq ue ly without choosing a spherical structure, cf. [70, 172]. The dimension satisﬁes d ( X ) ≥ 1 for every non-zer o X , and d ( X ) = 1 holds if and o nly if X is inv ertible. F urther more, o ne has [1 72] a ∗ - categorica l version of the quantization of the Jo nes index [1 26]: d ( X ) ∈ n 2 cos π n , n = 3 , 4 , . . . o ∪ [2 , ∞ ) . On the other hand, every tensor ∗ -categor y c an b e equippe d [286] with an (essentially) unique spherical str ucture such the trac e s and dimension deﬁned using the latter coincide with those of [172]. • In a C -linea r fusion categ ory (no ∗ -o p era tion req uired!) o ne has d 2 ( X ) > 0 for all X , cf. [84]. The following is a very useful application: If A ⊂ B is a full inclusion of C -linear fusion category then dim A ≤ dim B , a nd equality holds if and only if A ≃ B . • In a unitary categ o ry , dim C = F P − dim C . Ca tegories with the latter proper t y are called pseudo-unitary in [84], where it is shown that every pseudo-unitary c ategory admits a unique spher ical structure such that F P − d ( X ) = d ( X ) for all X . • Ther e are T annak a-st yle theorem for not necessar ily symmetric categorie s (Ulbrich [268], Y etter [2 88], Sc hauenburg [239]): Let C be a k -linear pivotal category with End 1 = k id 1 and let E : C → V ect k a ﬁber functor. Then the algebra A ( E ) deﬁned as abov e admits a copr o duct and a n antipo de, thus the structur e of a Hopf algebra H , and an equiv a lence F : C → Co mo d H such that E = K ◦ F , wher e K : Como d H → V ect k is the for g etful functor. (If C and E ar e symmetric , this H is a commutativ e Hopf a lgebra o f functions o n the g roup obtained earlier G .) W oro nowicz prov ed a similar res ult [280] for ∗ -categor ies, obtaining a compact q uant um g roup (as deﬁned b y him [279, 28 1]). Commutativ e compac t quantum groups are just algebra s C ( G ) for a compac t gr o up, th us one recovers T anna k a’s theorem. Cf. [131] for a n excellent introduction to the are a of T annak a-Krein reco ns truction. 19 • Given a ﬁb er functor, ca n o ne ﬁnd an algebra ic structur e whose r epr esentations (r ather than corepres en tations) are equiv alen t to C ? The answer is p ositive, provided one uses a slight generaliza tion of Hopf algebr as, to wit A. v an Daele’s ‘Algebr aic Quantum Groups’ [269, 2 7 0] (or ‘Multiplier Hopf algebr as with Haar functional’). The y are not neces sarily unital a lgebras equipp e d with a copro duct ∆ that takes v a lues in the mu ltiplier a lg ebra M ( A ⊗ A ) and with a left-inv aria n t Haar -functional µ ∈ A ∗ . A nice featur e of algebraic quantum groups is that they admit a nice version of Pon tryagin duality (whic h is not the case for inﬁnite dimensional ordinary Hopf algebra s). In [200] the fo llowing was s hown: If C is a semisimple spherical ( ∗ -)category and E a ( ∗ - )ﬁber functor then there is a dis crete mu ltiplier Hopf ( ∗ -)algebra ( A, ∆) and an equiv alence F : C → Rep( A, ∆) s uc h that K ◦ F = E , where K : Rep( A, ∆) → V ect is the for getful functor. (This ( A, ∆) is the Pontrjagin dual of the A ( E ) ab ov e.) This theor y exploits the semisimplicity fro m the very b eginning, which makes it quite transpar en t: One deﬁnes A = M i ∈ I End E ( X i ) and M ( A ) = Y i ∈ I End E ( X i ) ∼ = Nat E , where the summation is ov er the eq uiv alence cla sses of simple ob jects in C . No w the tensor structures o f C and E give r ise to a copro duct ∆ : A → M ( A ⊗ A ) in a very direct way . Notice: This reco ns truction is r elated to the preceding one as follows. Since H − co mod ≃ C is semisimple, the Ho pf algebra H ha s a left-inv aria n t integral µ , thus ( H , µ ) is a compact alge- braic quantum gro up, and the discrete alge braic quantum group ( A, ∆) is just the Pon trjagin dual of the latter. • In this situation, there is a bijection b etw een braiding s o n C and R-matr ices (in M ( A ⊗ A )), cf. [200]. But: The br aiding on C plays no ess en tial rˆ ole in the reconstructio n. (Since [200] works with the category of ﬁnite dimensional representations, which in ge ner al do es not con tain the left reg ular representation, this is more work than e.g. in [137] and r equires the use of semisimplicity .) • Summing up: Linear [braided] tenso r categories admitting a ﬁb er functor are (co)represe n ta- tion catego ries of [(co)quasi- tr iangular] discrete (compac t) quantum groups. Notice that her e ‘Quantum groups’ refer s to Hopf alg ebras and suitable generalizations thereof, but not necessarily to q-deformations of some str ucture arising fro m groups! • W ARNING: The non-uniqueness of ﬁb er functors means that there can be non-isomo r phic quantum gr oups whose (co)re pr esentation categ ories are equiv alent to the given C ! The study of this pheno menon leads to Hopf-Galois theory and is connected (in the ∗ -case) to the study of ergo dic actio ns of quantum g roups on C ∗ -algebra s. (Cf. e.g . Bichon, de Rijdt, V aes [28]). • Despite this non-uniquenes s, one may ask whether one can intrinsic al ly c hara cterize the tensor categorie s admitting a ﬁb er functor, thus b eing rela ted to quan tum gro ups . (Existence of a ﬁber functor is an extrinsic cr iterion.) The few known results to this q uestions are o f tw o types. On the one hand there are some recognition theor ems for certa in cla s ses of repres en tation categorie s of q uan tized en veloping algebra s, which will be discus sed so mewhat later . On the other hand, there are results based o n the regular r epresentation, to which we turn now. How ev er, it is only in the symmetric case that this lea ds to really satisfactory r esults. • The left regular repre sent ation π l of a compact gro up G (living on L 2 ( G )) has the following well known prop erties: π l ∼ = M π ∈ b G d ( π ) · π , (Peter-W eyl theorem) π l ⊗ π ∼ = d ( π ) · π l ∀ π ∈ Rep G. (absorbing pr ope r t y) . 20 • The second pro p erty genera lizes to any alg ebraic quantum g roup’ ( A, ∆), cf. [20 1]: 1. Let Γ = π l be the left r egular re presentation. If ( A, ∆) is discrete, then Γ carries a monoid structure (Γ , m, η ) with dim Hom( 1 , Γ) = 1 , which w e call the regular monoid . (Algebras in k -linear tensor categories satisfying dim Hom( 1 , Γ) = 1 hav e been called ‘s imple’ or ‘ha ploid’.) If ( A, ∆) is c o mpact, Γ has a como noid structur e. (And in the ﬁnite (=compact + discrete) case, the algebr a and coalge bra structures combine to a F rob enius algebr a, cf. [19 1], discussed below.) 2. If ( A, ∆) is a discrete algebra ic q uant um g roup, o ne has a monoid version o f the absorbing prop erty: F or every X ∈ Rep( A, ∆) o ne has an isomor phism (Γ ⊗ X , m ⊗ id X ) ∼ = n ( X ) · (Γ , m ) (3.2) of (Γ , m, η )-mo dules in Rep( A, ∆). (Here n ( X ) ∈ N is the dimension of the v ector space of the repr esentation X , which in general diﬀers from the categor ic a l dimension.) • The following theorem from [2 01] is motiv ated by Deligne’s [58]: Let C be a k -linear ca tegory and (Γ , m, η ) a monoid in C (more generally , in the asso cia ted catego r y Ind C of inductive limits) satisfying dim Hom( 1 , Γ) = 1 and (3.2) for so me function n : Ob j C → N . Then E ( X ) = Hom V ec t k ( 1 , Γ ⊗ X ) deﬁnes a faithful ⊗ -functor E : C → V ect k , i.e. a ﬁber functor. (One has dim E ( X ) = n ( X ) ∀ X and Γ ∼ = ⊕ i n ( X i ) X i .) If C is s ymmetric and (Γ , m, η ) commutativ e (i.e. m ◦ c Γ , Γ = m ), then E is symmetric. Remark: Deligne co nsidered this o nly in the symmetric case, but did not make the requirement dim Hom( 1 , Γ) = 1. This leads to a tensor functor E : C → A − Mo d, where A = Hom( 1 , Γ) is the c omm utative k -algebra of ‘elemen ts of Γ’ enco un tered earlier . • This gives rise to the following implications: There is a discr ete A QG ( A, ∆) such that C ≃ Rep( A, ∆) C admits an absorbing monoid There is a ﬁb er functor E : C → H      ✒ ❅ ❅ ❅ ❅ ❅ ❘ ✛ Remarks: 1. This ca n b e considered as an int rinsic characterization of quantum gr oup ca te- gories. (O r rather semi-intrinsic, since the regular mo noid lives in the Ind-category of C rather than C itself.) 2. The c a se of ﬁnite ∗ -ca tegories had b een treated in [170], using subfactor theory and a functional analys is. 3. This res ult is quite unsatisfactory , but I do ubt that a better result can b e obtained without restriction to sp e cial classes of c ate gorie s or a dopting a wide gener alization of the notion of quantum gr oups . Exa mples for b oth will b e given b elow. 4. F or a diﬀeren t a pproach, a ls o in terms o f the regular repr esentation, cf. [69]. 21 • No tice that ha ving a n absorbing monoid in C (or ra ther Ind( C )) means having an N -v a lued dimension function n o n the hypergr o up I ( C ) and a n asso ciative pro duct on the ob ject Γ = ⊕ i ∈ I n i X i . The la tter is a cohomolo gical c ondition. If C is ﬁnite, one can show using Perron-F rob enius theory that there is o nly one dimension function, namely the int rinsic one i 7→ d ( X i ). Thus a ﬁnite ca tegory with non-integer in trinsic dimensions canno t be T annakian (in the a bove sense). • W e no w turn to a very b eautiful result of Deligne [58] (simpliﬁed considerably by Bichon [2 7]): Let C b e a semisimple k -linear rigid ev en symmetric categor y satisfying End 1 = k , whe r e k is algebr aically closed of character is tic zer o. Then there is an abso rbing commutativ e monoid as ab ov e. (Th us w e have a symmetric ﬁb er functor, implying C ≃ Rep G .) Sketc h: The homomor phisms Π X n : S n → Aut X ⊗ n allow to deﬁne the idemp oten ts P ± ( X, n ) = 1 n ! X σ ∈ S n sgn( σ ) Π X n ( σ ) ∈ End( X ⊗ n ) and their images S n ( X ) , A n ( X ), which a re dir ect summands of X ⊗ n . Making cruc ia l use of the evenness assumption on C , one proves d ( A n ( X )) = d ( X )( d ( X ) − 1) · · · ( d ( X ) − n + 1) n ! ∀ n ∈ N . In a ∗ -catego ry , this must b e non-negative ∀ n , implying d ( X ) ∈ N , cf. [7 0]. Using this – o r assuming it as in [58] – one ha s d ( A d ( X ) ( X )) = 1, and A d ( X ) ( X ) is c alled the determi nan t of X . On the other hand, o ne can deﬁne a commutativ e monoid structure on S ( X ) = ∞ M n =0 S n ( X ) , obtaining the symmetric alg ebra ( S ( X ) , m, η ) o f X . Let Z be a ⊗ -generator Z of C sat- isfying det Z = 1 . Then the ‘interaction’ b etw een s y mmetrization (symmetric algebra ) a nd antisymmetrization (deter minan ts) a llows to construct a maxima l ideal I in the co mm utative algebra S ( Z ) such that the quotient algebra A = S ( Z ) /I ha s all desired proper ties: it is commutativ e, absorbing and satisﬁes dim Hom( 1 , A ) = 1. Q ED. Remarks: 1. The absorbing mono id A constructed in [58 , 27] did not satisfy dim Hom( 1 , A ) = 1. Therefore the construction considered ab ov e do es not give a ﬁber functor to V ect C , but to Γ A − Mo d, and one needs to quotient by a maximal ideal in Γ A . Showing that one ca n achiev e dim Ho m( 1 , A ) = 1 was p erhaps the main innov ation of [197]. This has the a dv antage that ( A, m, η ) actually is (isomorphic to) the regula r monoid of the group G = Nat ⊗ E . As a consequence, the latter g roup can b e o btained simply as the a utomorphism gr oup Aut(Γ , m, η ) ≡ { g ∈ Aut Γ | g ◦ m = m ◦ g ⊗ g , g ◦ η = η } of the mo noid – without even ment ioning ﬁb er functors! 2. Combining T annak a’s theorem with those o n ﬁb er functors from mo noids and with the ab ov e, o ne has the following bea utiful Theorem [70, 58]: Let k b e algebraically close d o f characteristic zero and C a semisimple k- linear rigid even symmetric category with E nd 1 = k . Assume that all ob jects hav e dimension in N . T he n there is a pr o-algebr a ic group G a , unique up to is o morphism, such that C ≃ Rep G a (ﬁnite dimensio nal ra tional r epresentations). If C is a ∗ -ca tegory then semisimplicity and the dimension condition are redundan t, and there is a unique compact group G c such that C ≃ Rep G c (contin uous unitary ﬁnite-dimensiona l representations). In this case, G a is the complexiﬁcation of G c . 22 3. If C is symmetric but not even, its symmetry can be ‘b osonized’ into an even one, c f. [70]. Then one applies the ab ov e result and obtains a group G . The Z 2 -grading o n C g iven by the t wist gives rise to an element k ∈ Z ( G ) satisfying k 2 = e . Thu s C ≃ Rep( G, k ) as symmetric category . Cf. a lso [60]. • The ab ov e res ult has several applica tions in pur e mathematics: It plays a big rˆ ole in the theory of motives [5, 166] and in diﬀere ntial Galois theory a nd the rela ted Riema nn Hilb ert problem, cf. [230]. It is used for the classiﬁcation of triang ular Hopf algebras in ter ms of Drinfeld twists of gro up algebr as (Eting of/Gelaki, cf. [100] and refere nc e s therein) and for the mo dularization of braided tensor categor ies [39, 190], cf. below. The w ork of Do plicher a nd Ro b erts [70] was motiv a ted by applications to quantum ﬁeld theor y in ≥ 2 + 1 dimensio ns [6 8, 7 1], where it leads to a Galois theo ry of quantum ﬁelds, cf. also [111]. • Thus, at least in characteristic zero (in the absence of a ∗ -op eratio n one needs to impo se int egra lit y of all dimensions ) rigid symmetric categories with End 1 = k id 1 are reasonably well unders too d in terms of compa ct o r pro-aﬃne groups. What ab out rela xing the last condition? The catego ry of a representations (on contin uous ﬁelds of Hilb ert spa ces) of a compact group oid G is a symmetric C ∗ -tensor category . Since a lot of infor mation is lost in passing from G to Rep G , there is no hope of reco nstructing G up to iso morphism, but one may hop e to ﬁnd a compa ct group bundle giving rise to the given categor y and pr oving that it is Morita equiv alent to G . Howev er, there seem to b e top o logical o bstructions to this b eing alwa ys the case, c f. [272]. • In this co n text, we ment ion related work by B rugui` eres/Ma ltsiniotis [18 4, 40, 37] o n T annak a theory for quasi qua n tum group oi ds in a pur e ly alg ebraic setting. • W e now turn to the characterization of certain sp e cia l cla sses of tensor categor ies: • Co m bining Doplicher-Rob erts reco nstruction with the mentioned r esult of McMullen and Handelman o ne obtains a simple prototype : If C is an e v en symmetric tensor ∗ -ca tegory with conjugates and End 1 = C who se fusion h yp ergro up is isomorphic to that of a connected compact Lie g roup G , then C ≃ Rep G . • K azhdan/W enzl [145]: Let C be a semisimple C - linear spheric a l ⊗ -categor y with End 1 = C , whose fusion h ype rgroup is isomorphic to that of sl ( N ). Then there is a q ∈ C ∗ such that C is eq uiv alent (as a tens or category) to the representation ca teg ory o f the Drinfeld/Jim b o quantum gr oup S L q ( N ) (or one of ﬁnitely many twisted versions of it). Here q is either 1 or not a ro o t of unity and unique up to q → q − 1 . (F or another a pproach to a c haracter ization of the S L q ( N )-categor ies, excluding the ro ot of unity case, cf. [228].) F urther more: If C is a semisimple C -linea r rigid ⊗ -categ o ry with End 1 = C , who se fusion hypergro up is isomorphic to that of the (ﬁnite!) representation category of S L q ( N ), where q is a primitive ro o t o f unity of order ℓ > N , then C is equiv alent to Rep S L q ( N ) (or one o f ﬁnitely many twisted versions). W e will say (a bit) more on qua n tum groups later . The rea s on that w e men tion the Ka zh- dan/W enzl result alrea dy here is that it does not require C to come with a bra iding. Un- fortunately , the proo f is not independent o f quantum group theory , nor does it pr ovide a c onstruction of the ca tegories. Beginning of pr o of: The assumption on the fusion rules implies that C has a multiplicative generator Z . Consider the full monoida l subca tegory C 0 with ob jects { Z ⊗ n , n ∈ Z + } . Now C is equiv alent to the idempo tent co mpletion (‘Karoubiﬁcation’) of C 0 . (Aside: T ensor categories with ob jects N + and ⊗ = + for ob jects appear quite often: The symmetric category S , the braid category B , PROPs [179].) A semisimple k -linea r catego r y with ob jects Z + is called a monoidal algebra , a nd is equiv alent to having a family A = { A n,m } o f vector spaces together with semisimple alg e bra structures on A n = A n,n and bilinear ope r ations ◦ : A n,m × A m,p → A n,p and ⊗ : A n,m × A p,q → A n + p,m + q satisfying obvious axioms. A mono idal a lgebra is 23 diagonal if A n,m = 0 for n 6 = m and o f t ype N if dim A (0 , n ) = dim A ( n, 0) = 1 and A n,m = 0 unless n ≡ m (mo d N ). If A is of t yp e N , there are exactly N mo noidal alg ebras with the same diago nal. The pos sible diago nals arising fro m type N monoidal alg ebras can be class iﬁed, using Heck e algebr as H n ( q ) (deﬁned later). • Ther e is an analog ous result (T uba /W enzl [259]) for catego r ies with the other cla ssical (BCD) fusion rings, but that do es requir e the catego ries to come with a bra iding. • F or fusion catego ries, there are a num ber o f cla ssiﬁcation results in the case of low rank (n umber of simple ob jects) (Ostrik: fusion catego ries of ra nk 2 [22 4], bra ided fusion categories of rank 3 [225]) or sp ecial dimensions, like p or pq (Etingof/Gelaki/ O strik [82]). F urthermore, one can c la ssify near group categories , i.e. fusion categ ories with a ll simple o b jects but one inv ertible (T ambara/Y a magami [254], Siehler [24 4]). • In another direction one may try to represent more tens o r ca tegories as mo dule categories by gener ali zing the notion of Hopf algebr as . W e hav e a lready encountered a very modest (but useful) genera lization, to wit V an Daele’s m ultiplier Hopf alg ebras. (But the main r ationale for the latter was to repair the breakdown of Pontrjagin dualit y for inﬁnite dimensiona l Ho pf algebras , which works so nicely for ﬁnite dimensiona l Hopf algebras .) • Dr infeld’s quasi-Hopf algebras [73] g o in a diﬀerent directio n: One considers an asso ciative unital algebra H with a unital algebr a homomorphism ∆ : H → H ⊗ H , where c o asso ciativity holds only up to conjugation with a n inv ertible element φ ∈ H ⊗ H ⊗ H : id ⊗ ∆ ◦ ∆( x ) = φ (∆ ⊗ id ◦ ∆( x )) φ − 1 , where (∆ , φ ) must sa tisfy so me ide ntit y in or der for Rep H with the tensor pro duct deﬁned in terms of ∆ to be (no n-strict) monoidal. Unfortunately , duals of quasi- Ho pf a lgebras a re no t quasi-Hopf alge bras. They are useful nevertheless, even for the pro of of r esults concerning ordinary Hopf algebra s, like the Ko hno-Drinfeld theorem fo r U q ( g ), cf. [73, 74] a nd [137]. Examples: Giv en a ﬁnite group G a nd ω ∈ Z 3 ( G, k ∗ ), there is a ﬁnite dimensional quasi Hopf algebra D ω ( G ), the twisted q ua n tum double of Dijkgr aaf/Pasquier/Ro che [66]. (W e will later deﬁne its representation categ ory in a purely ca teg orical wa y .) Recently , Naidu/Nikshyc h [205] ha ve given neces sary and suﬃcien t conditions on pairs ( G, [ ω ]) , ( G ′ , [ ω ] ′ ) for D ω ( G ) − Mo d , D ω ′ ( G ′ ) − Mo d to be equiv alent as braided tensor c ategories. But the question for which pair s ( G, [ ω ]) D ω ( G ) − Mo d is T annakian (i.e. admits a ﬁb er functor and ther efore is equiv a len t to the r epresentation category of an or dinary Hopf algebra) seems to b e still o p en. • Ther e ha ve b een v arious attempts at proving generalized T annak a reconstruction theorems in terms o f quasi- Hopf alge br as [182] and “weak quasi-Hopf a lgebras” . (Cf. e.g. [176, 113].) As it turned o ut, it is suﬃcient to co ns ider ‘w eak’, but ‘non-quasi’ Hopf alg ebras: • P receded b y Hay ashi’s ‘face alg ebras’ [115], which larg ely w ent unnoticed, B¨ ohm and Szlach´ an yi [35] and then Nikshyc h, V ainerman, L. Ka dis on int ro duced weak Hopf algebras , which may be consider ed as ﬁnite-dimensio na l quantum gro upo ids : They a r e asso cia tive unital alg ebras A with c o asso ciative algebra homomorphism ∆ : A → A ⊗ A , but the axioms ∆( 1 ) = 1 ⊗ 1 and ε ( 1 ) = 1 are weak ened. W eak Hopf algebras ar e closely related to Hopf a lgebroids and hav e v arious des irable prop- erties: Their duals are weak Hopf algebras, and Pon trjagin dualit y holds. The categ o rical dimensions of their representations can b e non-integer. And they ar e gener al enough to ‘ex- plain’ ﬁnite-index depth-t wo inclusions of von Neumann factors, cf. [215]. • F urthermo re, Ostrik [2 22] prov ed that every fusion categor y is the mo dule ca tegory of a semisimple weak Hopf alg e bra. (Again, there w as re lated earlier w ork by Hay ashi [116] in the context o f his face algebras [11 5].) Pro of idea: An R -ﬁb er functor on a fusio n catego ry C is a faithful tens o r functor C → Bimo d R , where R is a ﬁnite dir ect sum o f matr ix alg ebras. Szla c h´ an yi [2 52]: An R-ﬁb er 24 functor on C gives rise to an equiv alence C ≃ A − Mo d for a weak Hopf algebra (with base R ). (Cf. also [110].) Ho w to construct an R-ﬁber functor? Since C is se mis imple, we can c ho ose an algebra R s uc h that C ≃ R − Mo d (as ab elian categorie s). Since C is a mo dule category over itself, we hav e a C -mo dule structure on R − Mo d. Now use that, for C and R as above, there is a bijection b et ween R-ﬁb er functors and C -module category structures on R − Mo d (i.e. tensor functors C → End( R − Mo d). Remarks: 1. R is hig hly non-unique: The only requirement was that the num ber o f simple di- rect summands equals the n umber of simple ob jects o f C . (Th us there is a unique commutativ e such R , but even for that, there is no uniqueness of R -ﬁb er functor s.) 2. The ab ov e pro of uses semisimplicity . (A non-semisimple generalizatio n was a nnounced by Brugui` eres and Virezilier in 2 0 08.) • Le t C b e fusion categor y and A a weak Hopf alge bra such that C ≃ A − Mod. Since there is a dua l weak Hopf a lgebra b A , it is natural to ask how b C = b A − Mod is rela ted to C . (One may call s uc h a categ ory dual to C , but must keep in mind that there is o ne for every weak Ho pf algebra A such that C ≃ A − Mod.) • Answer: b A − Mo d is (weakly monoida lly) Morita e q uiv alent to C . This notion (M¨ uger [191]) was inspired by subfactor theory , in pa rticular idea s of Ocneanu , cf. [21 6, 217]. F o r this w e need the following: • A F rob eni us algeb ra in a strict tenso r ca tegory is a quintuple ( A, m, η , ∆ , ε ), where ( A, m, η ) is an algebra, ( A, ∆ , ε ) is a co algebra and the F r ob enius identit y m ⊗ id A ◦ id A ⊗ ∆ = ∆ ◦ m = id A ⊗ m ◦ ∆ ⊗ id A holds. Diagra mmatically: ☛ ✟ ✡ ✠ = ✡ ✠ ☛ ✟ = ☛ ✟ ✡ ✠ . A F rob enius alg ebra in a k -linear catego ry is called strongly separable if ε ◦ η = α id 1 , m ◦ ∆ = β id Γ , αβ ∈ k ∗ . The ro ots o f this deﬁnition g o quite far back. F. Quinn [231] discussed them under the name ‘ambialgebras’, and L. Abrams [1] prov ed that F rob enius algebr as in V ect ﬁn k are the usual F ro benius algebra s, i.e. k -algebr as V equipped with a φ ∈ V ∗ such that ( x, y ) 7→ φ ( xy ) is non-degenera te. F rob enius a lgebras play a central rˆ ole for top olog ical quantum ﬁeld theories in 1 + 1 dimensions, cf. e.e. [156]. • F rob enius alge bras ar is e from tw o-sided duals in tensor categ ories: Let X ∈ C with t wo-sided dual X , and deﬁne Γ = X ⊗ X . Then Γ carries a F ro benius algebr a structure, cf. [191]: m = X X ✛ ✘ e X X X X X ∆ = X X X X ✚ ✙ d ′ X X X η = X X ✚ ✙ d X ε = ✛ ✘ e ′ X X X V erifying the F ro benius identities and strong separability is a trivial exercis e. In view of End( V ) ∼ = V ⊗ V ∗ in the category of ﬁnite dimensional v ector spaces, the a bove F rob enius algebra is called an ‘endomorphism (F r ob enius) alg ebra’. • This leads to the questio n whether every (stro ngly separa ble ) F r ob e nius algebr a in a ⊗ - category arise in this wa y . The answer is , not quite, but: If Γ is a strongly separ able F rob enius algebra in a k -linea r spherical tenso r category A then there e xist – a spherical k - linear 2-categ ory E with t wo ob jects { A , B } , 25 – a 1- morphism X ∈ Ho m E ( B , A ) with 2-s ided dual X ∈ Ho m E ( A , B ), and therefore a F ro benius algebra X ◦ X in the ⊗ -ca tegory End E ( A ), – a monoidal equiv alence End E ( A ) ≃ → A mapping the the F rob enius a lgebra X ◦ X to Γ. Thu s every F r ob e nius alge br a in A a r ises from a 1 -morphism in a bicateg ory E co ntaining A as a co r ner. In this situation, the tensor categor y B = End E ( B ) is calle d weakly mono ida lly Morita equiv alen t to A and the bica tegory E is called a Morita cont ext. • The or iginal pr o of in [191] was tedious. Assuming mild tec hnical conditions o n A and strong separability of Γ, the bicategor y E can simply b e obtained as fo llows: Hom E ( A , A ) = A , Hom E ( A , B ) = Γ − Mo d A , Hom E ( B , A ) = Mo d A − Γ , Hom E ( B , B ) = Γ − Mo d A − Γ , with the comp ositio n of 1-morphisms giv en by the usual tensor products of (left and righ t) Γ-mo dules. Cf. [28 5]. (A discussio n fr e e of any technical a ssumptions o n A was recently given in [16 3].) • W eak monoida l Morita equiv alence of tensor ca tegories also admits a n interpretation in terms of mo dule categ ories: If A , B are o b jects in a bicatego ry E as ab ov e, the category Hom E ( A , B ) is a left mo dule category o ver the tensor category End E ( B ) and a right module categ ory ov er A = E nd E ( A ). In fact, the whole structur e can be formulated in terms of mo dule categories , thereby getting rid of the F ro benius algebra s, cf. [85, 8 4]: W r iting M = Hom E ( A , B ), the dual ca tegory B = End E ( B ) can be obta ined as the tensor category HOM A ( M , M ), denoted A ∗ M in [85], of right A -mo dule functors from M to itself. Since the tw o pictures are essentially equiv alen t, the choice is a matter of taste. The picture with F rob enius algebras and the bicategory E is clo s er to subfactor theory . What sp eaks in fav or of the mo dule catego ry picture is the fact that non-is omorphic alg ebras in A can hav e equiv a len t mo dule categor ies, th us give rise to the same A -mo dule categor y . (But not in the case of comm utative algebras!) • Mo rita eq uiv alence of tensor categor ie s indeed is an equiv alenc e rela tio n, denoted ≈ . (In particular, B contains a s trongly separable F ro b enius algebra b Γ suc h that b Γ − Mo d B − b Γ ≃ A .) • As men tioned ea rlier, the left r egular repr esentation of a ﬁnite dimensional Hopf algebra H gives rise to a F rob enius algebr a Γ in H − Mo d. Γ is stro ng ly separ able if and only if H is semisimple and c o semisimple. In this case, one ﬁnds for the ensuing Mor ita eq uiv alent category : B = Γ − Mod H − Mo d − Γ ≃ b H − Mo d . (This is a s ituation encountered earlier in subfactor theory .) Actually , in this ca se the Morita context E had been deﬁned indep endently by T a m bara [25 3]. The s ame works for weak Hopf algebr as, thus for a ny semisimple and co-semisimple weak Hopf alg e br a we have A − Mod ≈ b A − Mo d, provided the weak Hopf algebr a is F r ob enius, i.e. has a non-degene r ate integral. (It is unkno wn whether every weak Hopf algebra is F rob enius.) • The above concept of Morita eq uiv alence has impor tant applications : If C 1 , C 2 are Morita equiv a len t (spherica l) fusion categ ories then 1. dim C 1 = dim C 2 . 2. C 1 and C 2 give rise to the sa me triangula tion TQFT in 2+ 1 dimensions (as deﬁned by Ba rrett/W estbury [19] and S. Gelfand/Kazhdan [10 4], generalizing the T uraev/ Viro TQFT [265, 2 62] to no n-braided categ ories. Cf. also Ocneanu [2 18].) This ﬁts nicely with the known fact (Kup erb erg [159], Ba rrett/W estbury [18]) that, the spherical categories H − Mod and b H − Mo d (for a semisimple a nd co- semisimple Ho pf algebra H ) give rise to the same tria ngulation TQFT. 26 3. The braided cen ters Z 1 ( C 1 ) , Z 1 ( C 2 ) (to b e discussed in the next section) a re equiv alent as bra ided tensor catego r ies. This is quite immediate b y a result o f Schauen burg [240]. • W e emphas iz e that (just like V ect k ) a fusion catego ry ca n con tain many (strongly separable) F ro benius algebras, th us it can b e Mor ita equiv alent to man y other tensor ca tegories. In view of this, studying (F rob enius) algebras in fusion catego ries is an imp or ta n t and in teresting sub ject. (Even mor e so in the braided case.) • E xample: Commutativ e a lgebras in a representation categor y Rep G (for G ﬁnite) ar e the same as commutativ e alge bras ca rrying a G -a ction by a lgebra automorphisms. The condition dim Hom( 1 , Γ) = 1 means that the G -action is er go dic. Such a lgebras cor resp ond to closed subgroups H ⊂ G v ia Γ H = C ( G/H ). Cf. [1 55]. • Alg e bras in and mo dule categories over the category C k ( G, ω ) deﬁned in Section 1 w ere studied in [22 3]. • A group the o retical category is a fusion categor y that is weakly Mo r ita equiv alent (or ‘dual’) to a po in ted fusion category , i.e. one of the for m C k ( G, ω ) (with G ﬁnite and [ ω ] ∈ H 3 ( G, T )). (The origina l deﬁnition [222] was in terms of quadruples ( G, H , ω , ψ ) with H ⊂ G ﬁnite gr o ups, ω ∈ Z 3 ( G, C ∗ ) and ψ ∈ C 2 ( H, C ∗ ) suc h that dψ = ω | H , but the t wo notions ar e equiv a len t by Ostrik’s analysis of mo dule categor ies of C k ( G, ω ) [222].) F or mor e on group theoretical catego ries cf. [20 3, 101]. • The a bove considerations are closely rela ted to subfactor theory (a t ﬁnite Jones index): A factor is a von Neumann a lg ebra with center C 1 . F or an inclusion N ⊂ M o f facto rs, ther e is a notion of index [ M : N ] ∈ [1 , + ∞ ] (not necessa rily integer!!), cf. [126, 1 69]. One has [ M : N ] < ∞ if a nd only if the canonical N-M-bimo dule X has a dual 1-mor phism X in the bicategory of von Neumann algebra s , bimo dules and their intert winers. Motiv ated by Ocneanu’s bimo dule picture of subfactors [216, 2 17] o ne obser ves that the bica tegory with the ob jects { N , M } and bimo dules g enerated by X , X is a Morita con text. On the other hand, a single factor M gives rise to a certa in tenso r ∗ -catego ry C (consisting of M − M -bimo dules or the endomo rphisms End M ) s uc h that, by Lo ngo’s work [1 7 0], the F rob enius a lgebras (“Q- systems” [170]) in C are (roughly) in bijection with the subfacto r s N ⊂ M with [ M : N ] < ∞ . (Cf. also the intro ductio n of [191].) 4 Braided tensor categories • The symmetric groups hav e the well known pr e s en tation S n = { σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i when | i − j | > 1 , σ 2 i = 1 } . Dropping the last rela tion, one obtains the Braid groups : B n = { σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i when | i − j | > 1 } . They w ere int ro duced by Artin in 1 928, but had app ear ed implicitly in muc h ear lier work by Hurwitz, cf. [141]. They hav e a natural g eometric interpretation: σ 1 = • • • • • • • • · · · ✂ ✂ ✂ ✂ ✂ ✂ ❇ ❇ ❇ ❇ ❇ ❇ , σ 2 = • • • • • • • • · · · ✂ ✂ ✂ ✂ ✂ ✂ ❇ ❇ ❇ ❇ ❇ ❇ , σ n − 1 = • • • • • · · · • • • ✂ ✂ ✂ ✂ ✂ ✂ ❇ ❇ ❇ ❇ ❇ ❇ Note: B n is inﬁnite for a ll n ≥ 2, B 2 ∼ = Z . The repres e ntation theory of B n , n ≥ 3 is diﬃcult. It is known that all B n are linear , i.e. they have faithful ﬁnite dimensional repres e n tations B n ֒ → GL ( m, C ) for suita ble m = m ( n ). Cf. Kassel/T uraev [1 41]. 27 • Analo gously , one can dro p the condition c Y ,X ◦ c X,Y = id on a symmetric tensor ca tegory . This leads to the concept of a braiding , due to Joyal and Street [128, 13 2], i.e. a family of natural isomorphisms c X,Y : X ⊗ Y → Y ⊗ X satisfying t wo hexa gon iden tities but not necessarily the condition c 2 = id. Notice that w itho ut the latter condition, one needs to require tw o hexag on identit ies, the second being obtained fr om the ﬁrst o ne by the replacement c X,Y ❀ c − 1 Y ,X (whic h do es nothing when c 2 = id). (The latter is the non-strict generaliza tion of c X ⊗ Y ,Z = c X,Z ⊗ id Y ◦ id X ⊗ c Y ,Z .) A braided tens or category (BTC) now is a tensor category equipp ed with a braiding. • In analog y to the s ymmetric ca se, given a BTC C and X ∈ N , n ∈ Z + , one ha s a homomor - phism Π X n : B n → Aut( X ⊗ n ). • The most obvious example o f a BTC that is no t symmetric is provided b y the braid ca tegory B . In ana logy to the s y mmetric categ ory S , it is deﬁned by Ob j B = Z + , End( n ) = B n , n ⊗ m = n + m , while on the morphisms ⊗ is deﬁned by juxtapo s ition of braid dia grams. The deﬁnition of the braiding c n,m ∈ End( n + m ) = B n + m is illustrated b y the exa mple ( n, m ) = (3 , 2): c n,m =                      ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ • If C is a strict BTC and X ∈ C , ther e is a unique braided tensor functor F : B → C suc h that F (1) = X and F ( c 2 , 2 ) = c X,X . Thus B is the free braided tensor category g enerated by one ob ject. • Ce ntralizer and center Z 2 : If C is a BTC, w e say that tw o o b jects X , Y commute if c Y ,X ◦ c X,Y = id X ⊗ Y . If D ⊂ C is sub category (o r just subset o f Ob j C ), we deﬁne the centr alizer C ∩ D ′ ⊂ C as the full sub c ategory deﬁned b y Ob j ( C ∩ D ′ ) = { X ∈ C | c Y ,X ◦ c X,Y = id X ⊗ Y ∀ Y ∈ D } . Now, the cen ter Z 2 ( C ) is Z 2 ( C ) = C ∩ C ′ . Notice that C ∩ D ′ is monoidal and Z 2 ( C ) is symmetric! In fact, a BTC C is symmetric if and only if C = Z 2 ( C ). Apar t from ‘central’, the ob jects of Z 2 ( C ) have bee n called ‘deg enerate’ [232] or ‘transparent’ [3 9]. • W e thus see that STC ar e maximally commut ative BTCs. Do es it ma k e sense to s p eak of maximally non- commu tative BTCs? B is an example since Ob j Z 2 ( B ) = { 0 } . Br aided fusion categorie s with ‘trivia l’ center will turn o ut to b e just T ur aev’s mo dular categor ies, cf. Section 5. • Since the deﬁnition o f BTCs is quite na tural if one knows the braid groups, one may wonder why they a ppea red more than 20 years after symmetric categor ies. Most likely , this was a consequence of a lack of really interesting exa mples. When they ﬁnally app eare d in [12 8], this was ma inly motiv a ted by dev elopments in ternal to category theory (and homotopy theory). It is a rema rk a ble historical accident tha t this happ ened at the same time as (and independently from) the de velopment of quantu m groups , which dramatically gained in p opularity in the wak e of Drinfeld’s talk [72]. 28 • In 1971 it was shown [68] that ce rtain representation theoretic consider ations for quantum ﬁeld theories in spa c etimes of dimension ≥ 2 + 1 lea d to symmetric categories. Adapting this theory to 1 + 1 dimensions inevitably le a ds to braided ca tegories, a s was ﬁnally shown in 19 8 9, cf. [90]. That this w as not done right after the app eara nc e of [68] m ust be consider ed as a missed opp ortunity . • As promised, we will br ieﬂy lo ok at br aided ca teg orical gro ups. Consider C ( G ) for G ab elian. As shown in [132] – and in muc h more detail in the preprints [12 8] – the braide d catego rical groups C with π 0 ( C ) ∼ = G (isomo r phism clas ses of ob jects) and π 1 ( C ) ∼ = A (End 1 ) ar e classi- ﬁed by the group H 3 ab ( G, A ), where H n ab ( G, A ) refers to the Eilenberg -Mac Lane coho mo logy theory fo r ab elian gro ups, cf. [177]. (Whereas H 3 ( G, A ) ca n b e deﬁned in terms of top o- logical cohomolo gy theory a s H 3 ( K ( G, 1 ) , A ) of the Eilen b erg-Mac Lane space K ( G, 1), one has H 3 ab ( G, A ) := H 4 ( K ( G, 2 ) , A ). This group also has a description in terms o f quadratic functions q : G → A . The subgroup of H 3 ab ( G, A ) corresp onding to symmetric braidings is isomorphic to H 5 ( K ( G, 3 ) , A ), cf. [46].) • Duality: Contrary to the symmetric c ase, in the presence o f a (non-symmetric) braiding, having a left duality is not suﬃcient for a nice theory: If w e deﬁne a rig h t duality in ter ms of a left duality and the bra iding, the left and r ig h t tr aces will fail to have a ll the prop erties they do hav e in the s y mmetric case. Ther efore, some additional concepts are needed: • A t wist for a braided ca tegory with left dualit y is a natura l family { Θ X ∈ End X , X ∈ C } of isomor phisms (i.e. a natural iso morphism of the functor id C ) satis fying Θ X ⊗ Y = Θ X ⊗ Θ Y ◦ c Y ,X ◦ c X,Y , Θ 1 = id 1 , ∨ (Θ X ) = Θ ∨ X . Notice: If c Y ,X ◦ c X,Y 6≡ id then the natur al is o morphism Θ is not monoidal and Θ = id is not a leg al twist! • A ribb on category is a s trict braided tensor category equipp ed with a left dualit y and a t wist. • Le t C b e a ribb on categ ory with left dualit y X 7→ ( ∨ X , e X , d X ) and twist Θ. W e deﬁne a right duality X 7→ ( X ∨ , e ′ X , d ′ X ) by X ∨ = ∨ X and (2.2). No w one can show, cf. e.g. [1 3 7], that the maps End X → E nd 1 deﬁned as in (3.1) coincide a nd that T r( s ) := T r L ( s ) = T r R ( s ) has the trace proper t y and behav es w ell under tensor products, as previo usly in the symmetric case. W r iting X = ∨ X = X ∨ , one ﬁnds that C is a spherical category in the sense of [20]. Conv ersely , if C is spherica l and braide d, then deﬁning Θ X = (T r X ⊗ id X )( c X,X ) , { Θ X , X ∈ C } satisﬁes the axioms of a t wist and thus for ms a r ibb on structure tog ether with the left dualit y . (Cf. Y etter [289], base d on idea s of Deligne, and Bar r ett/W estbur y [2 0].) (Personally , I pr efer to consider the t wist as a de r ived structure, thus talking a bout spherica l categorie s with a braiding, ra ther than a bo ut ribb on ca teg ories. In so me situa tions, e.g. when the center Z 1 ( C ) is inv olved, this is adv antageous. This also is the appr o ach of the Rome school [71, 172].) • So far , our only example of a non-symmetric braided category is the free braided ca tegory B , which is not rigid. In the remainder of this section, we will consider three main ‘r outes’ to braided categ ories: (A) the top ologic a l r oute, (B) the “non- per tur bative approa c h” via qua n- tum doubles and categorical centers, and (C) the “per turbative approa ch” via deformation (or ‘quantization’) of symmetric catego ries. • W e brieﬂy men tion one construction o f an interesting braide d categ ory tha t do esn’t seem to ﬁt nicely into one o f o ur routes: While the usua l repres en tation categor y of a gr o up is symmetric, the category of representations of the general linear group GL n ( F q ) ov er a ﬁnite ﬁeld with the external tensor pro duct of representations tur ns out to b e bra ided and non-symmetr ic, cf. [133]. 29 4.1 Route A : F ree braided categories (tangles) and their quotients • Co m bining the ideas b ehind the T emp erley-Lieb categ o ries TL ( τ ) (which hav e duals) and the braid categ ory B (which is braided but ha s no duals), one arr ives at the categ ories of tangles (T ura ev [260], Y e tter [2 87]. See also [26 2, 137].) One must distinguish betw een ca tegories of unorien ted tangles having Ob j U − T AN = Z + with tensor pro duct (of o b jects) g iven by addition a nd orien ted tangles , based on Ob j O − T AN = { + , −} ∗ (i.e. ﬁnite words in ± , 1 = ∅ ) with conca tenation a s tensor pro duct. In either case, the morphisms ar e given a s sets of pictures as in Figure 1 , or else b y linear comb inations of such pictures with coe ﬃcie n ts in a commutativ e ring or ﬁeld. All this is just as in the discussion o f the free sy mmetr ic categorie s at the end o f Sectio n 2 . The only diﬀere nc e is that o ne must distinguish b etw een ov er- and under crossings of the lines; for technical r e asons it is more conv enient to do this in terms of pictures embedded in 3- space. Figure 1: An u n orien ted 3-5 tangle There also is a category O − T AN of oriented tangles, where the o b jects are ﬁnite words in ± , 1 = ∅ and the lines in the morphis ms are directed, in a way that is co mpatible with the signs of the ob jects. It is clear tha t the morphisms in E nd( 1 ) in U − T AN ( O − T AN ) are just the unoriented (o r ient ed) links. While the deﬁnition is in tuitively na tural, the details are tedious and we refer to the tex tb o oks [262, 137, 2 90]. In particular, we omit discussing ribb on tangles. • The tangle ca teg ories are pivotal, in fact spherica l, thus ribb on categor ies. O − T AN is the free ribb on category genera ted by one element , cf. [243]. • Le t C b e a ribb on categor y . Then one can deﬁne a catego ry C − T AN of C -labe le d orie nted tangles and a ribb on tensor functor F C : C − T AN → C . (This is the r igorous ratio nale b ehind the diag rammatic calculus for braided tensor categor ies!) Let C b e a ribb on categ ory and X a self-dua l ob ject. Giv en an unoriented tangle, w e can lab el every edge by X . This gives a comp osite ma p { links } ∼ = − → Hom U −T AN (0 , 0) − → Hom C −T AN (0 , 0) F C − → End C 1 . In particular, if C is k - linear with End 1 = k id, we obtain a map fr om { links } to k , which is easily see n to b e a knot inv aria n t. If C = U q ( sl (2)) − Mod and X is the fundamental ob ject, one ess en tially o btains the Jones po lynomial. Cf. [2 6 0, 234]. (The o ther ob jects of C g iv e rise to the color ed Jones p olynomia ls, which are m uc h studied in the con text of the volume conjecture for h yp erb olic knots.) • So far, a ll o ur exa mples of bra ided categ ories have co me from top olog y . In a s ense, they are quite trivial, since they ar e just the univ ersal braided (ribb on) categ o ries fre e ly generated by one ob ject. F urthermo re, w e are primarily in terested in linear categor ies. Of course, we can apply the k -linearizatio n functor C AT → k -lin . - C AT . But the categor ies we obtain hav e inﬁnite dimensio nal hom-s ets and are no t more in teresting than the original ones. (This should be contrasted to the symmetric case, where this construc tio n pro duces the re pr esentation categorie s of the clas sical groups, cf. Section 2.) 30 • Thus in order to obtain interesting k -linear r ibbo n ca tegories from the tangle categ ories, w e m ust reduce the inﬁnite dimensional hom-spaces to ﬁnite dimensiona l ones. W e co nsider the following analogo us situation in the context of asso cia tiv e a lgebras: The braid group B n ( n > 1) is inﬁnite, thus the g roup algebra C B n is inﬁnite dimensional. But this algebra has ﬁnite dimensiona l quotients, e.g . the Heck e algebra H n ( q ), the unital C -algebra generated by σ 1 , . . . , σ n − 1 , mo dulo the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i when | i − j | > 1 , σ 2 i = ( q − 1) σ i + q 1 . This algebra is ﬁnite dimensional for any q , and for q = 1 w e hav e H n ( q ) ∼ = C S n . In fact, H n ( q ) is isomorphic to C S n , th us s e misimple, whenev er q is not a r o o t of unit y , but this isomorphism is highly non-tr ivial. Cf. e.g . [164]. The idea no w is to do a similar thing on the level of ca tegories, or to ‘categorify’ the Hec ke algebras or o ther quotien ts of C B n like the Birman/Mura k ami/W enzl- (BMW-)-algebr as [2 9]. • W e hav e s een that ribb on categories give rise to knot inv aria n ts. One can go the other wa y a nd construct k -linea r ribbo n categor ies from link in v a riants. This appr oach was initiated in [262, Chapter XI I], where a to po lo gical constr uction of the repres e ntation category of U q ( sl (2)) was given. A more gener a l appr oach was studied in [267]. A k -v alued link inv a riant G is said to admit functorial extension to tangles if ther e exists a tensor functor F : U − T AN → k − Mo d who se restrictio n to End U −T AN (0) ∼ = { links } eq uals G . F or any X ∈ U − T AN , f ∈ E nd( X ), let L f be the link obtained by closing f on the rig ht, and deﬁne T r G ( f ) = G ( L f ). If C is the k -linea rization of U − T AN , it is shown in [267], under weak assumptions on G , that the idemp otent and direct sum completion of the quotient of C by the idea l of negligible morphisms is a semis imple r ibbon ca tegory with ﬁnite dimens ional hom-sets. Cf. [267]. Example: Applying the ab ov e pro cedure G = V t , the Jones p olyno mial, one obtains a T emp erley- Lieb ca tegory T L τ , whic h in turn is equiv alent to a category U q ( sl (2)) − Mo d. Cf. [262, Chapter XI I]. Applying it to the Kauﬀman po lynomial [14 2], one o btains the quan- tized BTCs of t yp es BCD, c f. [267]. The genera l theo ry in [2 67] is quite nice, but it s ho uld b e noted tha t the assumption of functorial extendability to tangles is rather strong : It implies that the r e sulting se misimple category a dmits a ﬁber functor and therefore is the represen- tation catego ry of a discrete quantum gr oup. F urther more, the application of the ge ne r al formalism of [267] to the Kauﬀman p olyno mia l used input fro m (q-deformed) quantum group theory for the pr o of of functorial extension to tang les and of mo dularity . This dr awbac k was repaired by Beliako v a /Blanchet, cf. [22, 23]. Blanchet [31] gave a similar co nstruction with HOMFL Y p olynomial [92], obtaining the type A catego ries. (The HOMFL Y po lynomial is an in v a riant for oriented links, th us one must work with oriented tangles.) Remark: The ribb on catego r ies o f BCD type arising from the Ka uﬀman p olynomia l give rise to topologica l quantum ﬁeld theories. The latter can even b e constructed direc tly from the Kauﬀman br ack et, b ypassing the ca tegories, cf. [32]. This construction a c tua lly preceded those mentioned ab ov e. • The prece ding constructio ns reinforce the close connec tio n b etw een br aided categ ories and knot inv a riants. It is important to realize that this reasoning is not circular , since the p oly- nomials of Jones, HOMFL Y, Kauﬀman can (now adays) be co nstructed in rather element ary wa ys, independently of categorie s and qua n tum gr oups, cf. e.g. [167]. Since the knot po lyno- mials are deﬁned in terms of skein r elations, we sp eak of the skein construction of the quantum categorie s, which arg uably is the simples t known so far. • In the case q = 1, the skein constructio ns of the ABCD categor ies reduce to the construction of the c a tegories arising from classical g roups men tioned in Section 2. (This happe ns since q = 1 corres p onds to parameter s in the knot p olynomials for whic h they fail to distinguish ov er- 31 from under-cr ossings. Then one can r eplace the tangle catego ries by sy mmetric categor ies of non-embedded cob ordis ms (oriented or not) as in [59].) • Co ncerning the exceptional Lie algebras and their quantum catego ries, inspired by w ork of Cvitanovic, cf. [5 2] for a bo o k -length treatment, and b y V og el [273], Deligne conjectured [59] that there is a one parameter family of symmetric tensor categorie s C t sp ecializing to Rep G for the exceptional Lie gr oups at c ertain v alues of t . This is still unprov en, but see [48, 63, 62] for work resulting from this conjecture. (F or the E n -categor ies, including the q -deformed ones, cf. [277].) • In a similar vein, Deligne deﬁned [61] a o ne parameter fa mily of rig id symmetric tensor categorie s C t such that C t ≃ Rep S t for t ∈ N . These c a tegories were s tudied further in [49]. (Recall that S n is considered a s the GL n ( F 1 ) where F 1 is the ‘ﬁeld with one elemen t’, cf. [248].) • Mo re generally , one can deﬁne linear categ ories b y generators and r e lations, cf. e.g. [160]. 4.2 Route B : Doubles and cen ters W e b egin with a brief lo o k at Ho pf algebra s. • Q uasi-triangular Hopf algebra s (Drinfeld, 19 86 [72]): If H is a Hopf algebra a nd R an inv ertible element of (po ssibly a completion of ) H ⊗ H , sa tisfying R ∆( · ) R − 1 = σ ◦ ∆( · ) , σ ( x ⊗ y ) = y ⊗ x, (∆ ⊗ id)( R ) = R 13 R 23 , (id ⊗ ∆)( R ) = R 13 R 12 . ( ε ⊗ id)( R ) = (id ⊗ ε )( R ) = 1 . If ( V , π ) , ( V ′ , π ′ ) ∈ H − Mod, the deﬁnitio n c ( V ,π ) , ( V ′ ,π ′ ) = Σ V ,V ′ ( π ⊗ π ′ )( R ) pro duces a braiding for H − Mo d. • B ut this has only shifted the problem: How to g et quas i-triangular Ho pf a lgebras? T o this purp ose, Drinfeld [72] g av e the q uant um double co nstruction H ❀ D ( H ), which a sso ciates a quasi-tria ng ular Ho pf algebra D ( H ) to a Hopf alg ebra H . Cf. also [13 7]. • So o n after, a n analogo us ca tegorical construction was given by Drinfeld (unpublished), Joyal/Street [130] and Ma jid [1 81]): The (braided) center Z 1 ( C ), deﬁned as follows. Let C be a strict tenso r categor y and let X ∈ C . A half braiding e X for X is a family { e X ( Y ) ∈ Hom C ( X ⊗ Y , Y ⊗ X ) , Y ∈ C } of isomo rphisms, natural w.r.t. Y , satisfying e X ( 1 ) = id X and e X ( Y ⊗ Z ) = id Y ⊗ e X ( Z ) ◦ e X ( Y ) ⊗ id Z ∀ Y , Z ∈ C . Now, the cen ter Z 1 ( C ) of C ha s as ob jects pair s ( X , e X ), where X ∈ C a nd e X is a half braiding for X . The morphisms ar e given b y Hom Z 1 ( C ) (( X, e X ) , ( Y , e Y )) = { t ∈ Hom C ( X, Y ) | id X ⊗ t ◦ e X ( Z ) = e Y ( Z ) ◦ t ⊗ id X ∀ Z ∈ C } . The tensor pro duct of o b jects is given by ( X, e X ) ⊗ ( Y , e Y ) = ( X ⊗ Y , e X ⊗ Y ), where e X ⊗ Y ( Z ) = e X ( Z ) ⊗ id Y ◦ id X ⊗ e Y ( Z ) . The tenso r unit is ( 1 , e 1 ) where e 1 ( X ) = id X . The co mpos itio n and tensor pro duct of morphisms ar e inherited fro m C . Finally , the braiding is g iven by c (( X, e X ) , ( Y , e Y )) = e X ( Y ) . (The author ﬁnds this deﬁnition is muc h more tra nsparent than that of D ( H ) even though a priori little is known ab out Z 1 ( C ).) 32 • J ust a s the cent ralizer C ∩ D ′ generalizes Z 2 ( C ) = C ∩ C ′ , there is a version of Z 1 relative to a sub c ategory D ⊂ C , cf. [181]. • Z 1 ( C ) is catego r ical version (genera lization) o f Hopf algebra quantum double in the following sense: If H is a ﬁnite dimensio nal Hopf a lgebra, there is an eq uiv alence Z 1 ( H − Mo d) ≃ D ( H ) − Mod (4.1) of braided tensor categor ies, cf. e.g. [13 7]. (If H is inﬁnite dimensional, one still has an equiv a lence b et ween Z 1 ( H − Mo d) and the categor y of Y etter- Dr infeld mo dules ov er H .) • If C is a category a nd D := Z 0 ( C ) = End( C ) is its tensor catego ry of endofunctors, then Z 1 ( D ) is trivial. (This ma y be considered a s the categor iﬁcation of the fact that the cen ter (in the usual sense) o f the endomor phism mono id End( S ) of a set S is trivial, i.e. equal to { id S } .) But in general, the braided cent er of a tensor c ategory is a non-trivial braided categ ory that is not symmetric. Unfortunately , this do esn’t seem to hav e b een studied thoro ug hly . Pres en tly , strong results on Z 1 ( C ) exist o nly in the case where C is a fusion catego r y . • Ther e are abstra ct categor ical consider ations, quite unrelated to top ology and q ua n tum groups, that provide ra tionales for studying BTCs: (A): A second, compatible, multiplication functor o n a tensor category gives rise to a braid- ing, and con versely , cf. [132]. (This is a hig her dimensional version of the Eckmann-Hilton argument mentioned ear lier.) (B): Recall that tenso r categorie s are bicategories with one ob ject. Now, braided tensor categorie s turn out to b e monoidal bicategor ies with one ob ject, which in turn ar e weak 3- categorie s with one ob ject and o ne 1-mor phism. Th us braided (and symmetric) categor ies really are a manifestation o f the existence of n -categor ies for n > 1! • B a ez-Dolan [10] c o njectured the following ‘p erio dic table’ of ‘k- tuply monoidal n-categorie s’: n = 0 n = 1 n = 2 n = 3 n = 4 k = 0 sets categorie s 2-catego ries 3-ca tegories . . . k = 1 monoids monoidal monoidal monoidal . . . categorie s 2-catego ries 3-ca tegories k = 2 commutativ e braided braided braided . . . monoids mo noidal monoidal monoidal categorie s 2-catego ries 3-ca tegories k = 3 symmetric ‘sylleptic’ ” monoidal monoidal ? . . . categorie s 2-catego ries k = 4 symmetric ” ” monoidal ? . . . 2-catego ries k = 5 symmetric ” ” ” monoidal . . . 3-catego ries k = 6 ” ” ” ” . . . In particular, one exp ects to ﬁnd ‘center constructions’ from eac h structure in the table to the one underneath it. F or the co lumn n = 1 these are the centers Z 0 , Z 1 , Z 2 discussed ab ov e. F or n = 0 they are given b y the endomo rphism monoid o f a set and the or dinary ce n ter of a monoid. The column n = 2 is also rela tiv ely well under s too d, cf. Crans [50]. Ther e is an accepted notion of a non- strict 3 -categor y (i.e. n = 3 , k = 0) (Gordon/Po wer/Street [1 08]), but there are many co mp eting deﬁnitions o f weak higher categor ies. W e refra in fr o m moving any fur ther into this sub ject. See e .g . [13]. 33 • With this heuristic preparation, one can give a high-br ow interpretation of Z 1 ( C ), cf. [132, 250]: Let C b e tensor categor y and Σ C the co rresp onding bicateg ory with one ob ject. Then the cat- egory End(Σ C ) of endofunctors of Σ C is a mo noidal bicatego ry (with natural trans formations as 1-morphisms and ‘mo diﬁcations’ as 2-morphisms ). Now, D = End End(Σ C ) ( 1 ) is a tensor category with tw o compatible ⊗ -structures (categorifying End 1 in a tensor categ ory), th us braided, and it is equiv a len t to Z 1 ( C ). • F or further abstract considerations o n the cen ter Z 1 , consider the w ork of Street [25 0, 25 1] and of Brugui` eres and Virelizier [41, 42]. • If C is braided there is a br aided e m b edding ι 1 : C ֒ → Z 1 ( C ), given by X 7→ ( X , e X ), where e X ( Y ) = c ( X, Y ). Deﬁning e C to b e the tensor c a tegory C with ‘oppos ite’ braiding e c X,Y = c − 1 Y ,X , there is an a nalogous embedding e ι : C ֒ → Z 1 ( C ). In fact, one ﬁnds that the ima ges of ι, ι ′ are each o thers’ centralizers: Z 1 ( C ) ∩ ι ( C ) ′ = e ι ( e C ) , Z 1 ( C ) ∩ e ι ( e C ) ′ = ι ( C ) . Cf. [192]. On the one hand, this is an instance of the do uble comm utan t principle, and on the other hand, this establishes one connection ι ( C ) ∩ e ι ( e C ) = ι ( Z 2 ( C )) = e ι ( Z 2 ( e C )) , betw een Z 1 and Z 2 which sugge sts that “ Z 1 ( C ) ≃ C × e C ” when Z 2 ( C ) is “trivial”. W e w ill return to b oth p oint s in the next sectio n. 4.3 Route C: Deformation of groups or symmetric categories • As for Route B, there is a mor e tra ditional approach via deformation of Hopf algebras and a somewhat mor e recent one fo cusing direc tly on defor mation of tenso r categor ies. • (C 1 ): The ea rlier approach to braided ca tegories relies on deformation of Hopf algebras r e- lated to groups. F or la c k of space we will limit o urselves to providing just eno ugh info r mation as needed for the discussion of the more catego rical approach. F or more, we refer to the text- bo o ks, in par ticular [137, 47, 124, 1 73]. In a n y case, one cho oses a simple (usually compact) Lie group G and considers either the env eloping algebra U ( g ) of its Lie algebra g in terms of Ser re’s g enerators and relations [24 2], or one departs fro m the algebra F un( G ) of reg ular functions on G , which can also b e describ ed in ter ms of ﬁnitely many relations, cf. e.g. [279]. In a n utshell, one inserts fa c tors o f a ‘defor mation pa r ameter’ q into the pre sent ation of U ( g ) or F un( G ) in s uc h a wa y that for q 6 = 1 o ne s till obtains a (non-trivial) Hopf algebr a. Quantum group theory beg an with the discov ery that this is p oss ible at a ll. • O b viously , this ‘deﬁnition’ is a farcical caricature . But there is so me tr uth in it: In the mathematical literature on quantu m groups, cf. e.g. [137, 47, 124, 173], it is all but imp oss ible to ﬁnd a comment on the origin of the presentation o f the quantum group under s tudy and of the underlying motiv atio n. While the initiators of quantum gro up theory from the Leningrad school (F a ddeev, Kulish, Semenov-Tian-Shansky , Sklyanin, Reshetikhin, Drinfeld and others) were v ery well a ware of these orig ins, this knowledge has now almost faded into obscurity . (This certainly has to do with the fact that the applications to theore tica l physics for which quantum g roups were in v ented in the ﬁrst pla c e are still exc lus iv ely pursued by physicists, cf. e.g . [1 06].) One p oint of this s ection will b e that – quite indep endently of the original ph ysical motiv a tion – the categoric a l approach to quantum deforma tion is mathematically better motiv ated. • In what follows, w e will concentrate on the env eloping algebr a a pproach. The usual Drinfeld- Jimbo pre sent ation of the quantized env eloping alg ebra is as fo llows, Consider the algebra U q ( g ) gener ated by elemen ts E i , F i , K i , K − 1 i , 1 ≤ i ≤ r , satisfying the relations K i K − 1 i = K − 1 i K i = 1 , K i K j = K j K i , K i E j K − 1 i = q a ij i E j , K i F j K − 1 i = q − a ij i F j , 34 E i F j − F j E i = δ ij K i − K − 1 i q i − q − 1 i , 1 − a ij X k =0 ( − 1) k  1 − a ij k  q i E k i E j E 1 − a ij − k i = 0 , 1 − a ij X k =0 ( − 1) k  1 − a ij k  q i F k i F j F 1 − a ij − k i = 0 , where  m k  q i = [ m ] q i ! [ k ] q i ![ m − k ] q i ! , [ m ] q i ! = [ m ] q i [ m − 1] q i . . . [1] q i , [ n ] q i = q n i − q − n i q i − q − 1 i and q i = q d i . This is a Hopf a lgebra with copro duct ∆ a nd counit ε deﬁned by ∆( K i ) = K i ⊗ K i , ∆( E i ) = E i ⊗ 1 + K i ⊗ E i , ∆( F i ) = F i ⊗ K − 1 i + 1 ⊗ F i , ε ( E i ) = ε ( F i ) = 0 , ε ( K i ) = 1 . One should distinguish b etw een Drinfeld’s [72] forma l approach, where one constructs a Hopf algebra H ov er the ring C [[ h ]] of forma l p ow er series in such a w ay that H /hH is isomorphic to the e n veloping a lgebra U ( g ), and the non-formal deformation of Jimbo [125], who obtains an honest quasi-tria ngular Hopf algebra U q ( g ) (ov er C ) for any v alue q ∈ C of a deformatio n parameter. (In this a pproach, the prop erties of the resulting Hopf alge br a dep end heavily o n whether q is a ro o t of unity or not. In the formal a pproach, this distinction obviously do es not arise.) The relation b etw een b oth approaches b ecomes clear b y inser ting q = e h in Jim bo ’s deﬁnition and considering the result a s a Hopf algebra over C [[ h ]]. • (C 2 ): As mentioned, one can obtain non-sy mmetric braided categor ies directly by ‘deforming’ symmetric categories . This a pproach was initiated by Cartier [45] a nd w orked o ut in mo re detail in [137, App endix] and [1 40]. (These works were all motiv ated by a pplica tions to V as s iliev link inv aria nts, which we cannot discuss here.) Let S b e a s trict symmetric Ab-ca tegory . Now an inﬁni tesimal braiding on S is a natural family of endomorphisms t X,Y : X ⊗ Y → X ⊗ Y satisfying c X,Y ◦ t X,Y = t Y ,X ◦ c X,Y ∀ X , Y , t X,Y ⊗ Z = t X,Y ⊗ id Z + c − 1 X,Y ⊗ id Z ◦ id Y ⊗ t X,Z ◦ c X,Y ⊗ id Z ∀ X , Y , Z. Strict symmetric Ab-catego r ies equipp ed with an inﬁnitesimal braiding were ca lle d i n ﬁni tes- imal s ymmetric . (W e would prefer to ca ll them sy mmetr ic categorie s equipped with an inﬁnitesimal br aiding.) • E xample: If H is a Hopf alg ebra, there is a bijection b etw een inﬁnitesimal bra idings t on S = H − Mod and elemen ts t ∈ Prim( H ) ⊗ P r im( H ) (where Pr im( H ) = { x ∈ H | ∆( x ) = x ⊗ 1 + 1 ⊗ x } ) s a tisfying t 21 = t and [ t , ∆( H )] = 0, given by t X,Y = ( π X ⊗ π Y )( t ). • Now we can deﬁne the forma l deformation of a symmetr ic catego ry asso ciated to an in- ﬁnitesimal bra iding: Let S b e a strict C -linear symmetric categ o ry with ﬁnite dimens io nal hom-sets a nd let t be an inﬁnitesimal braiding for S . W e write S [[ h ]] for the C [[ h ]]-linea r category obtained by extension of scalar s. (I.e. Ob j S [[ h ]] = Ob j S and Hom S [[ h ]] ( X, Y ) = Hom S ( X, Y ) ⊗ C C [[ h ]].) Also the functor ⊗ : S × S → S lifts to S [[ h ]]. F or ob jects X, Y , Z , deﬁne α X,Y ,Z = Θ K Z ( h t X,Y ⊗ id Z , h id X ⊗ t Y ,Z ) , e c X,Y = c X,Y ◦ e ht X,Y / 2 . Here Θ K Z is a Drinfeld asso ciato r [73], i.e. a formal p ow er ser ies Θ K Z ( A, B ) = X w ∈ { A,B } ∗ c w w in tw o non-commuting v ariables A, B , wher e c w ∈ C , satisfying cer tain identities. (Cf. [137, Chapter XIX, (8.27 )-(8.29)].) Then ( S [[ h ]] , ⊗ , 1 , α ) is a (non-strict) tens o r category with asso ciativity co nstraint α , trivial unit constraints a nd e c a braiding. If S is rig id, then ( S [[ h ]] , ⊗ , 1 , α, e c ) admits a ribbo n structure. 35 • Applica tio n: Let g be a simple Lie alg ebra / C . Let S = g − Mo d and deﬁne { t X,Y } b e as in the example, cor resp onding to t = ( P i x i ⊗ x i + x i ⊗ x i ) / 2, where x i , x i are dual bases of g w.r.t. the Killing form. Then [ t, ∆( · )] = 0 and o ne can prove ( S [[ h ]] , ⊗ , 1 , α, e c ) ≃ U h ( g ) − Mo d (4.2) as C [[ h ]]-linear ribb on categor ies. (The pro of is a coro llary of the pro of of the Ko hno-Drinfeld theorem [73, 74], cf. a lso [137].) Remark: 1. Obviously , we hav e c heated: The main diﬃcult y resides in the deﬁnition of Θ K Z ! Giving the latter and proving its prop erties requir es ca. 10 -15 pages of rather technical material (but no Lie theory). Le and Murak a mi explic itly wr ote do wn an asso ciator; cf. e.g. [13 7, Remark XIX.8.3 ]. Drinfeld a lso gave a non-constructive pro of of existence of a n asso ciator deﬁned ov er Q , cf. [7 4]. 2. The ab ov e is r e lev ant for a more co nceptual appro ach to the theory o f ﬁnite-type knot inv a riants (V a ssiliev inv aria n ts), cf. [4 5, 140]. 3. A disadv antage of the a bove is that we obtain only a formal defo r mation of S . If g is a simple Lie a lgebra and S = g − Mo d, w e know by (4.2 ), that we obtain the C [[ h ]]-ca tegory U h ( g ) − Mo d. On the other hand, tha nk s to the w ork of Jimbo [125] and o ther s [173, 12 4] w e know that ther e is a non-formal version U q ( g ) o f the quantum group with C -linear re pr esentation category . One would ther efore hop e tha t the C -linear categories U q ( g ) − Mo d ca n b e o bta ined directly as defor mations of the module categor ie s U ( g ) − Mod. Indeed, for numerical q ∈ C \ Q , with s ome more analytical eﬀor t one can ma k e sense of α q = Θ K Z ( h t X,Y ⊗ id Z , h id X ⊗ t Y ,Z ) as a n ele ment o f End( X ⊗ Y ⊗ Z ) and deﬁne a non-formal, C -linear catego ry C ( g , q ) and prov e an equiv alence C ( g , q ) = ( S , ⊗ , 1 , α q , e c q ) ≃ U q ( g ) − Mo d of C -line a r ribb on categories . This was done by Kazhda n and Lus z tig [14 4], but see also the nice recent expo sition by Neshv eyev/T uset [209]. • F act: If q ∈ C ∗ is g e ne r ic, i.e. not a ro ot of unity , then C ( g , q ) := U q ( g ) − Mo d is a s emisimple braided ribb on ca tegory whose fusio n hypergr oup is isomor phic to that of U ( g ), thus of the category of g -modules, cf. [12 4, 173]. But it is not symmetric for q 6 = 1, thus certainly not equiv a len t to the latter. In fact, U q ( g ) − Mo d and U ( g ) − Mo d ar e already inequiv alent as ⊗ -categor ies. (Reca ll that asso ciativ ity constraints α can b e considered as gener alized 3-co cycles, and the α q for diﬀerent q ar e not coho mo logous.) • W e ha ve brieﬂy discussed the Car tier/Kass el/T ur aev formal deforma tio n qua n tization of sy m- metric categor ies equipp ed with an inﬁnitesimal braiding . There is a coho mology theo r y for Ab- tens o r categ ories and tensor functor s that class iﬁes deformatio ns due to Davydov [53] and Y etter [2 9 0]. Deﬁnition: Let F : C → C ′ a tenso r functor. Deﬁne T n : C n → C by X 1 × · · · × X n 7→ X 1 ⊗ · · · ⊗ X n . ( T 0 ( ∅ ) = 1 , T 1 = id.) Let C n F ( C ) = End( T n ◦ F ⊗ n ). ( C 0 F ( C ) = End 1 ′ .) F or a fusion catego ry , this is ﬁnite dimensional. Deﬁne d : C n F ( C ) → C n +1 F ( C ) by d f = id ⊗ f 2 ,...,n +1 − f 12 , ··· ,n +1 + f 1 , 23 ,...,n +1 − · · · + ( − 1 ) n f 1 ,...,n ( n +1) + ( − 1) n +1 f 1 ,...,n ⊗ id , where, e.g., f 12 , 3 ,...,n +1 is deﬁned in terms of f using the isomorphism d F X 1 ,X 2 : F ( X 1 ) ⊗ F ( X 2 ) → F ( X 1 ⊗ F 2 ) coming with the tensor functor F . One has d 2 = 0, th us ( C i , d ) is a complex. Now H i F ( C ) is the cohomology o f this complex, and H i ( C ) = H i F ( C ) for F = id C . In low dimensio ns one ﬁnds that H 1 F classiﬁes deriv ations of the tensor functor F , H 2 F clas- siﬁes deformations of the tensor structure { d F X,Y } of F . H 3 ( C ) c la ssiﬁes deformations of the asso ciativity co nstraint α of C . Examples: 1 . If C is fusio n then H i ( C ) = 0 ∀ i > 0. This implies Ocnea n u rig idity , c f. [84]. 36 2. If g is a r eductive algebr aic gro up with Lie algebra g and C = Rep G (alg ebraic represen- tations). Then H i ( C ) ∼ = (Λ i g ) G ∀ i . If g is simple then H 1 ( C ) = H 2 ( C ) = 0, but H 3 ( C ) is one-dimensional, corr esp onding to a one-parameter family o f deformations C . Accor ding to [84] “it is ea sy to guess that this deformation comes from an a ctual deformation, namely the deformation of O ( G ) to the quan tum group O q ( G )”. It is not clea r to this author whether this sugg estion should be considered as prov en. If so, together with the o ne-dimensionality of H 3 ( g − Mo d) it provides a v ery satisfactor y ‘explana tion’ for the e x istence of the quan tized categorie s C ( g , q ) ≃ U q ( g ) − Mo d. • In analogy to the result of Kazhda n a nd W enzl men tioned in Section 3, T uba and W enzl [259] prov ed that a semisimple ribb on ca teg ory with the fusio n hyper g roup isomor phic to that of a simple cla ssical Lie alg e br a g of BCD type (i.e. orthogo nal or symplectic) is eq uiv alent to the categ ory C ( g , q ), with q = 1 or not a ro o t of unit y , or one o f ﬁnitely many twisted versions ther eof. Notice that in contrast to the Ka zhdan/W e nzl result [145], this result needs the category to b e braided! (Again, this is a characteriza tio n, not a construction of the categorie s.) • Finkelbe r g [89] prov ed a braided equiv alence betw een C ( g , q ), q = e iπ/mκ , wher e m = 1 for ADE, m = 2 for BCD and m = 3 for G 2 , and the r ibbo n ca tegory ˜ O κ of in tegrable representations of the aﬃne Lie a lg ebra ˆ g of cen tral charge c = κ − ˇ h , where ˇ h is the dual Coxeter num b er of g . The categ o ry ˜ O κ plays an impo rtant rˆ ole in confor mal ﬁeld theory , either in terms of ver- tex op era tor algebr as or via the representation theory of lo op groups (W asser mann [275], T o le dano-Laredo [25 6]). This is the main reason fo r the relev a nce of quantum gro ups to CFT. • Fina lly , we brieﬂy discuss the connection b etw een routes (B) and (C) to B TCs: In or der to ﬁnd an R-matrix for the Hopf algebra U q ( g ) one traditionally uses the quan tum do uble, app ealing to an isomo rphism U q ( g ) ∼ = D ( B q ( g )) /I , where B g ( g ) is the q- deformation o f a Borel suba lgebra of g and I an ideal in D ( B q ( g )). Now R U q ( g ) = ( φ ⊗ φ )( R D ( B q ( g )) ), where φ is the quotient map. Since a surjective Hopf a lgebra homomo rphism H 1 → H 2 corres p onds to a full monoidal inc lus ion H 2 − Mo d ֒ → H 1 − Mo d , a nd r ecalling the co nnection (4.1) betw een Drinfeld’s do uble constructio n a nd the braided center Z 1 , we co nclude that the BTC U q ( g ) − Mo d is a full monoidal subca tegory of Z 1 ( B q ( g ) − Mo d) (with the inherited braiding ). Therefore, als o in the deformation approach, the braiding can be understo o d as ultimately arising from the Z 1 center c o nstruction. • Q uestion: It is natur al to ask whether a similar observ ation also holds for q a r oo t of unity , i.e., whether the modular categor ies C ( g , q ), for q a roo t of unity , can be understoo d as full ⊗ -sub categor ies of Z 1 ( D ), where D is a fusion categor y cor resp onding to the deformed Borel subalgebra B q ( g ). V ery recently , Etingof and Gelak i [81] gav e an aﬃr mative ans w er in so me cases. Remark: In the next section, w e will discus s a criterion tha t a llows to recognize the qua n tum doubles Z 1 ( C ) of fusion categories . 5 Mo du lar categories • T ura ev [261, 26 2]: A mo dular categor y is a fusion catego ry that is ribb on (alternatively , spherical and braided) such that the matrix S = ( S i,j ) S i,j = T r X ⊗ Y ( c Y ,X ◦ c X,Y ) , i, j ∈ I ( C ) , where I ( C ) is the set of simple ob jects mo dulo isomor phism, is inv ertible. • A fusion categor y that is ribb on is mo dular if and only if dim C 6 = 0 and the c en ter Z 2 ( C ) is trivial. (In the sense of consisting only of the o b jects 1 ⊕ · · · ⊕ 1 .) (This was proven by Rehren [232] for ∗ -ca teg ories and by Beliako v a/ Blanchet [2 3] in general. Cf. also [39] and [2].) 37 Thu s: Mo dula r categories a re br aided fusion categories with tr iv ial center, i.e. the ma ximally non-symmetric ones . (This deﬁnition seems mor e conceptual than the original one in terms of inv ertibilit y of S .) • Why are these categories called ‘mo dular’ ? Let S as abov e and T = diag( ω i ), where Θ X i = ω i id X i , i ∈ I . Then S 2 = α C, ( S T ) 3 = β C, ( αβ 6 = 0 ) where C i,j = δ i,  , th us S, T give rise to a pro jective repr esentation of the mo dular gr oup S L (2 , Z ) (which has a presentation { s, t | ( st ) 3 = s 2 = c, c 2 = e } ). Cf. [2 32, 26 2]. • At ﬁrst sight, this is somewhat mysterious. Notice: S L (2 , Z ) is the mapping class group of the 2-torus S 1 × S 1 . Now, b y work of Reshetikhin/T ura ev [2 35, 262], providing a rigorous version of ideas of Witten, every mo dular categor y gives rise to a top ol ogical quan tum ﬁeld theory in 2 + 1 dimensions. Every such TQFT in turn gives rise to pro jective repr esentation of the mapping class g roups o f a ll clo sed surfa c es, and for the to rus one obtains just the ab ov e representation of S L (2 , Z ). Cf. [262, 15]. W e don’t hav e the time to say more ab out TQFTs. • T ura ev’s motiv ation came fro m conformal ﬁeld theory (CFT). (Cf. e.g. Moo re-Seib erg [18 9]). In fact, there is a (rigorous) deﬁnition of rational c hiral CFT s (using von Neumann alge- bras) and their repre s en tations, for which o ne can prov e that the latter are unitary mo dular (Kaw ahigashi, Lo ng o, M¨ uger [143]). Mos t of the examples considered in the (heuristic) ph ysics literature ﬁt in to this scheme. (E.g . the lo op gro up mo dels: [27 5, 282] and the minimal Vira- soro mo dels with c < 1 [168].) In the co n text o f vertex oper ator algebr as, simila r results w ere pr ov en by Huang [12 1]. • It is natur al to ask whether there are less complica ted ways to pro duce mo dular ca tegories? The answer is p ositive; we will reconsider o ur three r o utes to braided catego ries. • Ro ute A: Reca ll tha t the clas sical ca tegories can be o btained from the linearized tangle ca t- egories (type A: or ien ted tangles , t yp es BCD: unoriented tangles ), dividing by ideals deﬁned in terms of the knot p olynomials of HOMFL Y and Kauﬀman. At r o o ts of unity , this leads to mo dular catego ries, cf. [267, 31, 23]. • Ro ute C 1 : H. Andersen et a l. [4], T uraev/ W enzl [266] (and others): L e t g b e a simple Lie algebra and q a pr imitiv e ro ot of unit y . Then U q ( g ) − Mo d gives rise to a mo dular categor y C ( g , q ). (Using tilting mo dules, dividing b y negligible morphisms, etc.) • Le t q b e pr imitive ro ot of unity of o r der ℓ . Then C ( g , q ) has a p ositive ∗ -op eration (i.e. is unitary) if ℓ is even (Kirillov Jr. [152], W enzl [276]) a nd is not unitariz a ble for o dd ℓ (Row ell [236]). • Cha racterizatio n theorem: A braided fusion categ ory with the fusion h yp ergro up o f C ( g , q ), where g is a s imple Lie algebra of BCD type and q a ro ot o f unity , is equiv alent to C ( g , q ) or one of ﬁnitely many twisted versions. (T uba/W enzl [25 9]) • B e fore w e rec onsider Route B , we as sume that we alrea dy have a braided fusion category , or pre-mo dular category . As we have seen, failure of mo dularity is due to non-trivial c e n ter Z 2 ( C ). Idea: Given a br aided (but not symmetric) category with even center Z 2 ( C ), kill the latter, using the Deligne / Doplicher-Rober ts theorem: Z 2 ( C ) ≃ Rep G . The la tter contains a commutativ e (F rob enius) algebra Γ corr esp o nding to the r egular representation o f G . Now Γ − Mod C is mo dular. (Brugui` eres [39], M ¨ uger [19 0]) . This cons truction can be interpreted as Galois closure in a Galois theory for BTCs, cf. [19 0]. • Ro ute B to braided ca tegories: Quantum doubles: If G is a ﬁnite gr oup then D ( G ) − Mo d a nd D ω ( G ) − Mo d are modula r (Ba ntay [16], Altsch uler/ Coste [3 ]). If H is a ﬁnite-dimensional semisimple a nd co semisimple Hopf algebra then D ( H ) − Mo d is mo dular (Etingof/Gelaki [79]). If A is a ﬁnite-dimensio nal weak Hopf alge bra then D ( A ) − Mod mo dular (Nikshych/ T ur a ev/ V ainer man [214]). 38 • The center Z 1 of a left/r ig h t rigid, pivotal, spherical ca tegory has the s ame prop erties. In particular, the center of a spherical ca tegory is spherical and braided, th us a ribb on ca tegory . (Under weak er ass umptions , this is not true, and exis tence o f a t wist for the center, if desir ed, m ust b e enforced by a categ orical version o f the ribbo nization of a Hopf alg ebra, cf. [13 9].) • The braided cent er Z 1 : If C is spherical fusion ca tegory a nd dim C 6 = 0 then Z 1 ( C ) is mo dular and dim Z 1 ( C ) = (dim C ) 2 . (M ¨ uge r [192].) Comments on the pr oo f: Semisimplicity no t diﬃcult. Next, one ﬁnds a F rob enius alg ebra Γ in D = C ⊠ C op such that the dua l categor y Γ − Mod D − Γ is equiv alent to Z 1 ( C ), implying dim Z 2 ( C ) = (dim C ) 2 . Here Γ = ⊕ i X i ⊠ X op i , which is a gain a co end and can exist a lso in non-semisimple ca tegories. • This contains all the earlier mo dular it y re s ults on D ( G ) − Mo d and D ( H ) − Mo d, but also for D ω ( G ) − Mo d since: D ω ( G ) − Mo d ≃ Z 1 ( C k ( G, ω )) . (Using work by Hausser/Nill [1 14] or Panaite [226] on quantum double o f quasi Ho pf-algebras.) • Mo dular it y of Z 1 ( C ) a lso follows by combination of O s trik’s r esult that every fusion categor y arises from a weak Hopf alge bra A , combined with modula rity of D ( A ) − Mo d [214], provided one pr oves D ( A ) − Mo d ≃ Z 1 ( A − Mo d), generaliz ing the known result for Hopf algebr as. But the purely categor ical pro of av oiding weak Hopf a lgebras seems preferable, not leas t s ince it probably extends to ﬁnite non-semisimple categ ories. • In the Mor ita cont ext ha ving C ⊠ C op and Z 1 ( C ) as its corners, the tw o oﬀ-dia gonal categor ies are equiv alen t to C and C op , and their str uctures as C ⊠ C op -mo dule categor ies ar e the ob vious ones. Therefor e, the center can a lso be understo o d as (using the notation o f EO): Z 1 ( C ) ≃ ( C ⊠ C op ) ∗ C . A (somewha t sketc h y) pro of of this equiv alence can b e found in [223, Pr op. 2.5]. • W e give a nother example for a purely categ orical result tha t can b e proven using weak Hopf algebras : Radford’s form ula for S 4 has a g eneralization to weak Hopf algebra s [213], and this can b e used to prov e that in every fusion ca tegory , there exists a n isomorphism o f tensor functors id → ∗ ∗ ∗∗ , cf. [83]. (Notice that in every pivotal category we have id ∼ = ∗∗ , th us here it is imp ortant that w e understand ‘fusion’ just to mean existence o f t wo-sided duals. But in [8 4] it is conjectured that every fusion ca teg ory admits a piv otal s tructure.) • If C is already mo dula r then there is a braided equiv a lence Z 1 ( C ) ≃ C ⊠ C op , cf. [192]. Thus, every modular catego ry M is full sub catego ry of Z 1 ( C ) for so me fusion categor y . (This probably is not very useful for the classiﬁca tion of mo dular catego r ies, since there are ‘more fusion catego ries than mo dular categor ie s ’: Reca ll from Section 3 that C 1 ≈ C 2 ⇒ Z 1 ( C 1 ) ≃ Z 1 ( C 2 ). (F or converse, see be low.) • Ther e is a “Double commut ant theorem” for mo dular catego r ies (M¨ uger [193], inspired b y Ocneanu [21 9]): Let M a modular category and a C ⊂ M a replete full tensor sub categor y . Then: 1. ( M ∩ ( M ∩ C ′ ) ′ ) = C . 2. dim C · dim ( M ∩ C ′ ) = dim M , 3. If, in a ddition C is mo dular , then also D = M ∩ C ′ is mo dular and M ≃ C ⊠ D . (Thus every full inclusion of mo dular categ ories arises fro m a direct pro duct.) These results indicate that ‘mo dular ca teg ories are b etter behaved tha n ﬁnite g roups’. • Co rollary : If M is mo dular and S ⊂ M sy mmetric then S ⊂ M ∩ S ′ . Thus (dim S ) 2 ≤ dim S · dim( M ∩ S ′ ) = dim M , implying dim S ≤ √ dim M . Notice that the b ound is satisﬁed by Rep G ⊂ D ( ω ) ( G ) − Mo d. In fact, existence of a symmetric sub category a ttaining the b ound characterize s the representation categories of t wisted doubles, cf. b elow. 39 • O n the other hand, consider C ⊂ M with M modula r. W e hav e M ∩ C ′ ⊃ Z 2 ( C ), implying dim M ≥ dim C · dim Z 2 ( C ). This provides a lower bo und on the dimens io n of a mo dular category containing a given pre-mo dular sub categor y as a full tensor sub catego r y . In [19 3] it was conjectured that this bo und can alwa ys be attained. • It is natura l to ask how primality o f D ( G ) − Mo d is r elated to simplicity of G . It turns o ut that the t wo pr op erties are indep endent. On the one hand, ther e are no n- simple ﬁnite gr oups for which D ( G ) − Mod is prime. (This is a cor ollary of the cla s siﬁcation of the full fusio n sub c ategories of D ( G ) − Mo d given in [206].) On the other hand, for G = Z /p Z one ﬁnds that D ( G ) − Mo d is prime if and only if p = 2. F or p an odd prime, D ( G ) − Mo d has t wo prime factor s, bo th of which are mo dular categ ories with p inv ertible ob jects, cf. [193]. But for every ﬁnite simple non-a b elian G , one ﬁnds that D ( G ) − Mo d is prime. In fact, it has only one replete full tensor s ubca teg ory at all, namely Rep G . Th us all these categories are m utually ineq uiv alent: The classiﬁc a tion of prime modular categories contains that o f ﬁnite simple gro ups . • If C is symmetric and (Γ , m, η ) a comm utative algebra in C , then Γ − Mo d C is ag ain symmetric and dim Γ − Mo d C = dim C d (Γ) . (5.1) Now, if C is only bra ided, Γ − Mo d C is a fusion c a tegory satisfying (5.1 ), but in general it fails to b e bra ided! (Unless Γ ∈ Z 2 ( C ), a s was the case in the context o f mo dula rization.) • E xample: Given a BTC C ⊃ S ≃ Rep G , let Γ b e the regular monoid in S as considered in Section 3. Then C ⋊ S := Γ − Mo d C is fusion ca tegory , but it is braided only if S ⊂ Z 2 ( C ), a s in the discussion of mo dulariza tion. In general, one obtains a braided crossed G- category as deﬁned b y T uraev [263, 264] (cf. also Carra sco and Moreno [44]), i.e. a tensor categor y with G -grading ∂ on the o b jects, a G -action γ such that ∂ ( γ g ( X )) = g ∂ X g − 1 and a ‘br aiding’ c X,Y : X ⊗ Y ∼ = − → γ ∂ X ( Y ) ⊗ X . The degree zero part is Γ − Mo d C ∩S ′ ≃ Γ − Mo d 0 C (cf. below). (Kirillov Jr. [153, 15 4], M¨ uger [19 4]). This co nstruction has a n in teresting connection to conforma l orbifold mo dels ([196, 19 9]) . • E ven if Γ 6∈ Z 2 ( C ), there is a full tenso r sub categ ory Γ − Mo d 0 C ⊂ Γ − Mo d C that is bra ided. Calling a module ( X , µ ) ∈ Γ − Mo d C dyslectic if µ ◦ c X, Γ = µ ◦ c − 1 Γ ,X , one ﬁnds that the full sub catego ry Γ − Mo d 0 C of dysle ctic mo dules is not o nly monoida l, but also inherits the braiding fr om C , cf. Pareigis [227]. This was rediscov ered b y Kir illov and Ostrik [155] who in addition proved that if C is mo dular then Γ − Mo d 0 C is mo dular and the following identit y , similar to (5.1) but diﬀerent, holds: dim Γ − Mo d 0 C = dim C d (Γ) 2 . Remark: Analog ous results were previo us ly obtained by B¨ oc kenhauer, Ev ans and Kaw ahiga - shi [34] in an op erator alge br aic context. While the transpo sition of their work to tenso r ∗ -catego ries is immediate, r emoving the ∗ -assumption requir es some work. • The ab ov e implies (for ∗ -catego ries, but a lso in general ov er C b y [8 4]) that d (Γ) ≤ √ dim C for commutativ e F rob enius algebra s in mo dular categ ories. (The a bove b ound o n the dimension of full symmetric catego ries follows from this, since the regular monoid in S is a commutativ e F ro benius algebra Γ with d (Γ) = dim S .) • All these facts hav e a pplica tions to chiral conformal ﬁeld theor ies in the op erato r algebra ic framework, rev iew ed in more detail in [198]: 40 Longo/Rehr en [1 71]: Finite lo ca l extensions o f a CFT A are class iﬁed b y the ‘lo cal Q -systems’ ( ≈ commutativ e F rob enius alge br as) in Rep A , which is a ∗ -BT C . B¨ oc kenhauer/Ev ans [33], [198]: If B ⊃ A is the ﬁnite lo cal extens ion corresp onding to the commutativ e F ro benius algebra Γ ∈ Rep A , then Rep B ≃ Γ − Mo d 0 Rep A . Analogous res ults for vertex op erator alge br as were formu lated by Kirillov and Ostrik [1 55]. Remark: It is p erha ps not completely absurd to compar e these re s ults to lo cal class ﬁeld theory , where ﬁnite Galois extensions of a lo ca l ﬁeld k are shown to be in bijection to ﬁnite index subgro ups of k ∗ . • Dr infeld, Gelaki, Nikshyc h a nd Ostrik [75], and indep endently Kitaev and the a uthor, ob- served that every commutativ e F r obe nius algebra Γ in a mo dular categ ory M gives rise to a braided equiv alence Z 1 (Γ − Mo d M ) ≃ M ⊠ ^ Γ − Mo d 0 M . (5.2) T a k ing Γ = 1 , one recov ers the fact Z 1 ( M ) ≃ M ⊠ f M . The la tter raises the questio n whether one can ﬁnd a smaller fusion category C suc h that M ⊂ Z 1 ( C ). The answer given b y (5.2 ) is that the bigger a commu tative alge bra one can ﬁnd in M , the smaller one can take C to be. In pa rticular, if Γ − Mod 0 M is trivial (which is equiv a len t to d (Γ) 2 = dim M over C ) then M ≃ Z 1 (Γ − Mo d M ) is not just co n tained in a center of a fusio n ca tegory but is such a cen ter. In fact, this criter ion identiﬁes the modula r categ ories o f the form Z 1 ( C ) since, conv ersely , cf. [57], o ne ﬁnds that the cent er Z 1 ( C ) o f a fusion category contains a comm utativ e F rob enius algebra Γ o f the maximal dimension d (Γ) = p dim Z 1 ( C ) = dim C such that Γ − Mo d 0 Z 1 ( C ) trivial , Γ − Mo d Z 1 ( C ) ≃ C . • As an application one obtains that if M is modula r a nd S ⊂ M symmetric and even such that dim S = √ dim M then M ≃ D ω ( G ) − Mod, wher e S ≃ Rep G and ω ∈ Z 3 ( G, T ). This has an a pplication in CFT: If A is a ch iral CFT with trivial representation catego ry Rep A (i.e. A is ‘holomorphic’) acted upo n by ﬁnite gr oup G . Then Rep A G ≃ D ω ( G ) − Mo d. (T og ether with the results of [143], this pr ov es the fo lk conjecture, having its ro ots in [67, 66], that the represe ntation ca tegory of a ‘holomorphic chiral or bifold CFT’ is given by a ca tegory D ω ( G ) − Mo d.) • As sho wn in [1 91], a w eak monoidal Mo r ita equiv alence C 1 ≈ C 1 of fusion categories implies Z 1 ( C 1 ) ≃ Z 1 ( C 2 ). (This is an immediate co rollary of the deﬁnition of ≈ , combined with [2 40].) The con verse is true for gro up theoretical categories (Naidu/Niksh ych [205]), and a ge ner al pro of is a nnounced by Nikshyc h. • B y deﬁnit ion, a group theoretica l ca tegory C is w eakly Morita equiv alen t (dual) to C k ( G, ω ) for a ﬁnite group G and [ ω ] ∈ H 3 ( G, T ). Thus Z 1 ( C ) ≃ Z 1 ( C k ( G, ω )) ≃ D ω ( G ) − Mo d. The conv erse is also true . Therefore, with M mo dular and C fusion we hav e: contains M Z 1 ( C ) maximal comm. F A Γ M ≃ Z 1 ( C ) alwa ys true maximal STC S M ≃ D ω ( G ) − Mo d C is gr oup theoretical • What ca n w e say ab out non- c omm utative (F rob enius) alge bras in modula r categories? W e ﬁrst lo ok at the symmetric ca se. Le t th us C be a rigid symmetric k - linear tenso r category and 41 Γ a str o ngly separ able F ro benius algebra in C . Deﬁne p ∈ End Γ by p = (T r Γ ⊗ id Γ )(∆ ◦ m ◦ c Γ , Γ ) = Γ ☛ ✟ ✡ ✠ ☛ ✟ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✡ ✠ Γ = Γ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✡ ✠ Γ (5.3) (The fourfold vertex in the right diag ram repres en ts the mo rphism m (2) = m ◦ m ⊗ id.) Then p is idempotent (up to a scala r) and its kernel is an ideal. Thus the image of p is a co mm utative F ro benius subalgebr a of Γ. The latter is called the cen ter of Γ since it is the o rdinary cen ter in the case C = V ect ﬁn k . • Applica tio n to TQFT: Every ﬁnite dimensio nal semisimple k -algebr a A g ives rise to a TQFT in 1 + 1 dimensio ns via triangulation (F ukuma/ Hosono/Kawai [99]). B y the cla ssiﬁcation o f TQFTs in 1 + 1 dimensio ns [65, 1, 156], this TQFT cor r esp onds to a commutativ e F rob enius algebra B (in V e c t ﬁn k ), with A = V ( S 1 ) and the pro duct arising from the pan ts cobor dism. The latter is given b y the vector space asso ciated with the cir c le and the m ultiplication is given by the pants cobor dism. One ﬁnds B = Z ( A ), and B arises exactly as the imag e of A under the ab ove pro jection p . (This works sinc e every semisimple algebra is a F rob enius algebra.) • If C is br aided, but not symmetric, we must cho ose b etw een c Γ , Γ and c − 1 Γ , Γ in the de ﬁnitio n (5.3 ) of the idemp otent p . This implies that a no n- commut ative F rob enius algebra will t ypically hav e tw o diﬀerent centers, calle d the left and rig h t cen ters Γ l , Γ r . Remark ably , one then obtains an equiv alenc e E : Γ l − Mo d 0 C ≃ − → Γ r − Mo d 0 C of mo dular categories, cf. B¨ oc kenhauer, Ev ans, Kawahigashi [34], Ostrik [2 2 2] and F r¨ ohlich, F uchs, Runkel, Sch w eigert [98, 95]. Co nversely , if C is mo dular, every tr iple (Γ l , Γ r , E ) as ab ov e arises from a non-commutativ e algebra in C , [15 7]. (The latter is unique only up to Morita equiv alence.) • This is r elev a n t for the classiﬁca tion of CFTs in t w o dimens io ns: The latter are cons tructed from a pair ( A l , A r ) of chiral CFTs and so me algebraic datum (‘modular in v a riant’) sp ecifying how the two chiral CFTs a re glued together . In the left-rig h t symmetric case, where the tw o chiral theories coincide A l = A r = A , the ab ov e result indicates that F rob enius algebras in C = Rep A a r e the structure to use. This is substantiated b y a construction, using TQFTs, o f a ‘top ological from a mo dular ca tegory C a nd a F rob enius alg ebra Γ ∈ C , cf. F uchs, Runkel, Sch w eigert, cf. [9 7] and sequels . • The F ro benius algebras in / mo dule ca tegories of S U q (2) − Mod can be class iﬁe d in ter ms of ADE gr aphs. (Quantum MacKay cor resp ondence.) Cf. B ¨ oc kenhauer, E v ans [33], Kirillov Jr. and Ostrik [155], Etingo f/ Ostrik [86]. • Thes e results should be extended to other Lie groups. If S U (2) already leads to the ADE graphs (“ubiquitous” accor ding to [117]), the other classical groups should give r ise to very int eresting algebraic- combinatorial s tructures, cf. e.g. [220, 221]. • Mo re gener ally , when the t wo chiral theories A l , A r , and therefore the as so ciated mo dular categorie s C l , C r diﬀer, it is better to work with triples (Γ l , Γ r , E ), where Γ l/r ∈ C l/r are commutativ e a lgebras and E : Γ l − Mod 0 C l → Γ r − Mod 0 C r is a braided eq uiv alence. (By the ab ov e, in the left-right symmetric cas e C l = C r = C , this is equiv alent to the study of non- commutativ e F rob enius algebra s Γ ∈ C .) Now one ﬁnds [1 98] a bijection b etw een suc h triples 42 and c ommutative a lgebras Γ ∈ C l ⊠ e C r of the maxima l dimension d (Γ) = √ dim C l · dim C r . (This is a ca tegorical version of Rehr en’s appro ach [23 3] to the clas siﬁcation of mo dular inv a riants. It is based o n studying lo cal e xtensions A ⊃ A l ⊠ f A r , corres ponding to co mm utative algebras Γ ∈ C l ⊠ e C r .) • Ther e also is a concept o f a center of an algebra A in a not-neces sarily bra ided tensor category C , to wit the full cen ter deﬁned in [56] b y a univ ersal proper t y . While the full center is a commutativ e algebra in the br aided center Z 1 ( C ) of C , as appo sed to in C like the ab ov e notions of c e n ter, there are connectio ns b et ween these construc tio ns. • W e close this s e c tion giving thr e e more reasons why mo dular categ ories are interesting: 1. They hav e ma n y connectio ns with num be r theory : – Rehr e n [232], T uraev [2 62]: X i d 2 i = | X i d 2 i ω i | 2 . In the p ointed case (all simple o b jects hav e dimension one) this reduces to | P i ω i | = ± p | I | . F o r suitable C , this repro duces Gauss’ ev aluatio n of Gauss s ums. (Gauss actually also determined the sign of his s ums.) – The ele men ts of T matrix ar e ro o ts of unity , and the ele ments of S are cyclo tomic in tegers [36, 78]. – F or rela ted int egra lit y pr op erties in Y=TQ FSs , cf. Masbaum, Rob erts, W enzl [1 86, 187] and Brugui` er es [38]). – The congruence s ubgroup prop erty: Let N = o rd T ( < ∞ ). Then ker( π : S L (2 , Z ) → GL ( | I | , C )) ⊃ Γ( N ) ≡ ker( S L (2 , Z ) → S L (2 , Z / N Z )) . F or the modula r categorie s a rising from ra tional CFTs, this had b een known in many cases and widely believed to b e true in genera l. Conside r able pr o gress was made by Banta y [17], whose arguments w ere made rigoro us by Xu [283] using algebraic qua n tum ﬁeld theory . Banta y’s work inspired a pro of [247] by Sommerh¨ auser and Zhu for mo du- lar Hopf a lg ebras, using the higher F r ob e nius-Sch ur indicators deﬁned b y Kashina and Sommerh¨ auser [136]. Finally , Ng and Sc hauen burg pro ved the congruence pro per t y for all mo dular categ ories a lo ng similar lines, cf. [212], beg inning with a categ orical version of the hig her F rob enius-Sch ur indicators [2 11]. 2. A mo dular categor y M gives rise to a surger y TQFT in 2 + 1 dimensions (Reshetikhin, T ur a ev [235, 262]). In particular , this works for M = Z 1 ( C ) when C is spherica l fusion categorie s C with dim C 6 = 0. Since such a category C also deﬁnes a TQFT via triang ula tion [19, 104], it is natural to expect an isomorphism R T M = B W GK C of TQFTs. (When C is itself mo dular , this is indeed true b y Z 1 ( C ) ≃ C ⊠ e C and T uraev’s w ork in [262].) Recently , a general pro of of this result was announced by T uraev and Virelizier, based on the work of Brugui` eres and Virelizier [41, 42], partially joint with S. Lack. (Notice in any case that the surgery co nstruction provides mor e TQ FTs than the triang ulation approach, since not all mo dular catego ries are centers.) 3. W e close with the hyp o thetical application of mo dular categ ories to top olo gical quantum computing [274]. Ther e are a ctually t wo diﬀerent approaches to topo logical quan tum com- puting: The o ne initiated by M. F reedman, using TQFTs in 2 + 1 dimension and the one due to A. K itaev using d = 2 quan tum spin systems. Ho wev er, in b oth prop osals , the modula r representation categories are cen tral. Cf. also Z. W ang , E. Row ell et al. [120, 237]. 6 Some op en problems 1. Cha racterize the hyperg roups aris ing from a fusion categ ory . (Probably hop eless.) Or at least those cor resp onding to (connected) compact gro ups. 43 2. Find an algebraic structure whose r epresentation categor ie s give all semisimple piv otal cat- egories, gener alizing Os trik’s result [222]. Perhaps this will b e something like the quantum group oids deﬁned b y Lesieur and Eno ck [165]? 3. Cla ssify all prime mo dular categories. (The next ch allenge after the cla ssiﬁcation of ﬁnite simple gro ups ...) 4. Give a direct co nstruction of the fusio n ca tegories asso cia ted with the tw o Haag erup subfactors [109, 7, 8 ]. 5. P rov e that every br a ided fusion categ o ry C / C e mbeds fully into a mo dular ca tegory M with dim M = dim C · dim Z 2 ( C ). (This is the optim um allow ed by the double commutan t theor e m, cf. [19 3].) 6. Find the mo st genera l context in whic h an a nalytic (i.e. non-for ma l) version of the C a rtier/ Kassel/ T uraev [45, 1 40] formal deformation quantization of a s ymmetric tensor category S with inﬁnitesimal braiding can b e given. (I.e. give an abstra ct version of the Kaz hdan/Lusztig construction of Drinfeld’s ca teg ory [144] that do es not suppo se S = Rep G .) 7. Gener alize the pro of o f mo dularity of Z 1 ( C ) for semisimple fusion catego ries to not nece ssarily semisimple ﬁnite catego ries (in the sense of [85]), using Lyubashenko’s deﬁnitio n [175] of mo dularity . 8. Likewise for the triangula tio n TQ FT [265, 1 9, 104]. Genera lize the rela tion to surg ery TQFT to the no n-semisimple case. (F or the non-s emisimple version of the R T-TQFT in [151].) 9. Ha r d non-commutativ e analysis: E v ery countable C ∗ -tensor ca tegory with conjugates and End 1 = C embeds fully into the C ∗ -tensor category of bimodules ov er L ( F ∞ ) and, for any inﬁnite factor M , in to End( L ( F ∞ ) ⊗ M ). Here F ∞ is the free group with countably ma ny generator s and L ( F ∞ ) the type I I 1 factor as so ciated to its left regular repr esentation. (This would extend and co nceptualize the r esults of Popa/Shlyakhtenk o [229] on the universality of the factor L ( F ∞ ) in subfactor theor y .) 10. Giv e satisfac to ry categorical interpretations for v arious gener alizations of quasi-tria ng ular Hopf algebras, e.g. dynamical quantum g roups [77] a nd T oleda no-Laredo’s qua si-Coxeter al- gebras [2 57]. Soibelma n’s ‘mer omorphic tensor categorie s ’ and the ‘categories with cylinder braiding’ of tom Dieck and H¨ aring-Oldenburg [258] might be relev a n t – and in a n y case they deserve further study . A cknow le dgement : I thank B . Enr iques and C. Kassel, the or ganizers of the Rencontre “Group es quantiques dynamiques et cat´ egories de fusion” that to ok pla ce at C IRM, Mar s eille, fr o m April 14-18 , 2008, for the in vitation to give the lectures that gav e rise to these notes. (No pro ceedings were published for this meeting.) I am also gra teful to N. Andr us kiewitsch, F. F antino, G. A. Garc ´ ıa a nd M. Mombelli for the invitation to the “Collo quium on Ho pf algebra s, quantum gr o ups and tensor categ ories”, C´ ordoba , Argentina, August 31st to September 4th, 2009 , as well as for their willingness to publish these notes. Disclaimer : While the follo wing bibliography is quite extensive, it should be clear that it has no pretense whatso ever at co mpletenes s. Ther efore the abse nc e of this or that reference should no t b e construed as a judgmen t of its relev ance. The choice o f references was guided by the principal thrust of thes e lectures, namely linear categories. This means that the sub jects o f quantum gr oups and low dimensional topo logy , but a lso general catego rical algebr a are touched up on only tangentially . 44 References [1] L. Abrams: Two-dimensional top ologic al quantum ﬁeld theor ies and F rob enius algebras, J. Knot Th. Ramif. 5 , 5 69-58 7 (19 9 6). [2] D. Altsch uler, A. Br ug ui ` eres: Ha ndle slide for a sov ereign category . Prepr int, 2 001. [3] D. Altsc huler, A. Coste: Inv ar ia n ts of 3-manifo lds from ﬁnite groups. In: Pr o c. XXth In t. Conf. Diﬀ. Geo m. Meth. in Theor. Phys., New Y ork , 1991, pp. 219–2 33, W orld Scientiﬁc, 1992. Qua s i-quantum groups, knots, three-manifolds, and to p olo gical ﬁeld theor y . Co mm. Math. P hys. 150 , 83 -107 (1992). [4] H. H. Andersen; P . Polo; K. X. W en: Repr esentations of quantum a lgebras. In ven t. Math. 104 , 1-59 (1991). H. H. Andersen: T ensor pro ducts of quan tized tilting modules. Co mm. Math. P hys. 149 , 14 9-159 (1992). [5] Y. Andr´ e: Une intr o duction aux motifs (motifs purs, motifs mixtes, p ´ erio des) . So c. Math. F ra nce, 2004 . [6] A. Ar diz z oni: The catego ry of mo dules ov er a monoidal catego r y: Ab elian o r not? Ann. Univ. F erra ra - Sez. VI I - Sc. Mat., V ol. L , 167 -185 (2 0 04). [7] M. Asaeda, U. Haag erup: E xotic subfactors of ﬁnite depth with Jones indices (5 + √ 13) / 2 and (5 + √ 17) / 2. Comm. Ma th. Phys. 202 , 1- 63 (1999). [8] M. Asaeda , S. Y asuda: O n Ha a gerup’s list of p oten tial principal graphs of subfactor s. Comm. Math. P hys. 286 , 11 41-115 7 (2009 ). [9] J . C. Baez: Hig her-dimensional algebra. I I. 2- Hilbert spaces . Adv. Math. 127 , 1 25-18 9 (1997). [10] J. C. Ba e z , J . Dolan: Ca tegoriﬁcatio n. In: E. Getzler, M. Kapranov (eds.): Higher Cate gory The ory . Contemp. Math. 230 , 1 -36 (1998 ). [11] J. C. Baez; J. Dolan: Higher- dimensional alg e bra. I I I. n -categor ies a nd the algebra of op etop e s. Adv. Math. 135 , 1 45-206 (1998). [12] J. C. Baez; A. D. La uda: Hig her-dimensional algebra V’: 2 - groups. Theor. Appl. Categs. 1 2 , 423-4 91 (200 4). [13] J. C. Ba e z ; J. P . May (eds.): T owar ds higher c ate gories . Springer V erlag , 2010. [14] J. C. Baez; M. Sh ulman: Lectures on n -catego ries and cohomolo gy . In [13]. ( arXiv: math/0 608420 ) [15] B. Bak alov, A. Kir illov Jr.: L e ctur es on tensor c ate gories and mo dular functors . America n Mathematical So ciety , 2 0 01. [16] P . Banta y: Or bifolds and Hopf algebra s. Ph ys. Lett. B245 , 47 7 -479 (199 0). Orbifolds, Hopf algebras , and the Mo onshine. Lett. Math. Phys. 22 , 187- 194 (19 91). [17] P . Banta y: The kernel o f the mo dular representation and the Ga lois action in RCFT. Comm. Math. P hys. 233 , 42 3-438 (2003). [18] J. W. B arrett, B. W. W estbury : The equality of 3 - manifold inv ar ia n ts. Math. Pro c. Ca m br. Phil. So c. 118 , 503 -510 (1 9 95). [19] J. W. Barrett, B. W. W es tbur y: Inv ariants of piecewis e-linear 3- manifolds. T rans. Amer. Math. So c. 3 48 , 3997 -4022 (1996). [20] J. W. Bar rett, B. W. W estbury: Spherical categ o ries. Adv. Ma th. 143 , 3 57-375 (1999). [21] H. Baumg¨ artel, F. Lled´ o: Dualit y of co mpact groups and Hilb ert C*-systems for C* -algebra s with a non trivial cen ter. Int. J. Ma th. 15 , 7 59-812 (2004). [22] A. Beliakov a, C. Blanchet: Skein construction of idemp otents in Birman-Mura k ami-W enzl algebras . Math. Ann. 321 , 347-37 3 (20 01). 45 [23] A. Beliako v a, C. Blanchet: Mo dular categories of t yp es B, C and D. Comment. Math. Helv. 76 , 467 -500 (2 001). [24] J. Benab ou: Ca t´ eg ories avec multiplication. C. R. Acad. Sci. Paris 256 , 188 7-1890 (1 9 63). [25] J. Bena b ou: Alg` ebre ´ el´ ementaire dans le s ca t ´ e gories av ec multiplication. C. R. Aca d. Sci. Paris 258 , 771-77 4 (19 64). [26] J. Benab ou: Intro duction to bicategor ies. In : Rep orts o f the Midwest Categor y Seminar. Lecture Notes in Mathematics 47, pp. 1–77. Springer V er lag, 1967. [27] J. Bichon: T r ivialisations da ns les cat ´ e gories ta nnakiennes. Cah. T op ol. Geom. Diﬀ. Cat ´ eg. 39 , 243 –270 (1 998). [28] J. Bichon, A. De Rijdt, S. V aes: Ergo dic coactions with large m ultiplicit y and mono idal equiv a lences of quantum gro ups. Comm. Math. P h ys. 262 , 703-7 28 (200 6). [29] J. S. Birman, H. W enzl: Braids, link p olynomials and a new alg ebra. T ra ns. Amer. Math. So c. 313 , 2 4 9-273 (1989 ). [30] E. Bla nc hard: On ﬁniteness of the nu mber of N -dimensio nal Hopf C ∗ -algebra s. In: A. Gheon- dea, R. N. Golog an and D. Timotin (eds.): Op er ator the or etic al metho ds . (Timi¸ soa ra, 19 98), 39–46 , Theta F o und., Bucharest, 2000. [31] C. Blanchet: Heck e alg ebras, modular ca tegories a nd 3 - manifolds qua n tum in v a r iants. T op ol- ogy 39 , 193-2 23 (200 0). [32] C. Blanchet, N. Hab egger , G. Masbaum, P . V oge l: Three-manifold inv aria nts derived from the Kauﬀman bra cket. T op olog y 31 , 68 5-699 (1992 ). T op olog ical quantum ﬁeld theor ies derived from the Kauﬀman bracket. T o po logy 34 , 883-92 7 (1995 ). [33] J. B¨ o c kenhauer, D. E. Ev ans: Modula r in v ar iants, gr aphs a nd α -induction for nets of sub- factors I-I I I. Comm. Math. Phys. 197 , 3 61-386 (1998), 200 , 57-103 (1999 ), 205 , 183- 2 28 (1999). [34] J. B ¨ ock enhauer, D. E . Ev a ns, Y. Kawahigashi: O n α -induction, chiral genera tors and mo d- ular inv ariants for subfac to rs. Comm. Math. Ph ys. 2 0 8 , 429 -487 (199 9 ). Chiral structure of mo dular inv ariants for subfactors . Comm. Math. Phys. 2 1 0 , 733- 784 (20 0 0). [35] G. B¨ ohm, K. Szla c h´ anyi: A coasso ciative C ∗ -quantum group with nonin tegral dimensions. Lett. Math. Phys. 38 , 437 -456 (19 96). [36] J. de Boe r , J. Go er ee: Ma rko v tra ces and I I 1 factors in conformal ﬁeld theory . Comm. Math. Phys. 139 , 267-3 0 4 (19 91). [37] A. Brugui ` eres: Th´ e orie tannakienne non commut ative. Comm. Alg. 22 , 5817 –5860 (19 94). Dualit ´ e tannakienne pour les q uasi-gro upo ¨ ıdes quantiques. Comm. Alg. 25 , 73 7–767 (1997). [38] A. B rugui` eres, Alain: T res ses et s tructure ent i` ere sur la cat´ ego rie des rep´ es e n tations de SL N quantique. Co mm. Algebra 28 , 1989 – 2028 (2000), [39] A. B r ugui` eres: Cat´ egories pr´ e mo dulair es, mo dular is ations et in v ar iants de v ari´ et´ es de dimen- sion 3. Math. Ann. 316 , 215 -236 (2000). [40] A. Brugui ` eres, G. Maltsiniotis: Quas i- group o ¨ ıdes quantiques. C. R. Acad. Sci. P aris S ´ er. I Math. 31 9 , 933– 9 36 (199 4) [41] A. Brugui` eres, A. Virelizier: Catego rical centers and Reshetikhin-T uraev in v ar iants. Acta Math. Vietnam. 33 , 25 5-277 (2008). [42] A. Brugui` eres, A. Virelizier: Quantum double o f Hopf monads and categ orical cen ters. T o app ear in T rans . AMS. (Pre pr in t: The double of a Hopf mo na d. arXiv :0812. 2443 .) [43] D. Calaque, P . Etingof: Lectures on tensor categ ories. In: B. Enriquez (ed.): Quantum gr oups . IRMA Lec tur es in Mathematics and Theoretical Physics, 12 . Euro pea n Mathematical Society , 2008. 46 [44] P . Carrasco , J. M. Mor eno: Categoric al G-crossed mo dules and 2- fold extensions. J. Pure Appl. Alg. 163 , 23 5-257 (2001). [45] P . Cartier : Construction combinatoire des inv a riants de V as siliev-Kontsevich des nœuds. C. R. Acad. Sc i. Paris S´ e r . I Math. 316 , 1205 - 1210 (1 993). [46] A. M. Cegar ra, E. K hmaladze: Homoto p y classiﬁcation o f graded P icard catego ries. Adv. Math. 21 3 , 644-6 86 (200 7). [47] V. Chari, A. Pressle y : A gu ide to quantum gr ou ps . Cambridge Univ ersity Press, 199 5. [48] A. M. Cohen, R. de Man: Computational evidence for Deligne’s conjecture regarding excep- tional Lie gr oups. C . R. Acad. Sci., P aris, S´ er. I 322 , 4 27-43 2 (1996 ). On a tensor category for the exceptiona l Lie gro ups. In: P . Dr¨ axler et al. (eds.): Computational metho ds for r ep- r esentations of gr oups and algebr as . P rogr. Math. 17 3 , Birkh¨ auser V erlag, 1 999. [49] J. Comes, V. Ostrik: On blo cks of Deligne’s categ ory Rep( S t ). arX iv:091 0.5695 . [50] S. E. Crans: Gener alized centers of braided and sylleptic monoidal 2-categor ies. Adv. Ma th. 136 , 18 3-223 (1998). [51] G. Y. Chen: A new characteriza tion of ﬁnite s imple gr oups. Chinese Sci. Bull. 40 , 4 46-45 0 (1995). [52] P . Cvitanovi ´ c: Gr oup t he ory. Bir dtr acks, Lie’s, and exc eptional gr oups . Princeton Univ ersity Press, 20 08. [53] A. A. Davydov: Twisting of monoidal structures. q- alg/97 03001 . [54] A. A. Davydov: Monoidal categories . J. Math. Sci. (New Y ork) 88 , 457 - 519 (19 98) [55] A. A. Da vydov: Galois a lgebras and mono idal functors betw een c a tegories of representations of ﬁnite groups. J. Alg. 244 , 273-3 0 1 (2001 ). [56] A. A. Davydov: Centre of an algebra. T o app ear in Adv. Math. ( arXi v:0908. 1250 ) [57] A. A. Davydov, M. M¨ uger, D. Nikshyc h, V. O strik: In pr eparation [58] P . Deligne: Cat´ egor ie s tannakiennes. In : P . Car tier et al. (eds.): Gr othendie ck F est s chrift , vol. I I, pp. 11 1–195 . Birkh¨ a user V erla g, 1991. [59] P . Deligne: La s´ e r ie exceptionnelle de g roup es de Lie. C. R. Acad. Sci., P aris, S´ er. I 322 , 321-3 26 (199 6), [60] P . Deligne: Cat´ eg ories tensorielles. Moscow Math. J. 2 , 22 7-248 (2002). [61] P . Deligne: La cat´ e gorie des repr´ esentations du gro upe sym´ etrique S t , lorsque t n’es t pas un ent ier naturel. In: Algebraic gro ups and homogeneo us spac e s, 209– 2 73, T ata Inst. F und. Res. Stud. Math., T a ta Inst. F und. Res ., Mumbai, 2007 [62] P . Deligne, B. H. Gross: On the ex c eptional series, and its descendants. C. R. Math. Acad. Sci. Paris 335 , 877 -881 (2002). [63] P . Deligne, R. de Man: La s´ erie exc e ptio nelle de group es de Lie. I I. C. R. Acad. Sci., Paris, S´ er . I 323 , 5 77-58 2 (1996 ). [64] P . Delig ne, J. S. Milne: T anna kian categor ie s. Lecture notes in mathematics 900 , 101–2 28. Springer V erlag, 1 982. [65] R. Dijkgraaf: A Geometrical Approach to Two Dimensional Conformal Field Theor y , P h.D. thesis (Utrech t, 198 9). [66] R. Dijkgraaf, V. Pasquier, P . Ro che: Quasi Hopf algebras, group cohomolo gy and or bifold mo dels. Nucl. P h ys. B (Pro c. Suppl.) 18B , 60-72 (19 9 0). [67] R. Dijkgr aaf, C. V afa, E . V erlinde, H. V erlinde: The opera tor algebr a of orbifold mo dels. Comm. Math. Phys. 1 23 , 485- 527 (19 89). 47 [68] S. Doplicher, R. Haag, J . E. Ro ber ts: Lo ca l observ ables and par ticle statistics I & I I. Comm. Math. P hys. 23 , 199 -230 (1971), 35 , 4 9-85 (197 4). [69] S. Doplicher, C. Pinzari and J. E . Ro ber ts, An algebraic dua lit y theory fo r multiplicative unitaries. Int. J. Math. 12 , 415–459 (2001). C. Pinzari and J. E. Rob erts, Regular o b jects, m ultiplicative unitaries and conjugation. In t. J. Math. 13 (2002) 625 -665. [70] S. Doplicher, J. E. Rob erts: A ne w duality theory for co mpact gro ups. Inv ent . Math. 98 , 157–2 18 (198 9). [71] S. Doplic her, J. E . Robe rts: Why there is a ﬁeld algebra with a compa c t gauge group de- scribing the supers election structure in pa rticle physics. Comm. Math. Ph ys. 131 , 51-1 07 (1990). [72] V. G. Drinfeld: Quant um g roups. In: P ro c. In t. Co ngr. Math, Ber keley 198 6. [73] V. G. Drinfeld: Quasi-Hopf alge br as. Le ningrad Math. J. 1 , 1419 -1457 (199 0). [74] V. G. Drinfeld: On quasitriangular quasi- Ho pf a lgebras and on a gr o up that is closely con- nected with Gal( Q / Q ). L e ning rad Math. J . 2 , 8 29-860 (1991). [75] V. Drinfeld, S. Gelaki, D. Niksh ych, V. Ostrik: Group-theo retical prop erties of nilp oten t mo dular catego ries. math.QA /0704. 1095 . [76] S. Eilenberg , J. C. Mo ore : Adjoin t functor s and tr iples. Ill. J . Math. 9 , 381 -398 (19 65). [77] P . Etingo f; A. V ar chenk o: Solutions of the qua ntum dynamical Y a ng-Baxter equation and dynamical qua ntum groups. Comm. Math. P h ys. 196 5 91-640 (199 8 ). [78] P . E ting of: On V afa’s theorem for tensor categ o ries. Math. Res. Lett. 9 , 651-65 7 (20 02). [79] P . Eting of, S. Gelaki: Some pro p erties of ﬁnite-dimensiona l semisimple Hopf alge br as. Math. Res. Lett. 5 , 19 1 -197 (1 998). [80] P . E ting of, S. Gela k i: Iso categ o rical gr oups. Int. Math. Res. Not. 20 01:2, 59–76. [81] P . Etingof, S. Gela ki: The small quantum group a s a quantum double. J . Alg. 3 22 , 2580-2 585 (2009). [82] P . Etingo f, S. Gela ki, V. O strik: Classiﬁca tion o f fusion categ ories of dimension pq . Int . Ma th. Res. Not. 2004:57 , 30 41–305 6 [83] P . Etingo f, D. Nikshyc h, V. O strik: An analogue of Radford’s S 4 formula for ﬁnite tensor categorie s. Int. Math. Res . Not. 20 04:54, 2915 –2933 . [84] P . E ting of, D. Nikshych , V. Ostrik: On fusion categ ories. Ann. Ma th. 162 , 5 81-64 2 (2005). [85] P . E ting of, V. Os trik: Finite tensor categor ies. Mosc. Math. J. 4 , 6 27-654 , 782 –783 (2004). [86] P . Etingof, V. Ostrik: Mo dule categories over representations of SL q (2) and gr aphs. Ma th. Res. Lett. 1 1 , 10 3-114 (2004). [87] D. E . Ev ans , Y. Kaw ahigas hi: Quantum symm et ries on op er ator algebr as . Oxford Univ ersity Press, 19 98. [88] M. F ar ber : V on Neumann catego ries and extended L 2 -cohomolo gy . K -Theory 15 , 347-40 5 (1998). [89] M. Finkelberg: An equiv alence of fusio n categories . Geo m. F unct. Anal. 6 , 249 -267 (1 9 96). [90] K. F redenhag en, K .-H. Rehren, B. Schroer : Sup erselection sectors with braid gro up statistics and exchange algebras I. General theo ry . Comm. Math. Phys. 125 , 2 01-22 6 (19 8 9). [91] M. H. F reedma n, A. Kitae v , M. J. Lars en, Z. W ang: T op olog ical quantum computation. Mathematical challenges of the 21st century . Bull. Amer. Math. So c. 4 0 , 31 -38 (2003). [92] P . F reyd, D. Y etter, J. Hoste, W. B. R. Lick orish, K. Millett, A. Ocneanu: A new p olynomial inv a riant of knots and links. Bull. Amer. Math. Soc. 12 , 2 39-246 (1985). 48 [93] P . F reyd, D. Y etter: Bra ided compact clos ed catego ries with applications to low-dimensional top ology . Adv. Math. 77 , 15 6-182 (1989 ). [94] P . F r eyd, D. Y etter: Coherence theorems via knot theory . J. Pure Appl. Alg. 78 , 49-7 6 (1992). [95] J. F r ¨ ohlich, J. F uchs, I. Runkel, C. Sch weigert: Co rresp ondences of ribb on categories. Adv. Math. 19 9 , 192-3 29 (200 6). [96] J. F r¨ ohlich, T. K erler: Quantu m gr oups, quantum c ate gories and quantu m ﬁeld the ory . (Lec- ture Notes in Mathema tics vol. 1 542.) Springer V erlag , 1993. [97] J. F uchs, I. Runk el, C. Sch w eigert: TFT constr uction of RCFT correla tors. I. Partition functions. Nucl. Phys. B646 , 353-4 97 (2 002). [98] J. F uchs, C. Sch weigert: Categor y theory for confo r mal bounda ry conditions. Fields Inst. Commun. 39 , 25-7 1 (20 03). [99] M. F ukuma, S. Hos ono, H. Kaw ai: Lattice top ologic a l ﬁeld theory in tw o dimensions. Comm. Math. P hys. 161 , 15 7-175 (1994). [100] S. Gelaki: Semisimple tr iangular Hopf algebras and T annakia n categor ies. In : M. D. F ried, Y. Iha ra (eds.): A rithmetic fundamental gr oups and nonc ommu tative algebr a . P ro c. Symp os. Pure Math. 70 , 49 7-515 (2002) [101] S. Gelaki, D. Naidu: Some prop erties of group-theo retical categ ories. J. Alg. 3 2 2 , 2631-2 641 (2009). [102] S. Gelaki, D. Nikshyc h: Nilpo tent fusion categor ies. Adv. Math. 21 7 , 1053- 1071 (2 008). [103] S. Gelfand, D. Kazhdan: Examples of tensor ca tegories. Inv ent. Math. 109 , 5 95-617 (1992). [104] S. Gelfand, D. Kazhdan: Inv a riants of three - dimensional manifolds. Geom. F unct. Anal. 6 , 268-3 00 (199 6). [105] P . Ghez, R. Lima, J. E. Rob erts: W ∗ -categor ies. Pac. J. Ma th. 120 , 79-109 (198 5 ). [106] C. Go mez, M. Ruiz-Altaba, G. Sierra : Quantum gr ou ps in two-dimensional physics . Ca m- bridge University Pres s, 1996 . [107] F. Goo dman, H. W enzl: Ideals in T emp erley-Lieb ca tegory . App endix to: M. H. F reedman: A mag netic mo del with a p ossible Cher n- Simons pha se. Comm. Math. P h ys. 234 , 129 –183 (2003) [108] R. Gordon, A. J . Po wer, R. Street: Cohere nce for tric a tegories. Mem. Amer. Math. So c. 1 1 7 (1995). [109] U. Haagerup: Principa l graphs of subfactors in the index rang e 4 < [ M : N ] < 3 + √ 2. In: H. Ara ki, Y. Kaw ahigashi, H. Kosa ki (eds.): Subfactors . (Kyuzeso, 1993), 1–38, W orld Sci. Publ., River Edge, NJ, 199 4 . [110] P h ung Ho Hai: T anna k a-K rein dualit y for Hopf algebro ids. Israel J. Ma th. 167 , 193-2 25 (2008). [111] H. Halvorson: Algebraic Quantum Field Theo ry . In: J . Butterﬁeld and J. E arman (eds.): Handb o ok of the Ph ilosophy of Physi cs . Elsevier , 2 006. [112] D. Handelman: Repr esent ation rings as in v ar iants for compact g roups and limit ratio theo- rems for them. Int. J. Math. 4 , 5 9 -88 (1993). [113] R. H¨ aring - Oldenburg: Rec o nstruction of w eak quas i Hopf a lgebras. J. Alg. 194 , 14-35 (199 7). [114] F. Hausser, F r ank; F. Nill: Doubles of quasi-quantum groups. Comm. Math. Phys. 199 , 547-5 89 (199 9). [115] T. Ha yashi: F a ce algebra s. I. A genera lization o f quantum g roup theory . J. Math. So c. Japan 50 , 293 –315 (1 998). [116] T. Hay ashi: A canonical T annak a duality for ﬁnite semisimple tensor ca tegories. arXiv: math.Q A/9904073 . 49 [117] M. Hazewinkel, W. Hess e link , D. Siersma , F. D. V eldk amp: The ubiquit y of Coxeter-Dynkin diagrams (an introduction to the A − D − E problem). Nieu w Arch. Wisk. 25 , 257-30 7 (197 7). [118] E . Hewitt, K. A. Ros s: Abstr act harmonic analysis , V ol. I I. Spring er V erlag, 197 0. [119] F. Hiai, M. Izumi: Amenability and strong amenability for fusion algebra s with a pplica tions to subfactor theory . Int. J. Math. 9 , 669- 722 (1998). [120] Seung -mo on Hong, E. Rowell, Z. W ang : On exotic mo dular tensor categor ies. Commun. Contemp. Math. 10 , Suppl. 1, 1 049-10 74 (2008). [121] Y.-Z . Huang: V ertex oper ator algebra s , the V er linde conjecture, and modula r tensor cate- gories. Pr o c. Natl. Acad. Sci. USA 102 , 535 2-5356 (20 05). [122] M. Izumi; H. Ko saki, Hideki: On a subfactor analog ue o f the second cohomolo g y . Rev. Math. Phys. 14 , 733 - 757 (20 02). [123] G. James, M. Liebeck: R epr esentations and char acters of gr oups . Cambridge Univ ersity Press, 1993, 20 01. [124] J . C. Ja n tzen: L e ct u r es on qu antum gr oups . American Mathema tica l So ciety , 199 6. [125] M. Jimbo: A q -diﬀer e nce analogue of U ( g ) and the Y ang -Baxter equation. Lett. Math. P h ys. 10 , 6 3 -69 (1985 ). A q -analo g ue of U ( g l ( N + 1)), Hec ke a lgebra and the Y ang -Baxter equation. Lett. Math. Phys. 11 , 247 -245 (19 86). [126] V. F. R. Jones: Index for subfactor s. Inv en t. Math. 72 , 1-2 5 (1983). [127] V. F. R. Jones: A p olynomial inv ar iant for knots via von Neumann algebras . Bull. Amer . Math. So c. 1 2 , 103- 1 11 (1985). [128] A. Joyal, R. Str e et: Braided monoidal ca tegories. Macq uarie Mathematics Rep orts No s. 85006 7, Dec. 19 85 ( http: //www. math.mq.edu.au/ ∼ street/BMC850067.pdf ) and 86008 1, Nov. 1 986 ( http: //www. maths.mq.edu.au/ ∼ street/JS1.pdf ). [129] A. Joy al, R. Street: The geometry of tensor calculus. I. Adv. Math. 88 , 55-11 2 (19 91). [130] A. Joy al, R. Street: T ortile Y ang- Baxter o per ators in tensor categories. J. Pure Appl. Alg. 71 , 43- 51 (1 991). [131] A. Joy al, R. Street: An introduction to T a nnak a dua lit y and quant um gr oups. In: Pro ceedings of the confer ence on category theor y , Como 19 90. Lecture notes in mathema tics 1488 , 413– 492 (199 1). [132] A. Joy al, R. Street: Braided tensor categories. Adv. Ma th. 102 , 2 0-78 (1993). [133] A. Joyal, R. Street: The catego ry of representations of the general linea r g roups over a ﬁnite ﬁeld. J. Alg. 176 , 908–9 46 (199 5) [134] A. Joy al, R. Stree t, D. V er it y: T r aced monoida l categories. Ma th. Pr o c. Cambridge Philos. So c. 119 , 4 4 7–468 (1996 ). [135] M. Kar o ubi: K - the ory. An intr o duction . Springer V e r lag, 19 78. [136] Y. Kashina, Y. So mmerh¨ auser , Y. Zhu : On higher F r obe nius -Sch ur indicators. Mem. Amer. Math. So c. 1 81 (2006 ), no . 855 . [137] C. Kass el: Q u antum gr oups . Springer V erla g, 1995. [138] C. Kass el, M. Rosso and V. T uraev: Q uantum gr ou ps and knot invariants . SMF, 1997 . [139] C. Kassel, V. G. T uraev: Double constr uction fo r monoidal ca tegories. Acta Math. 1 75 , 1- 48 (1995). [140] C. Kassel, V. G. T uraev : Chord diagr am in v a r iants of tangles and gr aphs. Duk e Math. J. 92 , 497-5 52 (199 8). [141] C. Kass el, V. G. T ura e v : Br aid gr oups . Spr inger V er lag, 200 8. 50 [142] L. H. Kauﬀman: An in v a riant of reg ular isotopy . T rans . Amer. Math. Soc. 318 , 417 - 471 (1990). [143] Y. Kawahigashi, R. Longo, M. M ¨ uger: Multi-interv al subfactors and mo dula rity of represen- tations in conformal ﬁeld theory . Comm. Math. Phys. 219 , 631 -669 (2 001). [144] D. Kazhdan, G. Lusztig: T ensor str uc tur es arising fro m aﬃne Lie a lgebras I-IV. J . Amer. Math. So c. 6 , 905 -947, 9 49-10 11 (199 3 ); 7 , 335 –381, 383–45 3. (1994 ). [145] D. Kazhdan a nd H. W enzl, Reconstructing tens or ca tegories. Adv. Sov. Math. 16 (1993) 111–1 36. [146] G. M. Ke lly : On Mac Lane’s conditions for coher ence of na tural asso cia tivities, commutativ- ities, etc. J. Algebr a 1 , 39 7–402 (1964). [147] G. M. K elly: T ensor pro ducts in categ ories. J. Algebr a 2 , 15 –37 (1965). [148] G. M. Kelly , M. L. L aplaza: Coherence for compact clo s ed categories. J . Pur e Appl. Algebra 19 , 193 –213 (1 980). [149] G. M. Kelly , S. Mac Lane : Co herence in clo sed ca tegories. J . Pur e Appl. Algebr a 1 , 9 7–140 (1971) and erratum on p. 219. [150] G. M. Kelly , R. Street: Review of the ele ments of 2-catego r ies. Lecture Notes in Mathema tics 420, pp. 75-103 . Springer V er lag, 1974 . [151] T. Ker ler, V. V. Ly uba shenko: Non-semisimple top olo gic al qu ant um ﬁeld the ories for 3- manifolds with c orners . Spring er V erla g, 200 1. [152] A. A. Kirillov Jr. On an inner pro duct in mo dular tens o r categor ies. J . Amer. Math. So c. 9 , 11 35-11 6 9 (1996). On inner pro duct in mo dular tensor catego r ies. I I. Inner pro duct o n conformal blo c ks and aﬃne inner pro duct identities. Adv. Theor. Math. Phys. 2 , 155-1 8 0 (1998). [153] A. Kirillov Jr .: Mo dular categories and o rbifold mo dels I & I I. Comm. Math. Phys. 229 , 309-3 35 (200 2) & mat h.QA/0 110221 . [154] A. Kirillov Jr.: On G -eq uiv aria n t mo dular c a tegories. math. QA/040 1119 . [155] A. Kir illov Jr., V. Os tr ik: On q-analog of McKay corres p ondence and ADE classiﬁca tion of sl (2) conformal ﬁeld theories. Adv. Math. 171 , 183 -227 (20 02). [156] J . Ko ck: F r ob enius algebr as and 2D top olo gic al quantu m ﬁeld the ories . Cambridge Universit y Press, 20 04. [157] L. Kong, I. Runk el: Morita classes o f algebras in mo dula r tenso r categ ories. Adv. Math. 219 , 1548 -1576 (2008 ). Car dy alge br as and sewing c o nstraints. I. Comm. Math. Phys. 29 2 , 871-9 12 (200 9). [158] M. Kre ˘ ın: A pr inciple of dualit y for bicompact groups and quadr a tic blo ck alg ebras. Doklady Ak ad. Nauk SSSR (N.S.) 69 , 725-7 2 8 (19 49). [159] G. Kup erb erg: Involutory Hopf algebra s a nd 3-ma nifo ld in v a riants. Intern. J. Math. 2 , 41 -66 (1991). [160] G. Kup erb erg: Spider s for ra nk 2 Lie a lgebras. Comm. Ma th. Phys. 180 , 10 9-151 (1996). [161] G. Kupe rbe r g: Finite, connected, semisimple, rigid tensor categories are linea r. Math. Res. Lett. 10 , 411-42 1 (20 03). [162] S. Lack: A 2-catego ries companion. In [13]. ( mat h/07025 35 ) [163] A. D. Lauda : F ro b enius algebras and ambid extrous adjunctions. Theor y Appl. Categ. 1 6 , 84-12 2 (20 06). [164] G. I. Lehr er: A survey of Hecke algebras a nd the Artin braid groups. In: J. S. Birma n, A. Libgob er (eds.): Br aids , Contemp. Math. 78 , 3 65-38 5 (1988). 51 [165] F. Lesieur: Me asur e d qu antum gr oup oids . M ´ em. Soc. Math. F r. (N.S.) 109 (2007 ). M. Eno ck: Me asure d quantum gr oup oid s in action . M´ e m. So c. Math. F r. (N.S.) 1 14 (2008). [166] M. Levine: Mixe d motives . American Mathematical So ciet y , 19 98. [167] W. B. R. Lick orish: An intr o du ct ion to knot the ory . Spring e r V erlag , 1997 . [168] T. Loke: Op er ator algebr as and c onformal ﬁeld the ory of the discr ete series r epr esentations of Diﬀ( S 1 ), PhD thesis Cambridge (UK), 19 9 4. [169] R. Longo : Index of subfactors a nd statistics of quan tum ﬁelds I & II. Comm. Ma th. Phys. 126 , 21 7-247 (1989) & 130 , 285 -309 (1 9 90). [170] R. Longo: A duality for Ho pf algebr as and for subfactors I. Comm. Math. Ph ys. 159 , 1 33-150 (1994). [171] R. Longo, K.- H. Rehren: Nets of subfactor s. Rev. Math. Phys. 7 , 567- 597 (1995). [172] R. Longo, J. E. Ro b erts : A theory of dimension. K- Theory 11 , 10 3-159 (1997). [173] G. Lusztig: Int r o duction to Quant um Gr oups . Bir kh¨ auser V erlag , 1993 . [174] V. V. Lyubashenko: T angles and Hopf algebr as in braided categories . J . Pur e Appl. Alg. 9 8 , 245-2 78 (199 5). [175] V. V. Lyubashenk o: Mo dular transformations for tensor ca tegories. J. Pure Appl. Alg. 98 , 279-3 27 (199 5). [176] G. Mack, V. Schomerus: Quas i Hopf quantum s y mmetry in quantum theor y . Nucl. Phys. B370 , 18 5-230 (1992). [177] S. Mac Lane: Cohomolog y theory o f ab elian groups. Pro ceedings of the ICM 195 0, vol. I I, p. 8-14. [178] S. Mac Lane: Natural asso ciativity and commutativit y . Rice Univ. Studies 49 , 28-46 (1963). [179] S. Mac La ne: Categorica l algebra. Bull. Amer. Math. Soc. 71 , 40- 106 (1965). [180] S. Mac La ne: Cate gories for the working mathematician . 2nd ed. Spr inger V erlag, 1998. [181] S. Ma jid: Representations, duals a nd quantum doubles o f monoida l catego r ies. Rend. Circ. Mat. Palermo Suppl. 26 , 197 - 206 (19 91). [182] S. Ma jid: Reco nstruction theor ems and ratio nal confor mal ﬁeld theories. Internat. J. Mo dern Phys. A 6 , 43 59–43 74 (1991). [183] S. Ma jid: F oun dations of quantum gr oup the ory . Cambridge Universit y Press, 19 9 5. [184] G. Ma lts inio tis: Group o ¨ ıdes qua n tiques. C. R. Acad. Sci. Paris S´ er. I Math. 314 , 2 4 9–252 (1992). [185] G. Maltsinio tis : T races dans les cat´ egor ies mono ¨ ıdales, dualit´ e et ca t´ eg ories mo no ¨ ıdales ﬁbr¨ ees. Cah. T op ol. G´ eom. Diﬀ. Cat´ eg. 36 , 1 95–28 8 (1995 ). [186] G. Ma s baum, J. D. Rob e r ts: A simple proo f of in tegrality of quantum inv ar ia n ts at prime ro ots of unity . Math. P ro c. Cambr. P hilos. So c. 121 , 443-4 54 (199 7). [187] G. Masbaum, H. W enzl: In tegral mo dular categor ies and in tegrality of quantum in v a riants at ro ots of unit y of prime order. J. Reine Angew. Math. 505 , 209–2 35 (199 8). [188] J . R. McMullen: O n the dual ob ject of a compact connected group. Math. Z . 185 , 53 9 -552 (1984). [189] G. Mo or e, N. Seib erg : Classica l and quan tum conformal ﬁeld theor y . Comm. Math. P h ys. 123 , 17 7-254 (1989). [190] M. M¨ uger: Ga lois theory for bra ided tensor categorie s and the mo dular closure. Adv. Math. 150 , 15 1-201 (2000). [191] M. M¨ ug er: F ro m subfactors to ca tegories and top ology I. F rob enius algebr as in and Morita equiv a lence of tensor categor ie s. J. Pure Appl. Alg. 1 8 0 , 81-1 57 (2 0 03). 52 [192] M. M ¨ ug e r : F rom subfactors to categories a nd top olog y I I. The quantum double of tenso r categorie s and subfactors. J. Pure Appl. Alg. 180 , 159-2 1 9 (20 03). [193] M. M ¨ uger, On the structure o f modular categories . P ro c. Lond. Math. So c. 87 , 2 91-30 8 (2003). [194] M. M¨ ug er: Galois extensio ns of braided tensor categ ories a nd br aided crossed G-catego ries. J. Alg. 277 , 25 6 -281 (2 004). [195] M. M ¨ uger : On the ce n ter of a compact g roup. In tern. Math. Res. No t. 2004:51, 2751-27 56 (2004). [196] M. M ¨ uge r : Conformal orbifold theories and braided crossed G-c a tegories. Comm. Math. P hys. 260 , 72 7-762 (2005). [197] M. M ¨ ug e r: Abstract dua lit y for sy mmetric tensor ∗ -categ ories. Appendix to [111], p. 8 65-92 2 . [198] M. M¨ uger: O n Super selection theor y of quantum ﬁelds in low dimensions. Invited lecture at the XVIth In ternationa l Congress o n Mathematical Ph ysics. Pra gue, August 3-8, 2009 . T o app ear in the pro ceedings. ( arXiv:0 909.253 7 .) [199] M. M ¨ uger : On the structure of bra ided cr o ssed G-ca tegories. Appendix to V. T ura ev: Ho- motopy TQFTs . E urop ean Mathematical So ciety , to app ear 2010. [200] M. M ¨ ug er, J. E. Robe r ts, L. T uset: Repre sent ations of algebraic quant um gro ups and recon- struction theor ems for tensor categor ies. Alg. Repres. The o r. 7 , 5 1 7-573 (2004 ). [201] M. M¨ uger, L. T uset: Monoids, embedding functors a nd quantum groups. In t. J. Math. 19 , 1-31 (200 8). [202] M. M ¨ uger : Iso categor ical co mpa ct groups. In preparatio n. [203] D. Naidu: Catego rical Morita equiv alence for group-theo retical categories. Co mm. Alg. 35 , 3544- 3565 (200 7). [204] D. Naidu: Crossed p o in ted categories and their equiv a riantizations. Preprint, 20 09. Av a ilable from htt p://ww w.math.ta mu.edu/ ∼ dnaidu/publications.html . [205] D. Naidu, D. Nikshych: Lagrang ian sub catego ries and br a ided tensor equiv a le nces of twisted quantum doubles of ﬁnite groups. Comm. Math. P h ys. 27 9 , 845- 8 72 (200 8). [206] D. Naidu, D. Niksh ych, S. Withersp o on: F usion sub c ategories of repre s en tation ca teg ories of t wisted quantum doubles of ﬁnite groups. Int. Math. Res . Not. 22 , 4183-4 219(200 9). [207] S. Natale: F rob enius-Sch ur indicators fo r a c lass of fusion categor ies. Pac. J. Math. 221 , 353-3 77 (200 5). [208] S. Natale: O n the exp onent o f tensor categor ies coming from ﬁnite gro ups. Is r ael J. Math. 162 , 25 3-273 (2007). [209] S. Nes hvey ev, L . T uset: Notes on the Kazhda n-Lusztig theo r em o n e q uiv alence of the Drinfeld category and the categ ory of U q ( g )-mo dules. arXiv:0 711.43 02 . [210] S.- H. Ng, P . Schauenburg: F rob enius-Sch ur indica tors and exp onents of spherical categ ories. Adv. Math. 211 , 34–7 1 (2007). [211] S.- H. Ng, P . Schauenburg: Higher F ro benius - Sc hur Indica tors for Pivotal Categories. Con- temp. Math. 441 , 63 -90 (2007). [212] S.- H. Ng, P . Schauen burg: Congruence subgr oups and generalized F rob enius-Sch ur indicator s . arXiv: 0806.2 493 . [213] D. Nikshyc h: On the structure o f weak Hopf algebr as. Adv. Math. 1 70 , 257 - 286 (20 02). [214] D. Nikshyc h, V. T ura ev, L. V ainerman: In v a r iants of Knots and 3-manifolds from Quantum Group oids. T op ol. Appl. 127 , 91- 123 (2003). 53 [215] D. Nikshyc h, L. V ainerman: A c haracter ization o f depth 2 subfactors of I I 1 factors. J . F unct. Anal. 17 1 , 278-3 07 (2000). A Galois cor resp ondence for I I 1 factors a nd quantum group oids. J. F unct. Anal. 178 , 11 3-142 (2000 ). [216] A. Ocnea n u: Quantized gro ups, string alg ebras and Galo is theor y for algebr as. In: Opera tor algebras and applications, V ol. 2 , pp. 11 9-172, Cambridge Univ. Pres s, 1988. [217] A. Ocnean u (notes by Y. Kawahigashi): Quant um symmetry , diﬀeren tial geometr y of ﬁnite graphs and cla ssiﬁcation o f subfactors. Lectur es given at T o kyo Univ. 1990. [218] A. Ocnea n u: An in v a riant co upling betw een 3-manifolds and subfacto rs, with c onnections to top ological and co nformal quantum ﬁeld theor y . Unpublished manuscript. Ca. 1 991. [219] A. Ocneanu: Chirality for op era tor algebras. In : H. Araki, Y. Kaw ahigashi and H. Kosak i (eds.): Subfactors . pp 39- 63. W or ld Scientiﬁc Publ., 19 94. [220] A. Ocneanu: Oper ator algebras, to p olo gy and subgroups of quantum symmetry— construction of subgro ups of quan tum groups. In: T aniguchi Conferenc e on Mathema tics Nara 199 8, 235–26 3, Adv. Stud. Pur e Math., 31, Math. So c. Ja pan, T okyo, 2 001. [221] A. Ocnea nu: The classiﬁcatio n of s ubgroups of qua n tum SU( N ). In : Qua n tum symmetries in theoretical physics a nd ma thematics (Barilo che, 200 0), 13 3–159 , Contemp. Math. 294 , 2 002. [222] V. O strik: Mo dule categorie s, weak Hopf alg ebras and mo dular inv ariants. T ra nsf. Groups 8 , 177-2 06 (200 3). [223] V. Ostr ik : Mo dule categories ov er the Drinfeld double of a ﬁnite group. In t. Math. Res. Not. 2003:2 7, 1 5 07–15 20 [224] V. Ostrik: F usion categories of rank 2. Math. Res. Lett. 10 , 177-1 83 (200 3 ). [225] V. Ostrik: P re-mo dular categorie s of rank 3. Moscow Math. J. 8 , 111- 118 (2008). [226] F. Panaite, F. V an Oys taeyen: Quasi- Hopf algebr as and the centre of a tensor categ ory . In: S. Caenep eel et al. (eds.): Hopf algebr as and qu ant um gr oups . Lect. Notes Pur e Appl. Math. 209 , 22 1-235 (2000). [227] B . Pareigis: On braiding and dyslexia. J. Algebr a 1 71 , 413 - 425 (19 95). [228] C. Pinza ri: The repres en tation ca tegory of the W oronowicz quantum gr oup S U ( d ) as a braided tensor C ∗ -categor y . Int. J. Math. 18 , 113 -136 (2 007). [229] S. Popa, D. Shlyakhtenk o: Univ ersa l proper ties of L ( F ∞ ) in subfactor theory . Acta Math. 191 , 22 5-257 (2003). [230] M. v an der Put, M. F. Singer: Galois the ory of line ar diﬀer ential e qu ations . Springer V erlag , 2003. [231] F. Q uinn: Lectur es on axio matic to po lo gical quan tum ﬁeld theo ry . In: D. S. F ree d and K. K. Uhlen b eck (eds.): Ge ometry and quantum ﬁeld t he ory. IAS/Park Cit y Mathematics Series, 1. American Mathematical So ciety , 1995. [232] K .-H. Rehr en: Br aid gr oup statistics and their sup erselection rules. In : D. Kastler (ed.): The algebr aic the ory of sup ersele ction se ctors. In tr o duction and r e c ent r esults . W orld Scientiﬁc, 1990. [233] K .-H. Rehren: Lo ca lit y and mo dular inv aria nce in 2D conformal QFT. Fields Ins t. Commu n. 20 , 297 -319 (2 001). [234] N. Y u. Reshetikhin, V. G. T ur a ev: Ribb on gra phs and their in v a riants derived from quantum groups. Comm. Math. Phys. 127 , 1– 26 (1990). [235] N. Y u. Reshetikhin, V. G. T uraev: Inv aria n ts of 3 -manifolds via link p o lynomials a nd quan- tum gro ups. In ven t. Math. 103 , 547-5 98 (1 991). [236] E . Ro well: O n a family of non-unitar izable r ibbon categories . Math. Z. 250 , 745-7 7 4 (2005). 54 [237] E . Ro well, R. Stong, Z. W ang: O n classiﬁcation of mo dular tens o r categorie s . Comm. Ma th. Phys. 292 , 343-3 8 9 (20 09). [238] N. Saav eda Riv ano: Cat ´ egories T annakiennes . Springer V erlag, 1 972. [239] P . Sc hauenburg: T a nnak a dualit y for a rbitrary Hopf algebra s. Algebra Berich te, 6 6. V erlag Reinhard Fischer, Munich, 1992. [240] P . Schauenb urg: The monoidal cen ter constr uctio n and bimodules . J. P ure Appl. Alg. 15 8 , 325-3 46 (200 1). [241] P . Schauenburg: T urning mono idal categor ies in to strict ones. New Y o rk J. Math. 7 , 257-2 65 (2001). [242] J .-P . Serre: Complex S emisimple Lie Alg ebr as . Springer, 1987. [243] M. C. Shum: T ortile tenso r categories . J . Pure Appl. Alg. 9 3 , 57-11 0 (19 94). [244] J . Siehler: Nea r-gro up categor ies. Algebr. Geom. T op ol. 3 , 719 -775 (2003). [245] Sinh Hong Xuan: Gr-Cat ´ egories . PhD thesis Univ ersit ´ Paris VII , 1 975. [246] Y. Soib e lma n: Meromorphic tensor c ategories, quantum aﬃne and chiral algebr as I. Contemp. Math. 24 8 , 437-4 51 (199 8). [247] Y. Sommerh¨ a user, Y. Zhu: Hopf alge br as and congr uence subgro ups. a rXiv:0 710.07 05 [248] C. Soul´ e: Les v ari´ et´ e s sur le cor ps ` a un ´ el´ ement . Mosc. Math. J. 4 , 217- 244 (20 04). [249] D. S ¸ tefan: The set o f types o f n -dimensional semisimple and cosemisimple Hopf algebras is ﬁnite. J. Alg. 193 , 571-58 0 (1997 ). [250] R. Street: The quantum double and related constructio ns. J. Pure Appl. Alg. 132 , 1 95-206 (1998). [251] R. Street: The mo noidal centre as a limit. Theo ry Appl. Categ. 13 , 184 -190 (2004). [252] K . Szla ch´ anyi: Finite quan tum groupo ids and inclusions o f ﬁnite type. In: R. Longo (ed.): Mathematic al Physics in Ma thematics and Physics: Quantu m and Op er ator Algebr aic As- p e cts . Fields Inst. Co mm un. 20 , 297-3 19 (200 1). [253] D. T ambara: A dua lity for mo dules ov er monoida l categor ies of r epresentations of s emisimple Hopf alg ebras. J. Algebr a 2 41 , 515- 547 (2001). [254] D. T ambara, S. Y amag ami: T enso r ca teg ories with fusion rules of self-duality for ﬁnite ab elian groups. J. Alg . 209 , 692-70 7 (1998 ). [255] T. T a nnak a: ¨ Uber den Dualit¨ atssatz der nicht komm utativen top olog ischen Grupp en. Tˆ oho ku Math. Jo urn. 45 , 1–1 2 (1 939). [256] V. T oledano Laredo: Positive ener g y repr e s en tations of the loo p gr oups of non-simply c o n- nected Lie groups. Co mm. Math. Phys. 207 , 307 -339 (1999). [257] V. T oledano Laredo: Quasi-Coxeter algebra s, Dynkin diagram co homology , and quantum W eyl groups . Int. Math. Res. Pap. IMRP 200 8, Art. ID rpn009, 16 7 pp. [258] T. tom Dieck: Categor ies of ro oted cylinder r ibbons a nd their representations. J. Reine Angew. Math. 494 , 35-6 3 (1998 ). T. tom Diec k, T ammo; R. H¨ aring-O lden burg: Quantum groups and cy linder bra iding. F or um Math. 10 , 619 -639 (19 98). [259] I. T uba, H. W enzl: Hans On braided tensor c ategories of type B C D . J. Reine Angew. Math. 581 , 31 -69 (2005). [260] V. G. T uraev: Op erator inv aria n ts of tangle s , and R -matrices. Math. USSR-Izv. 35 , 4 11–44 4 (1990). [261] V. G. T uraev: Mo dular categor ies and 3- ma nifold inv aria n ts. Int. J. Mod. Phys. B6 , 1 807- 1824 (19 92). [262] V. G. T urae v: Quantum invariants of kn ots and 3-manifolds . W alter de Gruyter, 1994. 55 [263] V. G. T uraev : H omotopy ﬁeld theor y in dimension 3 and cr ossed group-ca tegories. math.G T/0005 291 . [264] V. G. T urae v: Homotopy TQFTs . B irkh¨ auser V erla g, 2010. [265] V. G. T uraev, O. Viro: Sta te sum in v a riants o f 3-manifolds and quantum 6 j -sym b ols. T op ol- ogy 31 , 865-9 02 (199 2). [266] V. G. T ura ev, H. W enzl: Quan tum in v ariants of 3-manifolds ass o c iated with classical simple Lie algebr as. In tern. Jo urn. Math. 4 , 323- 358 (1993). [267] V. G. T uraev , H. W enzl: Semisimple and mo dular c ategories from link inv ariants. Math. Ann. 309 , 411-46 1 (1997 ). [268] K .-H. Ulbrich: On Hopf algebra s and rigid mo noidal catego ries. Is rael J. Math. 72 , 252–25 6 (1990) [269] A. V a n Daele: Multiplier Hopf a lgebras. T rans. Amer. Math. So c. 342 , 917 -932 (1994). [270] A. V a n Daele: An algebra ic fra mew ork for group duality . Adv. Math. 140 , 323-3 66 (1998). [271] E . V as selli: Bundles of C ∗ -categor ies. J. F unct. Anal. 247 , 35 1-377 (2007). [272] E . V asse lli: Gro up bundle duality , inv ariants for cer tain C ∗ -algebra s, and twisted equiv a riant K-theory . In: D. Burghelea, R. Melrose, A. S. Mishc henko, E. V. T ro itsky (eds.): C ∗ -Algebr as and el liptic the ory II , pp. 28 1-295 , Birkh¨ a user V erla g, 200 8 . [273] P . V ogel: Algebraic structure on mo dules o f diagr ams. Pr eprint 19 97. [274] Z . W ang : T op olo gic al quant um c omputation . American Mathematica l So ciety , 201 0. [275] A. W ass e rmann: Op erator a lgebras and conforma l ﬁeld theory I I I. F usion o f po sitive ener g y representations o f S U ( N ) using bo unded op era to rs. Inven t. Ma th. 133 , 467–5 3 8 (1998 ). [276] H. W enzl: C ∗ -T e ns or ca tegories from quantum gro ups. J. Amer. Math. Soc. 11 , 261-2 8 2 (1998). [277] H. W enzl: On tenso r categories of Lie type E N , N 6 = 9. Adv. Ma th. 177 , 6 6-104 (2003 ). [278] N. J. Wildber ger: Finite commutativ e h yp ergro ups and applica tions from group theory to conformal ﬁeld theory . Contemp. Math. 1 83 , 413-4 34 (19 95). (Pro ceedings volume of the conference Applic ations of hyp er gr oups and r elate d me asure algebr as , Seattle, 1 993. [279] S. L. W o ronowicz: Compact matrix pse udo groups. Comm. Math. P h ys. 111 , 613-66 5 (1987). [280] S. L. W oronowicz, T annak a-Krein dua lit y for compact matrix pseudogroups . Twisted SU(N) groups. Inv ent. Math. 93 (19 8 8) 35–76. [281] S. L. W or o nowicz: Compact quan tum groups. In: A. Co nnes, K. Ga w edzki, J. Zinn-Justin (eds.): Sym´ etries quantiques. North-Holla nd, Amsterdam, 1 998. [282] F. Xu: New bra ided endomorphis ms from conformal inclusio ns. Comm. Ma th. Phys. 192 , 349-4 03 (199 8). [283] F. Xu: Some computations in the cyclic pe rmu tations of co mpletely rationa l nets. Comm. Math. P hys. 267 , 75 7-782 (2006). [284] S. Y amagami: C ∗ -tensor categories and free pro duct bimodules. J. F unct. Anal. 197 , 323-3 46 (2003). [285] S. Y amagami: F rob enius alg ebras in tensor catego ries and bimo dule e x tensions. In: G. Janelidze, B. Pareigis, W. Tholen (eds.): Galois t he ory, Hopf algebr as, and semiab elian c at- e gori es . Fields Inst. Commun. 43 , 551- 570 (2004). [286] S. Y a magami: F r o benius duality in C ∗ -tensor c a tegories. J. O per . Th. 52 , 3-20 (20 04). [287] D. N. Y etter: Marko v a lgebras. In: J. S. Birma n, A. Libgob er (eds.): Br aids . Contemp. Math. 78 , 705 -730 (1 988). 56 [288] D. N. Y etter: Qua n tum gro ups and repre s en tations of monoidal ca tegories. Math. Pro c. Cambridge P hilos. So c. 108 , 261-29 0 (19 90). [289] D. N. Y etter: F ramed ta ngles a nd a theorem o f Deligne on braided deformations of tannakian categorie s. Contemp. Math. 134 , 325 -350 (1 992). [290] D. N. Y etter: F un ct orial knot the ory. Cate gories of tangles, c oh er enc e, c ate goric al deforma- tions, and top olo gic al invariants . W orld Scien tiﬁc, 2 001. [291] P . Zito: 2- C ∗ -categor ies with non-simple units. Adv. Math. 210 , 122-1 6 4 (2007 ). 57

Tensor categories: A selective guided tour

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment