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math.CT/0701223 KCL-MTH-07-01 ZMP-HH/2007-03 January 2007 7 0 0 2 THE FUSION ALGEBRA OF BIMODULE CATEGORIES n a J 8 Ju¨rgen Fuchs 1 Ingo Runkel 2 Christoph Schweigert 3 ] T C . h t a m 1 Avdelning fysik, Karlstads Universitet [ Universitetsgatan 5, S–65188 Karlstad 1 2 Department of Mathematics v 3 King’s College London, Strand 2 GB– London WC2R 2LS 2 1 3 Organisationseinheit Mathematik, Universität Hamburg 0 7 Schwerpunkt Algebra und Zahlentheorie 0 Bundesstraße 55, D–20146 Hamburg / h t a m : v i X r a Abstract We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category. Introduction The Grothendieck ring K (C) of a semisimple monoidal category C encodes a considerable amount 0 of information about the structure of C. If C is braided, so that K (C) is commutative, then 0 upon complexification to K0(C)⊗Z almost all of this information gets lost: what is left is just C the number of isomorphism classes of simple objects. In contrast, in the non-braided (but still semisimple) case the complexified Grothendieck ring is no longer necessarily commutative and thus contains as additional information the dimensions of its simple direct summands. The following statement, which is a refined version of an assertion made in [O] as Claim 5.3, determines these dimensions for a particularly interesting class of categories: Theorem O. Let C be a modular tensor category, M a semisimple nondegenerate indecomposable module category over C, and C∗ the category of module endofunctors of M. Then there is an isomorphism M of -algebras C F ∼= MEndC(HomCM∗ (α+(Ui),α−(Uj))) (1) i,j∈I between the complexified Grothendieck ring F =K0(CM∗ )⊗ZC and the endomorphism algebra of the specified space of morphisms in C∗ . M Here I isthe(finite) set ofisomorphism classes of simple objects ofC, U andU arerepresentatives i j of the classes i,j∈I, and α± are the braided-induction functors from C to C∗ . For more details M about the concepts appearing in the Theorem see section 1 below, e.g. the tensor functors α± are given in formulas (5) and (6). In [O] the assertion of Theorem O was formulated with the help of the integers zi,j := dim HomC∗ (α+(Ui),α−(Uj)), (2) C M in terms of which it states that F is isomorphic to the direct sum Mat of full matrix Li,j zi,j algebras of sizes z , i,j∈I. In this form the statement had been established previously for the i,j particular case that the modular tensor category C is a category of endomorphisms of a type III factor (Theorem 6.8 of [BEK]), a result which directly motivated the formulation of the statement in [O]. Indeed, in [O] the additional assumption is made that the quantum dimension of any nonzero object of C is positive, a property that is automatically fulfilled for the categories arising in the framework of [BEK], but is violated in other categories (e.g. those relevant for the so-called non-unitary minimal Virasoro models) of interest in physical applications. In this note we derive the statement in the form of Theorem O, where this positivity require- ment is replaced by the condition that M is nondegenerate. This property, to be explained in detail further below, is satisfied in particular in the situation studied in [BEK]. We present our proof in section 2. As an additional benefit, it provides a concrete expression for the structure constants of the Grothendieck ring of C∗ in terms of certain endomorphisms of the tensor unit of M C, see formulas (29) and (30). Various ingredients needed in the proof are supplied in section 1. In section 3 we outline the particularities of the case that C comes from endomorphisms of a factor [BEK], and describe further relations between Theorem O and structures arising in quantum field theory; this latter part is, necessarily, not self-contained. 2 1 Bimodule categories and Frobenius algebras We start by collecting some pertinent information about the quantities used in the formulation of Theorem O. Modular tensor categories A modular tensor category C in the sense of Theorem O is a semisimple -linear abelian ribbon C category with simple tensor unit, having a finite number of isomorphism classes of simple objects and obeying a certain nondegeneracy condition. Let us explain these qualifications in more detail. A ribbon (or tortile, or balanced rigid braided) category is a rigid braided monoidal category with a ribbon twist, i.e. C is endowed with a tensor product bifunctor ⊗ from C×C to C, with tensor unit 1, and there are fami- lies of (right-)duality morphisms b ∈Hom(1,U⊗U∨), d ∈Hom(U∨⊗U,1), of braiding isomor- U U phisms c ∈Hom(U⊗V,V⊗U), and of twist isomorphisms θ ∈End(U) (U,V ∈Obj(C)) sat- U,V U isfying relations analogous to ribbons in three-space [JS]. (Details can e.g. be found in section 2.1 of [FRS1]; the category of ribbons indeed enjoys a universal property for ribbon categories, see e.g. chapter XIV.5.1 of [K].) A ribbon category is in particular sovereign, i.e. be- sides the right duality there is also a left duality, with evaluation and coevaluation morphisms ˜b ∈Hom(1,∨U⊗U) and d˜ ∈Hom(U⊗∨U,1), such that the two duality functors coincide, i.e. U U ∨U =U∨ and ∨f =f∨∈Hom(V∨,U∨) for all objects U of C and all morphisms f ∈Hom(U,V), U,V ∈Obj(C). Denoting by I the finite set of labels for the isomorphism classes of simple objects of C and by U representatives for those classes, the nondegeneracy condition on C is that the I×I-matrix i s with entries s := tr(c c ) i,j Ui,Uj Uj,Ui (3) ≡ (dUj ⊗d˜Ui)◦[idUj∨⊗(cUi,Uj◦cUj,Ui)⊗idUi∨]◦(˜bUj ⊗bUi) ∈End(1), i,j ∈ I, is invertible. We will identify End(1)= id1 with , so that id1=1∈ and also the morphisms C C C s are just complex numbers, and we agree on I∋0 and U =1. In a sovereign category one has i,j 0 s =s . The (quantum) dimension of an object U of a sovereign category is the trace over its i,j j,i identity endomorphism, dim(U)=tr(idU):=dU ◦(idU ⊗idU∨)◦˜bU, in particular dim(Ui)=s0,i. Module and bimodule categories The notion of a module category is a categorification of the one of a module over a ring. That is, an abelian category M is a (right) module category over a monoidal category C iff there exists an exact bifunctor ⊠ : M×C → M (4) together with functorial associativity and unit isomorphisms (M ⊠U)⊠V ∼=M ⊠(U ⊗V) and M ⊠1∼=1 for M ∈Obj(M), U,V ∈Obj(C), which satisfy appropriate pentagon and triangle coherence identities. The latter involve also the corresponding coherence isomorphisms of C and are analogous to the identities that by definition of ⊗ are obeyed by the coherence isomorphisms of C alone. (Any tensor category C is a module category over itself, much like as a ring is a module over itself.) A module category is called indecomposable iff it is not equivalent to a direct sum of two nontrivial module categories. 3 Amodule functor between twomodulecategoriesM andM overthesamemonoidalcategory 1 2 C is a functor F: M →M together with functorial morphisms F(M ⊠ U)→F(M)⊠ U for 1 2 1 2 M ∈Obj(M ) and U ∈Obj(C), obeying again appropriate pentagon and triangle identities. For 1 details see section 2.3 of [O]. The composition of two module functors is again a module functor. Thus the category C∗ :=Fun (M,M) of module endofunctors of a right module category M M C over a monoidal category C is monoidal, and the action of these endofunctors on M turns M into a left module category over C∗ . M If C has a braiding c then two functors α± from C to C∗ are of particular interest. They are M defined by assigning to U ∈Obj(C) the endofunctors α+(U) = α−(U) := ⊠U (5) of M. In order that α± are indeed functors to C∗ , we have to make α±(U) into module functors, M i.e. to specify morphisms α±(U)(M⊠V)→(α±(U)(M))⊠V, i.e. M ⊠(V ⊗U) ∼= (M ⊠V)⊠U −→ (M ⊠U)⊠V ∼= M ⊠(U ⊗V) (6) for M ∈Obj(M) and U,V ∈Obj(C); this is achieved by using the morphisms1 id ⊗c for α+ M V,U and id ⊗c−1 for α−, respectively. M U,V The functors α± defined this way aremonoidal; we callthem braided-induction functors. These functors, originally referred to as alpha induction functors, were implicitly introduced [LR] and heavily used [X, BE1, BEK, BE2] in the context of subfactors, and were interpreted as monoidal functors with values in a category of module endofunctors in [O]. Further, by setting (U,X,V) 7→ βǫǫ′(U,V) := αǫ(U)◦X ◦αǫ′(V) (7) for U,V ∈Obj(C) and X∈Obj(C∗ ), for any choice of signs ǫ,ǫ′∈{±} one obtains a functor M βǫǫ′: C×C∗ ×C→C∗ ; it can be complemented by two sets of associativity constraints (for the M M left and right action of C), both obeying separately a pentagon constraint and an additional mixed constraint that expresses the commutativity of the left and right action of C. This turns C∗ into M a bimodule category over C. Frobenius algebras Given a monoidal category C, the structure of a module category over C on an abelian category M is equivalent toa monoidal functor fromC to thecategory of endofunctors ofM. If C issemisimple rigid monoidal, with simple tensor unit and with a finite number of isomorphism classes of simple objects, and M is indecomposable and semisimple, then it follows [O, Theorem3.1] that M is equivalent to the category C of (left) A-modules in C for some algebra A in C. By a similar A reasoning C∗ is then equivalent to the category C of A-bimodules in C. The monoidal product M A|A of C is the tensor product over A. The algebra A such that C ≃M is determined uniquely up A|A A to Morita equivalence; it can be constructed as the internal End End(M) for any M ∈Obj(M) [O]. Recall that a (unital, associative) algebra A=(A,m,η) in a (strict) monoidal category C consists of an object A∈Obj(C) and morphisms m∈Hom(A⊗A,A) and η∈Hom(1,A) satisfying m◦(m⊗id )=m◦(id ⊗m) and m◦(η⊗id )=id =m◦(id ⊗η). If M is indecomposable A A A A A 1 suppressing, for brevity, the mixed associativity morphisms 4 then the category C∗ has simple tensor unit (it is thus a fusion category in the sense of [ENO]). M In terms of the algebra A, this property means that A is simple as an object of the category C A|A of A-bimodules in C; such algebras A are called simple. A left A-module is a pair M =(M˙ ,ρ) consisting of an object M˙ ∈Obj(C) and of a morphism ρ∈Hom(A⊗M˙ ,M˙ ) that satisfies ρ◦(id ⊗ρ)=ρ◦(m⊗id ) and ρ◦(η⊗id )=id . A right A M˙ M˙ M˙ A-module (M˙ ,̺), ̺∈Hom(M˙ ,M˙ ⊗A), is defined analogously, and an A-bimodule is a triple X=(X˙,ρ,ρ ) such that (X˙,ρ) is a left A-module, (X˙,ρ ) is a right A-module and the left and l r l r right A-actions commute. The morphism space Hom (M,N) in C consists of those morphisms in A A Hom(M˙ ,N˙) which commute with the left A-action, and an analogous property characterizes the morphism space Hom (X,Y) in C . A|A A|A Of interest to us is a particular class of algebras in C, the symmetric special Frobenius algebras. A Frobenius algebra in a monoidal category C is a quintuple A=(A,m,η,∆,ε) such that (A,m,η) is an algebra, (A,∆,ε) is a coalgebra, and the coproduct ∆ is a morphism of A-bimodules.2 A Frobenius algebra A in a sovereign monoidal category is symmetric iff the morphism (dA⊗idA)◦[idA∨⊗(∆◦η◦ε◦m)]◦(˜bA⊗idA)∈End(A) (which for any Frobenius algebra A is an algebra automorphism) equals id . A Frobenius algebra A is special iff ∆ is a right-inverse A of the product m and the counit ε is a left-inverse of the unit η, up to nonzero scalars. For a special Frobenius algebra one has dim(A)6=0, and one can normalize the counit in such a way that m◦∆=idA and η◦ε=dim(A)id1; below we assume that this normalization has been chosen. The structure and representation theory of symmetric special Frobenius algebras have been studied e.g. in [KO, FRS1, FrFRS1, FRS2]. The braided-induction functors α± which exist when C has a braiding can be described in terms of the algebra A as the functors α±: C→C ≃C∗ A A|A M that associate to U ∈Obj(C) the following A-bimodules α±(U): the underlying object is A⊗U, A the left module structure is the one of an induced left module, and the right module structure is given by the one of an induced right module composed with a braiding c −1 for α+ and c for U,A A A,U α−, respectively [O, FrFRS1]. In terms of A the numbers z in (2) are given by z =z(A) with A ij ij ij z(A) := dim Hom (α+(U ),α−(U )). (8) i,j A|A A i A j C Let us also mention that for modular C every A-bimodule can be obtained as a retract of a tensor product α+(U)⊗ α−(V) of a suitable pair of α+- and α−-induced bimodules (see the conjecture A A A A A [O, Claim5.2] and its proof in [FrFRS2]), and that the matrix z(A) is a permutation matrix iff the functors α± are monoidal equivalences, i.e. iff A is Azumaya [FRS2]. A Nondegeneracy To cover the terms used in the formulation of Theorem O we need to introduce one further notion, the one of nondegeneracy. This is done in the following Definition. (i) An algebra A in a sovereign tensor category is called nondegenerate iff the morphism [(ε♮◦m)⊗idA∨]◦(idA⊗bA) ∈ Hom(A,A∨) (9) with ε♮:=dA◦(idA∨⊗m)◦(˜bA⊗idA)∈Hom(A,1) is an isomorphism. (ii) A semisimple indecomposable module category M over a semisimple sovereign tensor category 2 Inthe classicalcase ofFrobenius algebrasinthe categoryofvectorspacesovera field, the Frobenius property can be formulated in several other equivalent ways. The one given here does not require any further structure on C beyond monoidality. Also note that neither ∆ nor the counit ε is required to be an algebra morphism. 5 C that has simple tensor unit and a finite number of isomorphism classes of simple objects is called nondegenerate iff there exists a nondegenerate algebra A in C such that M≃C . A It is not difficult to see that the property of an algebra to be nondegenerate is preserved under Morita equivalence. But it is at present not evident to ushow to give adefinition of nondegeneracy for module categories which does not make direct reference to the corresponding (Morita class of) algebra(s); we plan to come back to this problem elsewhere. That such a formulation must exist is actually the motivation for introducing this terminology. In contrast, for algebras nondegeneracy is not a new concept, owing to Lemma 1 An algebra in a sovereign tensor category is nondegenerate iff it is symmetric special Frobenius. Proof. That a nondegenerate algebra in a sovereign tensor category is symmetric special Frobenius has beenshowninlemma3.12of[FRS1]. TheconverseholdsbecauseforasymmetricspecialFrobenius algebra one has ε♮=ε, and as a consequence (dA⊗idA)◦(idA∨ ⊗(∆◦η))∈Hom(A∨,A) is inverse 2 to the morphism (9). Fusion rules We denote by [U] the isomorphism class of an object U. If an abelian category C is rigid monoidal, thenthetensorproductbifunctorisexact, sothattheGrothendieckgroupK (C)hasanaturalring 0 structure given by [U]∗[V]:=[U ⊗V]. A distinguished basis of K (C) is given by the isomorphism 0 classes of simple objects of C; in this basis the structure constants are non-negative integers. If M is a module category over C, then the Grothendieck group K (M) is naturally a K (C)-module, 0 0 in fact a based module over the based ring K (C). 0 From now on, unless noted otherwise, C will stand for a modular tensor category, M for a semisimple nondegenerate indecomposable module category over C, and C∗ =Fun (M,M) will M C beregardedasabimodulecategoryoverC viathefunctorβ+−. ThenthecategoryC∗ issemisimple M -linear abelian rigid monoidal and has finite-dimensional morphism spaces and a finite number C of isomorphism classes of simple objects. For simplicity of notation we also tacitly take C to be strict monoidal; in the non-strict case the relevant coherence isomorphisms have to be inserted at appropriate places, but all statements about Grothendieck rings remain unaltered. Analogously as K (C) also the Grothendieck group K (C∗ ) has a natural ring structure. How- 0 0 M ever, C∗ is not, in general, braided, and hence, unlike K (C), the ring K (C∗ ) is in general not M 0 0 M commutative. The complexified Grothendieck ring F := K0(CM∗ )⊗Z C (10) of C∗ is a finite-dimensional semisimple associative -algebra, and hence a direct sum of full matMrix algebras, F ∼= Mat for some finite indCex set P, with Mat the algebra of n×n- Lp∈P np n matrices with complex entries. We denote the product in F by ∗. The algebra F has a standard basis {e }, F ∼= np e , with products e ∗e =δ δ e . The elements e := pn;pαβe areLthpe∈PpLrimαi,tβi=v1eCidepm;αβpotents projectingpo;αnβto thq;eγδsimpp,lqe sβu,γmmp;αaδnds Mat of F. p Pα=1 p;αα np Given any other basis {x |κ∈K}, there is a basis transformation κ np x = up;αβe , e = (u−1)κ x , (11) κ X X κ p;αβ p;ab X p;αβ κ p∈P α,β=1 κ∈K 6 satisfying Pp,α,β(u−1)κp;′αβupκ;αβ=δκ,κ′ and Pκuqκ;αβ(u−1)κp;γδ=δp,qδα,γ δβ,δ, so that the structure constants of F in the basis {x } can be written as κ np N κ′′ = up;αβup;βγ(u−1)κ′′ . (12) κκ′ X X κ κ′ p;αγ p∈P α,β,γ=1 For the particular case that {x } is the distinguished basis of the underlying ring given by the κ isomorphism classes [X ] of simple objects of C∗ , it has become customary in various contexts κ M to refer to the structure constants N κ′′ of F (or also to the algebra F, or to the Grothendieck κκ′ ring itself) as the fusion rules of the category under study. The formula (12) then constitutes a block-diagonalization of the fusion rules of C∗ . Also note that if F is commutative, then (12) M reduces to N κ′′= up up (u−1)κ′′, which is sometimes referred to as the Verlinde formula. κκ′ Pp∈P κ κ′ p In terms of these ingredients, the assertion of Theorem O amounts to the claim that the index set P and the dimensions n2 of the simple summands of F are given by p P = {(i,j)∈I×I|z 6=0} and n = z (13) i,j (i,j) i,j with the integers z the dimensions defined in formula (2). i,j We will establish the equalities (13) by finding an explicit expression for the matrix elements up;αβ≡u(i,j);αβ of the basis transformation (11) for the case that the basis {x |κ∈K} is the κ κ κ distinguished basis of isomorphism classes [X ] of the simple objects. κ 2 The structure of the algebra F The considerations above show in particular that a nondegenerate semisimple indecomposable module category M over a modular tensor category C is equivalent, as a module category, to the category C for an appropriate simple symmetric special Frobenius algebra A in C. Moreover, A the category C∗ of module endofunctors is equivalent to C as a monoidal category (and as a M A|A bimodule category over C and left module category over M). We may thus restate Theorem O as Theorem O′. Let C be a modular tensor category and A a simple symmetric special Frobenius algebra in C. Then the complexified Grothendieck ring F =K0(CA|A)⊗Z and the endomorphism algebra C E := End (Hom (α+(U ),α−(U )). (14) M A|A i j C i,j∈I are isomorphic as -algebras. C We will prove Theorem O′ by constructing an isomorphism from F to E. A map Φ from F to E To simplify some of the expressions appearing below, we replace j∈I in (2) by , defined as the unique label in I such that U ∼=U∨, and instead of the spaces Hom (α+(U),α−(V∨)) we  j A|A prefer to work with the isomorphic spaces Hom (U ⊗+A⊗−V,A), where for any A-bimodule A|A X=(X˙,ρ,ρ ) the A-bimodules U ⊗+X and X ⊗−V are defined as U ⊗+X :=(U⊗X˙,(id ⊗ρ)◦ l r U l 7 (c −1⊗id ),id ⊗ρ ) and as X ⊗−V :=(X˙⊗V,ρ⊗id ,(ρ ⊗id )◦(id ⊗c−1)), respectively. Ac- U,A X˙ U r l V r V X˙ A,V cordingly instead of with (14) we work with the isomorphic algebra End (Hom (U ⊗+A⊗−U ,A)) ∼= E (15) M A|A i j C i,j∈I which by abuse of notation we still denote by E. For any U,V ∈Obj(C) and X∈Obj(C ) we denote by DUV the mapping A|A X ϕ ∈ Hom (U ⊗+A⊗−V,A) A|A 7−→ DUV(ϕ) := (id ⊗d˜ )◦ [id ⊗ρ ⊗(ε◦ϕ)]◦[(∆◦η)⊗c ⊗id ⊗id ] ⊗id X A X˙ h(cid:16) A l U,X˙ A V (cid:17) X˙∨i ◦ id ⊗ [(ρ ◦(ρ ⊗id ))⊗id ]◦[id ⊗id ⊗(∆◦η)] ⊗id ⊗id h U (cid:16) r l A A A X˙ (cid:17) V X˙∨i ◦hidU ⊗idA⊗(cid:16)[cX−˙1,V ⊗idX˙∨]◦[idV ⊗bX˙](cid:17)i (16) One checks that DUV(ϕ), which by construction is a morphism in Hom(U ⊗A⊗V,A), is actually X again in the subspace Hom (U ⊗+A⊗−V,A), so that DUV is an endomorphism of the vector A|A X space Hom (U ⊗+A⊗−V,A). Further, we consider the linear map A|A Φ : F =K0(CA|A)⊗Z → E (17) C defined by Φ([X]) := D = DUiUj. (18) X M X i,j∈I It must be admitted that the expression (16) for the map DUV is not exceedingly transparent. X However, with the help of the graphical notation for monoidal categories as described e.g. in [JS, Ma, K], it can easily be visualized. Indeed, using U V W V V′ g idU = f = f g◦f = V f ⊗f′ = f f′ (19) f U U U U U′ foridentitymorphisms, generalmorphismsf ∈Hom(U,V),andforcompositionandtensorproduct of morphisms of C, U U∨ ∨U U b = ˜b = V U U V U U c = c−1 = (20) U,V U,V d = d˜ = U V V U U U U∨ U U ∨U for braiding and duality morphisms of C, as well as 8 A A A X˙ X˙ m = ∆ = A A A (21) ρ = ρ = l r A ε = η = A X˙ X˙ A A for the structural morphisms of the algebra A and of the bimodule X, the definition (16) amounts to A ϕ DUV(ϕ) = (22) X X˙ U A V With the help of this graphical description it is e.g. easy to verify that DUV only depends on the X isomorphism class [X] of the bimodule X: Given any bimodule isomorphism f ∈Hom (X,X′) A|A one may insert id =f−1◦f anywhere in the X-loop and then ‘drag f around the loop’ and use X f ◦f−1=idX′, thereby replacing the X-loop by an X′-loop. Φ is an algebra morphism Lemma 2 The map Φ defined in (18) is a morphism of unital associative -algebras. C Proof. We need to show that Φ(1 )=1 and that Φ([X])◦Φ([Y])=Φ([X]∗[Y]) for all X,Y ∈Obj(C ), F E A|A or in other words, that D = id and D ◦D = D , (23) A X Y X⊗AY where ⊗ is the tensor product over A. The latter equality is seen as follows. Using the defining A property of the counit, the unitality of the left A-action ρ, as well as the functoriality of the l 9 braiding and the fact that the left and right A-actions on the bimodule Y commute, one obtains A ϕ X˙ D ◦D (ϕ) = (24) X Y Y˙ U A V The assertion then follows immediately by the fact [KO, FrFRS2] that the morphism (ρX ⊗ρY)◦(id ⊗(∆◦η)⊗id ) = A r l X˙ Y˙ (25) X˙ Y˙ is the idempotent corresponding to the epimorphism that restricts X⊗Y to X⊗ Y. A To show also the first of the equalities (23), first note that for X=A the left and right A-actions are just given by the product m of A. One can then use the various defining properties of A and the fact that ϕ∈Hom (U ⊗+A⊗−V,A) intertwines the left action of A to arrive at A|A 10