On ribbon categories for singlet vertex algebras
aa r X i v : . [ m a t h . QA ] J u l On ribbon categories for singlet vertex algebras
Thomas Creutzig, Robert McRae and Jinwei Yang
Abstract
We construct two non-semisimple braided ribbon tensor categories of modulesfor each singlet vertex operator algebra M ( p ), p ≥
2. The first category consists ofall finite-length M ( p )-modules with atypical composition factors, while the secondis the subcategory of modules that induce to local modules for the triplet vertexoperator algebra W ( p ). We show that every irreducible module has a projectivecover in the second of these categories, although not in the first, and we compute allfusion products involving atypical irreducible modules and their projective covers. Contents W -algebras 63 Tensor categories of singlet modules 134 Rigidity 255 Projective modules and fusion rules 47 The singlet vertex algebras M ( p ), p ∈ Z ≥ , first appeared in the physics literature inthe early 1990s [Ka]. Together with their simple current extensions, the triplet algebras W ( p ), they are the first examples of vertex algebras associated to logarithmic conformalfield theories. While the triplet algebras have been thoroughly studied [AM3, AM4, CF,FHST, FGST1, FGST2, NT, TW], especially the monoidal structure on their modulecategories [TW], the representation theory of the singlet algebras is not yet completely un-derstood, although see [Ad1, AM2, CM1, CMR, CGR]. The triplet algebras are C -cofinite[AM3, CF] and hence have only finitely many inequivalent simple modules; moreover ev-ery (grading-restricted generalized) W ( p )-module has finite length and the full categoryof W ( p )-modules is a braided tensor category [Hu4]. The singlet algebras, on the other1and, are not C -cofinite. They have uncountably many inequivalent simple modules, andindecomposable modules do not necessarily have finite length. These properties make un-derstanding the representation theory of the singlet algebras a rather challenging problem.This problem is not only interesting in its own right: representation categories of singletalgebras have important connections and applications to higher-dimensional supersymmet-ric gauge theories, subregular W -algebras at admissible levels, and 3-manifold invariants.We now describe our main results and then comment on the implications of our work tothese applications. In this paper, we study two locally-finite categories of grading-restricted generalized M ( p )-modules. To define these categories, let O p denote the category of C -cofinite grading-restricted generalized modules for the Virasoro vertex operator algebra L ( c p ,
0) at centralcharge c p = 13 − p − p − . The category O p and its direct limit completion Ind( O p ) arebraided tensor categories [CJORY, CMY]. Then the singlet algebra M ( p ) is a commu-tative algebra in the braided tensor category Ind( O p ) [HKL], and we have an associatedbraided tensor category Rep M ( p ) of generalized M ( p )-modules which are objects ofInd( O p ) when viewed as L ( c p , M ( p )-modules inRep M ( p ) are precisely the atypical irreducibles M r,s , indexed by r ∈ Z , 1 ≤ s ≤ p .We can then view the triplet algebra W ( p ) as a commutative algebra in Rep M ( p ),and we have an associated tensor category Rep W ( p ) of not-necessarily-local W ( p )-moduleswhich are objects of Rep M ( p ) when viewed as M ( p )-modules. As shown in [KO, CKM1],there is an induction tensor functor F W ( p ) : Rep M ( p ) → Rep W ( p ). Now we can defineour two locally-finite categories of M ( p )-modules: • The category C M ( p ) is the full subcategory of finite-length grading-restricted gen-eralized M ( p )-modules whose composition factors come from the M r,s for r ∈ Z ,1 ≤ s ≤ p . • The category C M ( p ) is the full subcategory of generalized M ( p )-modules in Rep M ( p )that induce to (local) grading-restricted generalized W ( p )-modules.Our main results are summarized in the following theorem: (1) The category C M ( p ) is a proper subcategory of C M ( p ) with the same irreducible objectsas C M ( p ) .(2) The categories C M ( p ) and C M ( p ) are braided ribbon tensor categories, with the vertexalgebraic braided tensor category structure of [HLZ1]-[HLZ8] and with duals given bythe contragredient modules of [FHL].(3) Every irreducible module M r,s has a projective cover P r,s in C M ( p ) .
4) The fusion rules of the irreducible modules in C M ( p ) and C M ( p ) are as follows: M r,s ⊠ M r ′ ,s ′ = (cid:18) min { s + s ′ − , p − − s − s ′ } M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ M r + r ′ − ,ℓ (cid:19) ⊕ (cid:18) p M ℓ =2 p +1 − s − s ′ ℓ + s + s ′ ≡ P r + r ′ − ,ℓ (cid:19) for r, r ′ ∈ Z , ≤ s, s ′ ≤ p , with sums taken to be empty if the lower bound exceedsthe upper bound. For the fusion products of the projective covers P r,s with the irreducible modules andwith each other, see Theorem 5.2.1 below. Note that although the modules P r,s are pro-jective in C M ( p ) , they are not projective in the larger tensor category C M ( p ) ; this is onereason for introducing the smaller category C M ( p ) .The proof of our main theorem uses the existence of braided tensor category structureon the direct limit completion Ind( O p ) [CJORY, CMY] as well as three important resultson the triplet W -algebras. First, we use the construction of certain logarithmic modules R r,p − for r = 1 , M ( p )-module summands theprojective covers P r,p − for r ∈ Z . Secondly, [NT, Theorem 5.9] says that the R r,p − areprojective W ( p )-modules; we use this to show that the M ( p )-modules P r,p − are projectivein C M ( p ) . Finally, we use the fusion rules calculated in [TW] for the simple W ( p )-module W , with the remaining simple W ( p )-modules; these fusion rules are needed for computingfusion products of M ( p )-modules, for proving rigidity, and for constructing the remainingprojective covers in C M ( p ) .To prove that our tensor categories of M ( p )-modules are rigid, we first prove rigidity forthe simple modules M r,s . This part of the argument is essentially the same as the rigidityproof for W ( p ) in [TW]: we use BPZ equations to show that M , is rigid and then usefusion rules to get rigidity for the remaining simple modules. Then we prove rigidity forarbitrary finite-length modules using the following general theorem (Theorem 4.4.1 in themain text), which we expect to have many future applications to non-semisimple modulecategories for vertex operator algebras. Indeed, as we mention in Remark 4.4.6 below, thistheorem, combined with results from [McR, CJORY], implies rigidity for the category O of C -cofinite Virasoro modules at central charge 1: Assume that V is a self-contragredient vertex operator algebra and C isa category of grading-restricted generalized V -modules such that:(1) The category C is closed under submodules, quotients, and contragredients, and everymodule in C has finite length.(2) The category C has braided tensor category structure as in [HLZ8].(3) Every simple module in C is rigid.Then C is a rigid tensor category.
3e should mention that our category C M ( p ) of singlet modules is not the full category offinite-length M ( p )-modules since it lacks most irreducible typical M ( p )-modules (whichare Fock modules for the rank-one Heisenberg extension of M ( p )). Conjecturally, thecategory of finite-length M ( p )-modules agrees with the C -cofinite module category (see forexample [CMR]), in which case this category would have braided tensor category structurevia the methods of [CJORY, CY]. In Proposition 3.1.1 below, we show that if the entirecategory of C -cofinite M ( p )-modules indeed has braided tensor category structure, then C M ( p ) embeds as a tensor subcategory. In particular, the fusion rules of our main theoremdo not depend on our choice of subcategory. There is an intimate connection between topology, geometry, quantum groups, representa-tion theory of vertex algebras, and physics. The singlet algebras and their representationcategories serve as an important modern example:Motivated by three-dimensional N = 2 supersymmetric gauge theories, new and onlyconjectural invariants of 3-manifolds, denoted by ˆ Z a ( q ), were introduced in [GPPV]. Theˆ Z a ( q ) are q -series labeled by abelian flat connections, and they often turn out to be es-sentially mock modular forms or false theta functions. But the characters of the singletmodules M r,s are of the false theta function type, and indeed it turns out that in somecases they coincide with these new conjectural 3-manifold invariants [CCFGH].In addition, physics associates a tensor category to any 3-manifold. This tensor categoryshould be an appropriate non-semisimple, non-finite generalization of a modular tensorcategory, presumably a locally-finite rigid braided tensor category that is non-degeneratein the sense that its M¨uger center is trivial. Extended topological field theories in dimension1+1+1 and non-semisimple quantum invariants of closed 3-manifolds are constructed fromsuch non-semisimple tensor categories [CGP1, DeR], especially the categories of weightmodules of unrolled quantum groups at odd roots of unity [DGP]. In the vertex algebrasetting, already the simplest invariants are meaningful, that is, a general expectation isthat open Hopf links are related to analytic properties of characters of modules for avertex operator algebra [CG]. This is useful as it provides a way to compute fusion rules:for strongly-rational vertex operator algebras, this is the celebrated Verlinde formula [Ve]proven by Huang [Hu2, Hu3]. For the singlet algebra, the conjectural Verlinde formulainvolves regularized false theta functions [CM], and the normalized S -matrix coefficientscoincide with open Hopf links of the unrolled restricted quantum group of sl at a 2 p throot of unity [CMR]. Our fusion rule computations confirm the Verlinde conjecture of[CM] for the subcategory C M ( p ) , and they coincide with tensor products of modules forthe unrolled restricted quantum group of sl at a 2 p th root of unity computed in [CGP2].Conjecturally, this is not a coincidence:Two major sources of braided tensor categories are modules for quantum groups andfor vertex operator algebras. It is then natural to ask if there are equivalences of categoriesassociated to quantum groups and vertex operator algebras, and this firmates under thename Kazhdan-Lusztig correspondence since they proved a braided equivalence of ordinary4ighest-weight modules of affine vertex algebras at generic level with corresponding quan-tum group modules [KL1]–[KL5]. The first representation theory statements concerningthe triplet algebras W ( p ) were made under the assumption that there is an equivalencewith the category of weight modules for the restricted quantum group of sl at a 2 p th rootof unity [FHST, FGST1, FGST2], and an equivalence of abelian categories was stated to betrue in [NT]. A braided equivalence must fail, however, since the quantum group categoryturns out to be non-braidable [KS]. On the other hand, the category of weight modules ofthe unrolled restricted quantum group of sl at 2 p th root of unity is conjecturally equiv-alent to a category of generalized modules for the M ( p )-algebra [CGP2, CMR], and thetriplet algebra is a simple current extension of the singlet algebra. Translating back tothe quantum group side has led to a quasi-Hopf algebra whose underlying algebra is therestricted quantum group and whose representation category is a finite tensor category[CGR].Given that singlet and triplet algebras are by far the best understood vertex algebraswith non-semisimple representation theory, it is fair to say that a major problem in thiscontext is the conjectural correspondence with (quasi-Hopf modifications of the) quantumgroups. A first step towards a proof is the existence of a tensor category on the vertexalgebra side, that is, our main theorem. Next, one would like to prove that our category C M ( p ) is braided equivalent to the corresponding category of the unrolled quantum group.Uprolling, that is, performing the simple current extension, then immediately gives the cor-respondence between the representation category of W ( p ) and the quasi-Hopf modificationof the restricted quantum group; see [CGR] for details on this idea.A second family of vertex algebras with usually non-semisimple representation theoryis affine vertex algebras and W -algebras at non-positive-integer levels, for example admis-sible but non-integral levels. The best-understood examples are the affine vertex algebrasof sl at admissible levels [AM1, CRi2]. Even for sl , the generic module has neitherlower-bounded conformal weights nor finite-dimensional conformal weight spaces, that is,two essential finiteness conditions needed for the existence of tensor category structure in[HLZ1]-[HLZ8] fail. As a consequence, braided tensor category structure is only known toexist on the subcategory of ordinary grading-restricted modules [CHY, CY], and rigidityfor this subcategory is known only in the simply-laced case [CHY, Cr3].Our results now allow the study of tensor categories that include relaxed highest-weightmodules of special subregular W -algebras, namely the simple subregular W -algebras of sl p − at level k = − ( p −
1) + p − p . These coincide with the B p -algebras (for p ≥
3) of[CRW] by [ACGY, Corollary 16] together with [ACKR, Theorem 5]. These B p -algebrasare defined as extensions of M ( p ) tensored with a rank-one Heisenberg algebra. One canthus apply the theory of vertex algebra extensions [CKM1] to our category C M ( p ) tensoredwith a category of Fock modules for the Heisenberg algebra to obtain a braided tensorcategory of B p -algebra modules. The procedure will be very analogous to [ACKR], andas a consequence one finally has examples of rigid braided tensor categories of W -algebramodules that include relaxed highest-weight modules. We note that the B -algebra is thesimple affine vertex algebra of sl at level − [Ad2], and a Z -orbifold of the B -algebra5s the simple affine vertex algebra of sl at level − . Another family of W -superalgebrasthat have M ( p ) as a Heisenberg coset are the simple principal W -superalgebras of sl p − | at level k = − ( p −
2) + pp − [CGN], so it is now also possible to study braided tensorcategories for these superalgebras. The B p -algebras are also important in physics sincethey appear as chiral algebras of certain four-dimensional supersymmetric gauge theories,called the ( A , A p − )-Argyres-Douglas theories, by [ACGY, Corollary 16] together with[Cr1, Theorem 4.1]; representation theory data of the vertex algebras relate to interestinggauge theory data (see for example [BN, CS]).Finally, let us mention that there are higher-rank analogues of triplet, singlet, and B p -algebras [FT, CM2, Cr2], and they are expected to enjoy similar relations to topology,physics, and quantum groups; see [AMW, BMM, Cr2, CM2, CRu, FL, Le, Pa, Ru, Su] forsome results. In order to extend our work to higher rank, one first needs to show that thecategory of C -cofinine modules for principal W -algebras at appropriate levels have vertextensor category structure, that is, one needs to generalize [CJORY] beyond the Virasorocase. In Section 2, we provide background on the representation theory of the singlet and tripletvertex operator algebras. In Section 3.1, we obtain singlet module categories from the directlimit completion of the category of C -cofinite Virasoro modules, and then we computesome fusion rules in Section 3.2. In Section 3.3, we show that the categories C M ( p ) and C M ( p ) of singlet modules are braided tensor categories, and then we obtain some projectivemodules in C M ( p ) . We establish rigidity for both C M ( p ) and C M ( p ) in Section 4. Finally, inthe last section, we use rigidity to finish the proof of our main theorem, that is, we constructthe remaining projective covers in C M ( p ) and compute the remaining fusion rules. Acknowledgements
TC acknowledges support from NSERC discovery grant RES0048511. RM thanks theUniversity of Alberta for its hospitality during the visit in which this work was begun. W -algebras In this section, we recall the definitions of the singlet and triplet W -algebras, as well asresults from the representation theory of the singlet and triplet that we will use later. Formore details, see for example the references [Ad1, AM3, AM4, NT, TW, CRW]. For an integer p ≥
2, fix a rank-one lattice L = Z α with h α, α i = 2 p. h = α √ p ∈ R α , so that h h, h i = 1.We denote the lattice vertex operator algebra associated to L by ( V L , Y, , ω ). As avector space, V L = U ( b h < ) ⊗ C [ L ] , where C [ L ] is the group algebra of L and b h is the affinization of the abelian Lie algebra h = C α . The vacuum vector of V L is = 1 ⊗
1, and we use the modified conformal vector ω = 14 p α ( − + p − p α ( − = 12 h ( − + p − √ p h ( − . (2.1)The vertex operator algebra V L has finitely many irreducible modules up to equivalence,parametrized by cosets in L ◦ /L , where L ◦ = Z α p is the dual lattice of L . Specifically, for λ + L ∈ L ◦ /L , V λ + L = U ( b h < ) ⊗ e λ C [ L ]has the structure of an irreducible V L -module. Taking λ = 0 recovers V L itself, while thefull space V L ◦ = L λ + L ∈ L ◦ /L V λ + L has the structure of a generalized vertex algebra [DL].The Virasoro algebra acts on each V λ + L with the central charge c p := 13 − p − p − = 1 − p − p . Let H be the Heisenberg vertex operator algebra associated with b h , with the sameconformal vector (2.1). For λ ∈ C , let F λ denote the irreducible Fock H -module generatedby a highest-weight vector v λ such that h ( n ) v λ = δ n, λv λ , n > . In particular, F = H itself. As a vector space, F λ = U ( b h < ) ⊗ C v λ , and the lowest conformal weight of F λ is h λ = 12 λ ( λ − α ) , (2.2)where α = p p − p /p. The lattice vertex operator algebra V L is an extension of H and decomposes as an infinitedirect sum of Fock spaces as an H -module: V L = M µ ∈ L F µ , µ = nα ∈ L with n √ p ∈ R . Similarly, V λ + L = M µ ∈ λ + L F µ for λ ∈ L ◦ .Now define the screening operator e − α/p = Res x Y ( e − α/p , x )where Y is the vertex operator for the generalized vertex algebra V L ◦ . Then the singletvertex operator algebra M ( p ) is the vertex operator subalgebraker | F e − α/p of F , and the triplet vertex operator algebra W ( p ) is the vertex operator subalgebraker | V L e − α/p of V L . By [Ad1, Theorem 3.2], the singlet M ( p ) is generated as a vertex algebra by ω and H = S p − ( α ) , where S k ( α ) is the Schur polynomial in the variables α ( − , α ( − , . . . defined by exp ∞ X n =1 α ( − n ) n x n ! = ∞ X k =0 S k ( α ) x k . Introduce α + = √ p , α − = − p /p (corresponding to α, − p α ∈ L ◦ ), and define α r,s = 1 − r α + + 1 − s α − , for r, s ∈ Z . Note that α r,s is periodic: α r +1 ,s + p = α r,s and that α = α + + α − . Defining h r,s to be the conformal weight h α r,s of (2.2), we calculate h r,s = r − p − rs −
12 + s − p − . (2.3)Now from [Ad1, Theorem 2.1], the singlet M ( p ) decomposes into an infinite direct sum ofirreducible modules for its Virasoro subalgebra: M ( p ) = ∞ M n =0 L ( c p , h n +1 , ) , (2.4)where L ( c, h ) is the irreducible Virasoro module of central charge c and lowest conformalweight h . By [AM3, Theorem 1.1], the triplet W ( p ) decomposes into an infinite direct sumof the same irreducible Virasoro modules, but with different multiplicities: W ( p ) = ∞ M n =0 (2 n + 1) L ( c p, , h n +1 , ) . (2.5)8 .2 Representations of the triplet algebra The triplet W -algebra W ( p ) has 2 p simple modules up to isomorphism, labeled by W r,s for r = 1 , ≤ s ≤ p . Recalling the correspondence α r,s ∈ R ←→ ( p (1 − r ) − (1 − s )) α p ∈ L ◦ , (2.6)we have W r,p = V α r,p + L , while for 1 ≤ s ≤ p −
1, there are non-split short exact sequences0 −→ W r,s −→ V α r,s + L −→ W − r,p − s −→ . (2.7)The lowest conformal weight of W r,s is h r,s , and the lowest conformal weight space isone-dimensional for r = 1 and two-dimensional for r = 2. As L ( c p , W r,s = ∞ M n =0 (2 n + r ) L ( c p , h n + r,s ) (2.8)for r = 1 , ≤ s ≤ p .Let C W ( p ) denote the category of grading-restricted generalized (that is, logarithmic) W ( p )-modules. Since W ( p ) is C -cofinite [AM3, Theorem 2.1], every simple module W r,s in C W ( p ) has a projective cover R r,s [Hu4, Theorem 3.23]. The lattice modules W r,p = V α r,p + L are their own projective covers [NT, Section 5]. The projective covers R r,p − wereconstructed explicitly in [AM4], and we recall some details of this construction.Set V = V α , + L ⊕ V α ,p − + L = V L ⊕ V − α/p + L . By [Li1], V has a vertex operator algebra structure with vertex operator Y V (cid:0) ( u , u ) , x (cid:1) ( v , v ) = (cid:0) Y ( u , x ) v , Y ( u , x ) v + Y ( u , x ) v (cid:1) , (2.9)where Y is the vertex operator for the generalized vertex algebra V L ◦ . Moreover, W = V α , + L ⊕ V α ,p − + L = V α/ L ⊕ V ( p − α/ p + L is a V -module with vertex operator Y W (cid:0) ( v , v ) , x (cid:1) ( w , w ) = (cid:0) Y ( v , x ) w , Y ( v , x ) w + Y ( v , x ) w (cid:1) (2.10)for v ∈ V L , v ∈ V − α/p + L , w ∈ V α , + L , and w ∈ V α ,p − + L . Now we deform the vertexoperators Y W and Y V using Li’s ∆-operators [Li2]: for u ∈ V (1) , set∆( u, x ) = x u exp ∞ X n =1 u n − n ( − x ) − n ! , e Y W ( v, x ) = Y W (∆( e − α/p , x ) v, x ) , e Y V ( v, x ) = Y V (∆( e − α/p , x ) v, x )for v ∈ V . We restrict e Y W to W ( p ) to get a W ( p )-module R ,p − with underlying vectorspace W and Y R ,p − = e Y W | W ( p ) , and we restrict e Y V to W ( p ) to get a W ( p )-module R ,p − with underlying vector space V and Y R ,p − = e Y V | W ( p ) .Since Y V | V − α/p + L ⊗ V − α/p + L = 0 and W ( p ) = ker | V L e − α/p , we have Y R ,p − ( v, x ) = Y W ( v, x ) + ∞ X n =1 ( − n +1 n x − n Y W ( e − α/pn v, x ) Y R ,p − ( v, x ) = Y V ( v, x ) + ∞ X n =1 ( − n +1 n x − n Y V ( e − α/pn v, x )for v ∈ W ( p ). From this together with (2.9) and (2.10), it is clear that for r = 1 ,
2, thereis an exact sequence of W ( p )-modules0 → V α r,p − + L → R r,p − → V α − r, + L → . These exact sequences are non-split, and the R r,p − are logarithmic W ( p )-modules, because Y R ,p − ( ω, x )( w , w ) = (cid:0) Y W ( ω, x ) + x − Y W ( e − α/p , x ) (cid:1) ( w , w )= (cid:0) Y ( ω, x ) w , Y ( ω, x ) w + x − Y ( e − α/p , x ) w (cid:1) , (2.11)and similarly for Y R ,p − . In particular, L (0)( w , w ) = (cid:0) L (0) w , L (0) w + e − α/p w (cid:1) , which is non-semisimple (see the discussion in [AM4, Section 4] for more details).In [AM4, Section 5], it is shown that R r,p − has Loewy diagram W r,p − R r,p − : W − r, W − r, W r,p − . In particular, there are non-split exact sequences0 −→ Y r,p − −→ R r,p − −→ W r,p − −→ −→ W r,p − −→ Y r,p − −→ W − r, ⊕ W − r, −→ , (2.13)where Y r,p − is the maximal submodule generated by V α r,p − + L and W − r, ⊆ V α − r, + L .According to [NT, Theorem 5.9], R r,p − for r = 1 , C W ( p ) and isa projective cover of W r,p − . (For the case p = 2, one can also show these modulesare projective by using the isomorphism of W (2) with the even subalgebra of the vertexoperator superalgebra of one pair of symplectic fermions to identify the R r, with projectivemodules for the symplectic fermion superalgebra.) For the remaining projective covers ofirreducible W ( p )-modules, see the constructions in [NT, Section 4.1], but we shall not needthe detailed structure of these modules. In fact, one could use the techniques of Section 5.1below together with the fusion rules involving W , from [TW] to construct the remainingprojective covers recursively starting from R r,p − .Now because W ( p ) is C -cofinite, [Hu4, Theorem 4.13] shows that C W ( p ) has braidedtensor category structure as developed in [HLZ1]-[HLZ8]. In [TW], Tsuchiya and Wooddetermined fusion products in C W ( p ) and showed that it is a rigid tensor category. Here wesummarize some fusion rules in C W ( p ) that we will need for studying the singlet algebra: ([TW]) . (1) The simple module W , is a self-dual simple current with W , ⊠ W r,s = W − r,s for r = 1 , and ≤ s ≤ p .(2) The simple module W , is rigid with fusion rules W , ⊠ W r,s = W r, if s = 1 W r,s − ⊕ W r,s +1 if ≤ s ≤ p − R r,p − if s = p The singlet vertex operator algebra M ( p ) has infinitely many simple modules, first classi-fied in [Ad1]. Here, we use [CRW, Section 2] as a reference. For λ ∈ C \ L ◦ , the HeisenbergFock module F λ restricts to an irreducible M ( p )-module. For r ∈ Z and 1 ≤ s ≤ p onthe other hand, F α r,s is not usually irreducible as an M ( p )-module, but it contains theirreducible M ( p )-module M r,s = Soc( F α r,s ). We have M r,p = F α r,p , while for 1 ≤ s ≤ p −
1, there are non-split short exact sequences0 −→ M r,s −→ F α r,s −→ M r +1 ,p − s −→ . (2.14)These exact sequences arise from the action of a certain screening operator Q [ s ] − that mapsthe Fock spaces F α r,s for r ∈ Z to F α r +1 ,p − s , that is, M r,s = ker Q [ s ] − | F αr,s and M r +1 ,p − s =11m Q [ s ] − | F αr,s . Moreover, M r,s = W ¯ r,s ∩ F α r,s where ¯ r = 1 or 2 according as r is odd or even,and the diagram M r,s / / (cid:15) (cid:15) F α r,s Q [ s ] − | F αr,s / / (cid:15) (cid:15) M r +1 ,p − s ⊆ F α r +1 ,p − s (cid:15) (cid:15) W ¯ r,s / / V α ¯ r,s + L Q [ s ] − | Vα ¯ r,s + L / / W − ¯ r,p − s ⊆ V α − ¯ r,p − s + L (2.15)commutes, where the unlabeled arrows are inclusions.The lowest conformal weight of M r,s is h r,s if r ≥ h − r,s if r ≤ s = p ). As Virasoro modules, M r,s = ∞ M n =0 L ( c p , h r +2 n,s ) ( r ≥ , ≤ s ≤ p ) (2.16) M r +1 ,p − s = ∞ M n =0 L ( c p , h r − n,s ) ( r ≤ , ≤ s ≤ p ) (2.17)(note that these expressions for r = 1 and r = 0 agree).For r ∈ Z , we also have logarithmic M ( p )-modules P r,p − which satisfy a non-splitexact sequence 0 −→ F α r,p − −→ P r,p − −→ F α r − , −→ . (2.18)As vector spaces, P r,p − = F α r,p − ⊕ F α r − , ⊆ V α r,p − + L ⊕ V α r − , + L = V α ¯ r,p − + L ⊕ V α − ¯ r, + L = R ¯ r,p − , where ¯ r = 1 or 2 according as r is odd or even. The vertex operator for P r,p − is given by Y P r,p − = Y R ¯ r,p − | M ( p ) ⊗P r,p − , so that we have a commutative diagram F α r,p − / / (cid:15) (cid:15) P r,p − / / (cid:15) (cid:15) F α r − , (cid:15) (cid:15) V α ¯ r,p − + L / / R ¯ r,p − / / V α − ¯ r, + L (2.19)of M ( p )-module homomorphisms.To see that P r,p − is indeed an M ( p )-module, note first that (2.11) shows Y P r,p − ( ω, x )( w , w ) = (cid:0) Y ( ω, x ) w , Y ( ω, x ) w + x − Y ( e − α/p , x ) w (cid:1) (2.20)for w ∈ F α r − , , w ∈ F α r,p − , and the components of Y ( e − α/p , x ) indeed map the Fockspace F α r − , into F α r,p − since (cid:0) p (1 − ( r − − (1 − (cid:1) α p − αp = (cid:0) p (1 − r ) − (1 − ( p − (cid:1) α p H = S p − ( α ) of M ( p ), we can use formulasfrom [Ad1] to calculate Y P r,p − ( H, x )( w , w ) = Y W ( H, x ) + ∞ X n =1 ( − n − n x − n Y W ( e − α/pn H, x ) ! ( w , w )= Y ( H, x ) w , Y ( H, x ) w − p − X n =1 ( − n pp − x − n Y ( S p − − n ( α ) e − α/p , x ) w ! (2.21)for w ∈ F α r − , , w ∈ F α r,p − . Since ω and H generate M ( p ) as a vertex algebra, (2.20)and (2.21) show that indeed Y P r,p − preserves P r,p − for r ∈ Z ; they also show that (2.18)is non-split exact, and that P r,p − is a logarithmic M ( p )-module.It was conjectured in [CGR, Section 5] that P r,p − is the projective cover of M r,p − in asuitable subcategory of grading-restricted generalized M ( p )-modules. We shall prove thisin Section 3.3. In this section, we will construct several tensor categories of generalized M ( p )-modules.Although by the Main Theorem of [Mi2], the category of C -cofinite (grading-restrictedgeneralized) modules for a vertex operator algebra is closed under the vertex algebraictensor product of [HLZ3], this is not sufficient for existence of associativity isomorphismsas in [HLZ6]. If in addition the C -cofinite module category agrees with the category offinite-length modules, however, then the method in [CJORY] shows that this category hasbraided tensor category structure. For the singlet algebra, all finite-length modules are C -cofinite by [CMR, Theorem 13], but unfortunately the converse remains open. Thushere, we will need to restrict our attention to M ( p )-modules which live in the direct limitcompletion of the C -cofinite module category for the Virasoro vertex operator algebra L ( c p , Let O p denote the category of C -cofinite grading-restricted generalized modules for thesimple Virasoro vertex operator algebra L ( c p , L ( c p , h r,s ) for r ∈ Z , 1 ≤ s ≤ p . It was shown in [CJORY] that all modules in O p are of finite length and that O p has vertex algebraic braided tensor category structure asin [HLZ1]-[HLZ8].From (2.4) and (2.5), we see that the singlet and triplet algebras are not objects of O p ,but they are objects of the direct limit completion Ind( O p ) introduced in [CMY]. This isthe full subcategory of generalized L ( c p , C -cofinite submodules. By [CMY, Theorem 7.2], Ind( O p ) has vertex algebraicbraided tensor category structure extending that on O p , and then [CMY, Theorem 7.7]13ays that M ( p ) is a commutative algebra in the braided tensor category Ind( O p ). Thismeans that we can apply the extension theory of [CKM1] to study M ( p )-modules: inparticular, [CMY, Theorem 7.9] (see also [CKM1, Theorem 3.65]) says that the categoryRep M ( p ) of generalized M ( p )-modules in Ind( O p ) has vertex algebraic braided tensorcategory structure. From (2.16) and (2.17), we see that the irreducible M ( p )-modules M r,s for r ∈ Z , 1 ≤ s ≤ p are objects of Rep M ( p ), although the irreducible Fock modules F λ for λ / ∈ L ◦ are not.Now since the triplet algebra is in Ind( O p ) and is a module for its singlet subalgebra,we see that W ( p ) is a commutative algebra in the braided tensor category Rep M ( p ).This means we have a braided tensor category Rep W ( p ) of generalized W ( p )-modules inRep M ( p ), as well as the larger tensor category Rep W ( p ) of not-necessarily-local W ( p )-modules in Rep M ( p ). From [KO, CKM1], there is an induction functor F W ( p ) : Rep M ( p ) → Rep W ( p ) M
7→ W ( p ) ⊠ Mf id W ( p ) ⊠ f. Induction is a monoidal functor, so that in particular F W ( p ) ( M ⊠ M ) ∼ = F W ( p ) ( M ) ⊠ W ( p ) F W ( p ) ( M ) for generalized modules M , M in Rep M ( p ).We are mainly interested in locally-finite categories of M ( p )-modules, and Rep M ( p )is too large for these purposes (for example, it is closed under arbitrary direct sums).Probably the largest interesting “small” subcategory we could consider is the category of C -cofinite modules in Rep M ( p ). Unfortunately, we do not know that modules in thiscategory necessarily have finite length. So we will instead need to consider possibly smallerlocally-finite tensor subcategories of Rep M ( p ). However, since we do expect that the C -cofinite category should ultimately be the “right” tensor category of M ( p )-modules, wewill at least show here that the tensor product ⊠ C of [HLZ3, Mi2] for C -cofinite modulesagrees with the tensor product ⊠ on Rep M ( p ) for C -cofinite modules in Rep M ( p ).That is, if in the future it is established that the category of C -cofinite M ( p )-modules hasvertex algebraic braided tensor category structure, then the categories we construct herewill be tensor subcategories: If M , M are C -cofinite M ( p ) -modules in Rep M ( p ) , then M ⊠ M ∼ = M ⊠ C M .Proof. It is enough to show that the pair ( M ⊠ M , Y M ,M ) satisfies the intertwining op-erator universal property of a tensor product in the category of C -cofinite M ( p )-modules,where Y M ,M is the tensor product intertwining operator of type (cid:0) M ⊠ M M M (cid:1) in Rep M ( p ).Thus suppose M is any C -cofinite M ( p )-module and Y is an intertwining operator oftype (cid:0) M M M (cid:1) .Since M is C -cofinite as an M ( p )-module, it is in particular an N -gradable weak L ( c p , C -cofinite L ( c p , W ⊆ M and W ⊆ M ,the Key Theorem of [Mi2] implies that im Y | W ⊗ W is a C -cofinite L ( c p , . Then since M and M are objects of Rep M ( p ), they are the unions of their C -cofinite L ( c p , Y is the union of its C -cofinite L ( c p , Y is an object of Rep M ( p ).Now the universal property of the tensor product in Rep M ( p ) shows that there is aunique M ( p )-module homomorphism f : M ⊠ M → im Y ⊆ M such that f ◦ Y M ,M = Y . Since Y M ,M is a surjective intertwining operator, f is theunique map M ⊠ M → M with this property, so ( M ⊠ M , Y M ,M ) satisfies the universalproperty for a tensor product of C -cofinite M ( p )-modules.Now here are the categories of M ( p )-modules that we will consider in the remainderof the paper: The category C M ( p ) is full subcategory of finite-length grading-restrictedgeneralized M ( p )-modules whose composition factors come from the M r,s for r ∈ Z ,1 ≤ s ≤ p . The category C M ( p ) is the full subcategory of generalized M ( p )-modules M in Rep M ( p ) such that F W ( p ) ( M ) is an object of C W ( p ) (that is, a grading-restrictedgeneralized W ( p )-module).We will show that C M ( p ) is a subcategory of C M ( p ) , and that both are braided ten-sor subcategories of Rep M ( p ), in Section 3.3 below after deriving some fusion rules inRep M ( p ). For now, we show: The category C M ( p ) is a full subcategory of Rep M ( p ) .Proof. We need to show that every module in C M ( p ) is an object of the Virasoro directlimit completion Ind( O p ). Since modules in C M ( p ) are finite length by definition, we canprove this by induction on the length. For the base case, the irreducible modules M r,s in C M ( p ) are objects of Ind( O p ) by (2.16) and (2.17).For the inductive step, we note that because all modules in C M ( p ) have finite length,they are in particular grading-restricted generalized L ( c p , → A → W → B → L ( c p , A and B objects of Ind( O p ), theextension W is also an object of Ind( O p ).To prove the claim, we may assume that the conformal weights of W are containedin h + N for some h ∈ C , since in general W is a direct sum of such modules. Take an L (0)-eigenvector w ∈ W with minimal conformal weight, and let W be the Virasorosubmodule of W generated by w . Then W is an extension of W /A ∩ W by A ∩ W ,which are submodules of B and A , respectively. Since Ind( O p ) is closed under submodules,these two are in Ind( O p ). They are also subquotients of a Verma module for the Virasoro15lgebra, and since non-zero submodules of Virasoro Verma modules are never unions of C -cofinite submodules, A ∩ W and W /A ∩ W must be submodules of proper quotientsof Verma modules. Such proper subquotients have finite length, so they are modules in O p . Thus W is also a finite-length L ( c p , O p .Now the quotient W/W is an extension of W/ ( A + W ) ∼ = ( W/A ) / (( A + W ) /A ) by( A + W ) /W ∼ = A/A ∩ W , which are quotient modules of B and A , respectively. SinceInd( O p ) is closed under taking quotients, these two are objects of Ind( O p ). So we can usethe above argument again to obtain a finite-length L ( c p , W /W of W/W generated by an L (0)-eigenvector with minimal conformal weight. Continuing this way, weobtain an ascending chain of Virasoro submodules0 ⊆ W ⊆ W ⊆ · · · ⊆ W i ⊆ · · · of W such that the W i +1 /W i are finite-length L ( c p , i increases. So each W i has finite length and is anobject of O p . Also, because W has finite-dimensional conformal weight spaces, any vector w ∈ W is contained in W i for i sufficiently large, that is, W = S i ≥ W i . Since each W i isin O p , this means W is in Ind( O p ). The same argument using (2.8) shows that the category C W ( p ) of grading-restricted generalized W ( p )-modules is a full subcategory of Rep W ( p ), since every modulein C W ( p ) has finite length by [Hu4, Proposition 4.3]. Moreover, [CKM1, Theorem 3.65]shows that C W ( p ) is also a tensor subcategory of Rep W ( p ); in particular, the fusion rulesof Theorem 2.2.1 are valid in Rep W ( p ). The exact sequences (2.14) and (2.18) show that the M ( p )-modules F α r,s for r ∈ Z , 1 ≤ s ≤ p and P r,p − for r ∈ Z are objects of C M ( p ) and thus also of Rep M ( p ). In this section, we calculate some fusion products of M ( p )-modules in the tensor categoryRep M ( p ). We start with the action by vertex operator algebra automorphisms of R /L ◦ ∼ = U (1) on V L such that for u ∈ R and λ = n √ p ∈ L , (cid:18) u + 1 √ p Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F λ = e πiuλ id F λ . This restricts to a U (1)-action on W ( p ), and since F equals the U (1)-fixed points in V L , M ( p ) = W ( p ) U (1) . Then from [DLM], the triplet W ( p ) decomposes as a direct sum of irreducible U (1) ×M ( p )-modules: W ( p ) = M n ∈ Z C α n +1 , ⊗ M n +1 , , α n +1 , represents the irreducible U (1)-character corresponding to − nα ∈ L .From [CKLR, Theorem 3.1] (whose proof is an adaptation of the argument for finitecyclic automorphism groups from [Mi1, Proposition 20] and [CM, Theorem 4.2], moregenerally, see [McR]), we have: For n ∈ Z , the M ( p ) -modules M n +1 , are simple currents with fusionproducts M n +1 , ⊠ M n ′ +1 , = M n + n ′ )+1 , . (3.1)As a consequence of this, we can prove: The induction functor F W ( p ) : Rep M ( p ) → Rep W ( p ) is exact.Proof. Since induction is the functor W ( p ) ⊠ • , it is right exact by [HLZ3, Proposition 4.26].So we just need to show that if f : M → M is an injective M ( p )-module homomorphismin Rep M ( p ), then id W ( p ) ⊠ f is still injective. As a homomorphism of M ( p )-modules,id W ( p ) ⊠ f can be naturally identified with M n ∈ Z id M n +1 , ⊠ f : M n ∈ Z M n +1 , ⊠ M → M n ∈ Z M n +1 , ⊠ M . Since each M n +1 , is a necessarily rigid simple current and since tensoring with rigidobjects is exact, each id M n +1 , ⊠ f is injective. Thus id W ( p ) ⊠ f is injective as well.Next we look at how the simple currents M n +1 , fuse with other simple modules inRep M ( p ), and we calculate inductions: For r ∈ Z , ≤ s ≤ p , and n ∈ Z , M n +1 , ⊠ M r,s = M n + r,s , (3.2) and for r ∈ Z , ≤ s ≤ p , F W ( p ) ( M r,s ) = W ¯ r,s (3.3) where ¯ r = 1 or according as r is odd or even.Proof. If we compose the injections M n +1 , → F α n +1 , and M r,s → F α r,s of (2.14) withthe tensor product intertwining operator for Heisenberg Fock modules, we get an M ( p )-module intertwining operator of type (cid:0) F α n + r,s M n +1 , M r,s (cid:1) . Since Fock modules are irreducible as H -modules, [DL, Proposition 11.9] shows that this intertwining operator is non-zero, so weget a non-zero M ( p )-module homomorphism M n +1 , ⊠ M r,s → F α n + r,s . Since M n +1 , isa simple current, M n +1 , ⊠ M r,s is simple [CKLR, Proposition 2.5(3)]. So the image of thishomomorphism is a simple submodule of F α n + r,s , which must be M n + r,s = Soc( F α n + r,s ).This proves (3.2).Now (3.2) says that the M n +1 , ⊠ M r,s for n ∈ Z are distinct, so by [CKM1, Propo-sition 4.4], F W ( p ) ( M r,s ) is an irreducible W ( p )-module. Then Frobenius reciprocity ap-plied to the embedding M r,s → W ¯ r,s implies there is a non-zero W ( p )-homomorphism F W ( p ) ( M r,s ) → W ¯ r,s , which must be an isomorphism.17ow we use Proposition 3.2.3 to calculate fusion of the simple currents with reducibleFock modules: For r ∈ Z , ≤ s ≤ p − , and n ∈ Z , M n +1 , ⊠ F α r,s = F α n + r,s , (3.4) and for r ∈ Z , ≤ s ≤ p − , F W ( p ) ( F α r,s ) = V α ¯ r,s + L (3.5) where ¯ r = 1 or according as r is odd or even.Proof. We use the diagram of M ( p )-module homomorphisms0 / / W ( p ) ⊠ M r,s / / (cid:15) (cid:15) W ( p ) ⊠ F α r,s / / (cid:15) (cid:15) W ( p ) ⊠ M r +1 ,p − s / / (cid:15) (cid:15) / / W ( p ) ⊠ W ¯ r,s / / (cid:15) (cid:15) W ( p ) ⊠ V α ¯ r,s + L / / (cid:15) (cid:15) W ( p ) ⊠ W − ¯ r,p − s / / (cid:15) (cid:15) / / W ¯ r,s / / V α ¯ r,s + L / / W − ¯ r,p − s / / W ( p )-module vertex operators (which are also M ( p )-moduleintertwining operators). All three rows are exact, the first two by Proposition 3.2.2. Theupper squares commute by (2.15), and the lower squares commute because the lower ver-tical arrows come from W ( p )-module vertex operators while the bottom two rows comefrom W ( p )-module homomorphisms.Now, the compositions of vertical arrows in the diagram are the W ( p )-module homo-morphisms induced by Frobenius reciprocity. Since the first and third are isomorphismsby Proposition 3.2.3, the middle homomorphism F W ( p ) ( F α r,s ) → V α ¯ r,s + L is also an isomor-phism by the short five lemma, proving (3.5). For (3.4), we project all homomorphismsin the commutative diagram to the summands involving M n +1 , ⊆ W ( p ), leading to thecommutative diagram0 / / M n +1 , ⊠ M r,s / / ∼ = (cid:15) (cid:15) M n +1 , ⊠ F α r,s (cid:15) (cid:15) / / M n +1 , ⊠ M r +1 ,p − s / / ∼ = (cid:15) (cid:15) / / / / M n + r,s / / F α n + r,s / / M n + r +1 ,p − s / / M n +1 , ⊠ F α r,s ∼ = F α n + r,s again by the short five lemma.Using Proposition 3.2.4, the commutative diagram (2.19), and an argument similar tothe proof of Proposition 3.2.4, we can prove:18 .2.5 Proposition. For r, n ∈ Z , M n +1 , ⊠ P r,p − = P n + r,p − , (3.6) and for r ∈ Z , F W ( p ) ( P r,p − ) = R ¯ r,p − (3.7) where ¯ r = 1 or according as r is even or odd. Now we present the singlet algebra analogue of Theorem 2.2.1. However, in the spiritof the proof of the corresponding result for W ( p ) in [TW], which used rigidity of W , toprove the s = p case of Theorem 2.2.1(2), we shall defer the proof of the s = p case of (3.9)to Section 4.2.3, after we have proved that M , is rigid and P r,p − is projective in C M ( p ) . In the category
Rep M ( p ) ,(1) The simple module M , is a simple current such that M , ⊠ M r,s = M r +1 ,s (3.8) for r ∈ Z , ≤ s ≤ p .(2) For r ∈ Z and ≤ s ≤ p , M , ⊠ M r,s = M r, if s = 1 M r,s − ⊕ M r,s +1 if 2 ≤ s ≤ p − P r,p − if s = p. (3.9) Proof.
As in the proof of Proposition 3.2.3, there is a non-zero M ( p )-module homomor-phism f : M , ⊠ M r,s → F α r +1 ,s induced by a non-zero intertwining operator M , ⊗ M r,s ֒ → F α , ⊗ F α r,s → F α r +1 ,s { x } . As im f is a non-zero submodule of F α r +1 ,s , it is either M r +1 ,s or F α r +1 ,s . Since inductionis a tensor functor, F W ( p ) ( M , ⊠ M r,s ) ∼ = F W ( p ) ( M , ) ⊠ W ( p ) F W ( p ) ( M r,s ) ∼ = W , ⊠ W ( p ) W ¯ r,s ∼ = W − ¯ r,s by Proposition 3.2.3 and Theorem 2.2.1. Thus because induction is exact, we get an exact W ( p )-module sequence0 −→ F W ( p ) (ker f ) −→ W − ¯ r,s −→ W − ¯ r,s or V α − ¯ r,s + L −→ . This forces F W ( p ) (ker f ) = 0 and F W ( p ) (im f ) = W − ¯ r,s , so f is an isomorphism onto M r +1 ,s , proving (3.8). 19ow the s = 1 case of (3.9) follows from (3.2) and (3.8), as well as the associativityand commutativity of ⊠ .For 2 ≤ s ≤ p −
1, we have a non-zero M ( p )-module homomorphism f : M , ⊠ M r,s →F α r,s +1 induced by an intertwining operator M , ⊗ M r,s ֒ → F α , ⊗ F α r,s → F α r,s +1 { x } , with im f either F α r,s +1 or M r,s +1 . Since induction is monoidal, Theorem 2.2.1 implies F W ( p ) ( M , ⊠ M r,s ) ∼ = F W ( p ) ( M , ) ⊠ W ( p ) F W ( p ) ( M r,s ) ∼ = W ¯ r,s − ⊕ W ¯ r,s +1 . So using exactness of F W ( p ) , there is an exact sequence of W ( p )-modules0 → F W ( p ) (ker f ) → W ¯ r,s − ⊕ W ¯ r,s +1 → W ¯ r,s +1 or V α ¯ r,s +1 + L → . This forces im f = M r,s +1 and F W ( p ) (ker( f )) ∼ = W ¯ r,s − . Again because F W ( p ) is exact,ker( f ) has to be a simple M ( p )-module and therefore equals M r +2 n,s − for some n ∈ Z .Since h r +2 n,s − − h r,s +1 ≡ sp (mod Z ), any extension of M r,s +1 by M r +2 n,s − for 2 ≤ s ≤ p − M , ⊠ M r,s = M r,s +1 ⊕ M r +2 n,s − .Now we show that n = 0 by constructing a non-zero map M , ⊠ M r,s → M r,s − . First,we have the M ( p )-module map f : M , ⊠ F α r − ,p − s → F α r − ,p − s +1 induced by a non-zerointertwining operator M , ⊗ F α r − ,p − s ֒ → F α , ⊗ F α r − ,p − s → F α r − ,p − s +1 { x } . Since M , is generated by a highest-weight vector for the Heisenberg algebra H , im f contains a highest-weight vector in F α r − ,p − s +1 . If r ≤
0, then M r − ,p − s +1 does not containa highest-weight vector for H , so f is surjective in this case. Consequently, we get asurjective homomorphism¯ f : M , ⊠ F α r − ,p − s → F α r − ,p − s +1 ։ M r,s − using (2.14). Moreover, because we have seen any M ( p )-module map M , ⊠ M r − ,p − s →M r,s − is zero, right exactness of M , ⊠ • means ¯ f induces a non-zero M ( p )-module map g : M , ⊠ M r,s → M r,s − as in the diagram: M , ⊠ M r − ,p − s (cid:15) (cid:15) ( ( PPPPPPPPPPPP M , ⊠ F α r − ,p − s (cid:15) (cid:15) ¯ f / / M r,s − M , ⊠ M r,s g ♥♥♥♥♥♥ (cid:15) (cid:15) r ≤
0, 2 ≤ s ≤ p − r > r ≤ s = p case of (3.9) in Section 4.2.3 below.20 .3 Locally-finite tensor categories of singlet modules We have now derived enough fusion rules and induction relationships to show that C M ( p ) and C M ( p ) are tensor subcategories of Rep M ( p ). In the next section, we will use thefusion rules of Theorem 3.2.6 to show that these tensor subcategories are rigid. The category C M ( p ) is a tensor subcategory of Rep M ( p ) and a subcat-egory of C M ( p ) .Proof. The category C M ( p ) is closed under the tensor product on Rep M ( p ) because in-duction is monoidal and because C W ( p ) is a tensor subcategory of Rep W ( p ). This meansthat C M ( p ) is a monoidal subcategory of Rep M ( p ). To show that C M ( p ) is an abeliancategory and therefore a tensor subcategory of Rep M ( p ), it is enough to show that C M ( p ) is closed under submodules and quotients. Consider an exact sequence0 → N → M → N → M ( p ) where M is a module in C M ( p ) . Because induction is exact by Proposition3.2.2, F W ( p ) ( N ) is a W ( p )-submodule of F W ( p ) ( M ) and F W ( p ) ( N ) is a quotient. Since C W ( p ) is closed under submodules and quotients, this means that indeed N and N aremodules in C M ( p ) .Now to show that C M ( p ) is a subcategory of C M ( p ) , we need to show that every module M in C M ( p ) has finite length with composition factors M r,s for r ∈ Z , 1 ≤ s ≤ p . Wefirst show that M contains a simple submodule. Indeed, since F W ( p ) ( M ) is a grading-restricted generalized W ( p )-module, it contains an irreducible submodule W r,s . Since W r,s contains M r,s as an M ( p )-submodule and since F W ( p ) ( M ) ∼ = L n ∈ Z M n +1 , ⊠ M as an M ( p )-module, we have a non-zero (necessarily injective) homomorphism M r,s → M n +1 , ⊠ M for some n ∈ Z . Then because tensoring with the simple current M − n +1 , is exact, wecan use (3.2) and the associativity of ⊠ to get an injection M r − n,s → M .Now since C M ( p ) is closed under quotients, we can iterate to get an ascending chain ofsubmodules 0 ⊆ M ⊆ M ⊆ M ⊆ ... ⊆ M (3.10)such that for each i , M i +1 /M i ∼ = M r i ,s i for r i ∈ Z , 1 ≤ s i ≤ p . Because F W ( p ) is exact,inducing yields an ascending chain of submodules in F W ( p ) ( M ) such that F W ( p ) ( M i +1 ) / F W ( p ) ( M i ) ∼ = F W ( p ) ( M i +1 /M i ) ∼ = F W ( p ) ( M r i ,s i ) ∼ = W ¯ r i ,s i . Since F W ( p ) ( M ) is a finite-length W ( p )-module in C W ( p ) , it follows that the chain (3.10)must terminate at some finite i , so that M has a finite-length composition series withcomposition factors M r i ,s i . The category C M ( p ) is a tensor subcategory of Rep M ( p ) . roof. Because C M ( p ) is the category of all finite-length grading-restricted generalized M ( p )-modules with composition factors M r,s for r ∈ Z , 1 ≤ s ≤ p , it is closed undersubmodules and quotients. This means that C M ( p ) is an abelian category. We also showedin Proposition 3.1.3 that C M ( p ) is a full subcategory of Rep M ( p ). To show that C M ( p ) is a tensor subcategory, therefore, we just need to show that it is closed under the tensorproduct on Rep M ( p ).Since every module M in C M ( p ) has finite length ℓ ( M ), we can use induction on ℓ ( M ) + ℓ ( M ) to prove that M ⊠ M has finite length with composition factors M r,s for any twomodules M , M in C M ( p ) . For the base case ℓ ( M ) = ℓ ( M ) = 1, both M and M aresimple modules, and (3.3) shows that they are modules in C M ( p ) . Thus Theorem 3.3.1shows that M ⊠ M is a (finite-length) module in C M ( p ) in this case.For the inductive step, assume without loss of generality that ℓ ( M ) ≥
2, so that thereis an exact sequence 0 → A f −→ M g −→ B → A , B in C M ( p ) satisfying ℓ ( A ) , ℓ ( B ) < ℓ ( M ). Then since • ⊠ M is rightexact, we have an exact sequence A ⊠ M f ⊠ id M −−−−→ M ⊠ M g ⊠ id M −−−−→ B ⊠ M → . Thus M ⊠ M is an M ( p )-module extension of B ⊠ M by A ⊠ M / ker( f ⊠ id M ). Since A ⊠ M and B ⊠ M are modules in C M ( p ) by the inductive hypothesis, it follows that M ⊠ M is in C M ( p ) as well.The reason we have introduced the category C M ( p ) in addition to C M ( p ) is that, aswe shall show here and in Section 5 below, the irreducible modules M r,s have projectivecovers in C M ( p ) but not in C M ( p ) . The following lemma relates projective objects in C W ( p ) to projective objects in C M ( p ) : If F W ( p ) ( P ) is projective in C W ( p ) for some module P in C M ( p ) , then P isprojective in C M ( p ) .Proof. Consider a surjection p : M ։ N in C M ( p ) and a homomorphism q : P → N . Sinceall three modules here are objects of C M ( p ) , we can induce to a diagram of homomorphismsin C W ( p ) . Since F W ( p ) is exact, F W ( p ) ( p ) is still surjective and projectivity of F W ( p ) ( P ) in C W ( p ) implies there is a homomorphism f : F W ( p ) ( P ) → F W ( p ) ( M ) such that the diagram F W ( p ) ( P ) f v v ♥♥♥♥♥♥♥♥♥♥♥♥ F W ( p ) ( q ) (cid:15) (cid:15) F W ( p ) ( M ) F W ( p ) ( p ) / / F W ( p ) ( N )commutes. 22ow for a module X in C M ( p ) , since F W ( p ) ( X ) ∼ = L n ∈ Z M n +1 , ⊠ X as an M ( p )-module,we have M ( p )-module homomorphisms ι ( n ) X : M n +1 , ⊠ X → F W ( p ) ( X ) , π ( n ) X : F W ( p ) ( X ) → M n +1 , ⊠ X such that π ( m ) X ◦ ι ( n ) X = δ m,n id M n +1 , ⊠ X , X n ∈ Z ι ( n ) X ◦ π ( n ) X = id F W ( p ) ( X ) . Note that the infinite sum here is well defined since it is finite when acting on any vectorin F W ( p ) ( X ).Since F W ( p ) ( g ) = id W ( p ) ⊠ g for any morphism g in C M ( p ) , we have π (0) N ◦ F W ( p ) ( p ) ◦ ι ( n ) M = δ ,n (id M , ⊠ p ) , π (0) N ◦ F W ( p ) ( q ) ◦ ι (0) P = id M , ⊠ q. Thus if we set e f = π (0) M ◦ f ◦ ι (0) P , we getid M , ⊠ q = π (0) N ◦ F W ( p ) ( q ) ◦ ι (0) P = π (0) N ◦ F W ( p ) ( p ) ◦ f ◦ ι (0) P = X n ∈ Z π (0) N ◦ F W ( p ) ( p ) ◦ ι ( n ) M ◦ π ( n ) M ◦ f ◦ ι (0) P = (id M , ⊠ p ) ◦ e f . Since we can identify e f with a homomorphism g : M → N such that q = p ◦ g , this showsthat P is projective in C M ( p ) .As a consequence: For r ∈ Z , the M ( p ) -modules P r,p := F α r,p = M r,p and P r,p − areprojective in C M ( p ) .Proof. We have F W ( p ) ( F α r,p ) ∼ = V α r,p + L by (3.5) and F W ( p ) ( P r,p − ) ∼ = R ¯ r,p − by (3.7), bothof which are projective in C W ( p ) according to [NT, Section 5.1]. So M r,p and P r,p − areprojective in C M ( p ) by Lemma 3.3.3.Since M r,p is irreducible and projective in C M ( p ) , it is obvious that it is its own projectivecover in C M ( p ) . However, we now show that M r,p is not projective, and in fact has noprojective cover, in C M ( p ) . For any n ∈ Z + , let F ( n ) α r,p denote the indecomposable Heisenbergmodule induced from an n -dimensional lowest conformal weight space on which α (0) actsby the indecomposable Jordan block A n = p p α r,p · · · α r,p . . . ...... ... . . . 10 0 · · · α r,p . n ≥
2, we have a non-split extension of H -modules0 −→ F ( n − α r,p −→ F ( n ) α r,p −→ F α r,p −→ . (3.11)Now we consider F ( n ) α r,p as an M ( p )-module (see [AM2, Theorem 6.1] for the n = 2 case).On the lowest conformal weight space, L (0) and the degree-preserving component H (0) of Y ( H, x ) act by the matrices L (0) = 14 p A n − p − p A n , H (0) = (cid:18) A n p − (cid:19) . Analysis of these matrices shows that L (0) acts indecomposably for r = 1, while for r = 1, H (0) is a nilpotent indecomposable matrix. Thus F ( n ) α r,p is also singly-generatedindecomposable as an M ( p )-module.Since each F ( n ) α r,p is an object of C M ( p ) , the non-split exact sequence (3.11) shows that M r,p = F α r,p is not projective in C M ( p ) . But since M r,p is projective in C M ( p ) , this meansthat for n ≥ F ( n ) α r,p is an object of C M ( p ) that is not in C M ( p ) . In particular, F W ( p ) ( F ( n ) α r,p )must be a non-local finite-length W ( p )-module that has local composition factors. We canalso see that M r,p fails to have a projective cover in C M ( p ) because a projective cover wouldhave to surject onto F ( n ) α r,p for all n , which is impossible for a finite-length M ( p )-module.We now discuss properties of P r,p − in C M ( p ) : For r ∈ Z , P r,p − is a projective cover of M r,p − in C M ( p ) , and it hasLoewy diagram M r,p − P r,p − : M r +1 , M r − , M r,p − . Proof.
From (2.18) and (2.14), P r,p − has a submodule/subquotient structure as illustratedby the diagram 0 (cid:15) (cid:15) (cid:15) (cid:15) M r,p − ' ' PPPPPP (cid:15) (cid:15) M r − , (cid:15) (cid:15) / / F α r,p − (cid:15) (cid:15) / / P r,p − ' ' PPPPPPP / / F α r − , / / (cid:15) (cid:15) M r +1 , (cid:15) (cid:15) M r,p − (cid:15) (cid:15) W ( p )-submodules of R ¯ r,p − with P r,p − : G r,p − whose underlying vector space is M r,p − ⊕ M r − , and which contains M r,p − as a submodule, and Z r,p − = Y r,p − ∩ P r,p − .By (3.7) and exactness of induction, semisimple submodules of P r,p − induce to semisim-ple submodules of R ¯ r,p − with the same length. Since Soc( R ¯ r,p − ) = W ¯ r,p − , this meansthat Soc( P r,p − ) = M r,p − . Next, restricting the exact sequences (2.12) and (2.13) to M ( p )-submodules, we get exact sequences0 −→ Z r,p − −→ P r,p − −→ M r,p − −→ −→ M r,p − −→ Z r,p − −→ M r +1 , ⊕ M r − , −→ . The first sequence does not split because induction is exact and R ¯ r,p − ∼ = F W ( p ) ( P r,p − ) isindecomposable. This together with the second sequence impliesSoc( P r,p − / M r,p − ) = Z r,p − / M r,p − ∼ = M r +1 , ⊕ M r − , , and then clearly Soc( P r,p − / Z r,p − ) = P r,p − / Z r,p − ∼ = M r,p − . This verifies the row structure of the Loewy diagram for P r,p − . To complete the verificationof the arrow structure, we need to show that G r,p − and P r,p − / G r,p − are indecomposable.For G r,p − , we note that a similar proof to that of (3.5) shows that F W ( p ) ( G r,p − ) is isomor-phic to the corresponding indecomposable W ( p )-submodule of R ¯ r,p − (called M in [AM4,Lemma 5.2]); this means G r,p − is indecomposable. Then P r,p − / G r,p − is indecomposablebecause it induces to the indecomposable W ( p )-module R ¯ r,p − / F W ( p ) ( G r,p − ).Now we can show that P r,p − is a projective cover of M r,p − . Note that the Loewydiagram implies that P r,p − is generated by any vector not in Z r,p − , that is, Z r,p − is theunique maximal proper submodule of P r,p − . Also, there is a surjection q : P r,p − → M r,p − such that ker q = Z r,p − . Now suppose P is any projective object in C M ( p ) with surjectivemap e q : P → M r,p − . Then there is a map f : P → P r,p − such that the diagram P f x x ♣♣♣♣♣♣♣♣♣♣♣♣ e q (cid:15) (cid:15) P r,p − q / / M r,p − commutes. We need to show that f is surjective. Indeed, otherwise we would have im f ⊆Z r,p − , so that q ◦ f = 0, contradicting the surjectivity of e q . In this section, we prove that C M ( p ) is a rigid tensor category, using the method of [TW]for proving rigidity of C W ( p ) . The steps of the proof are the following:251) First prove that M , is rigid and self-dual using the fusion rules (3.9) and Belavin-Polyakov-Zamolodchikov differential equations. This proof is exactly the same as therigidity proof in [TW] for the W ( p )-module W , , but we provide a more detailedexposition, especially for p = 2, in which case we do not assume the fusion rule (3.9).(2) Next, use rigidity of M , and of the simple currents M r, , together with the fusionrules from the previous section, to show that all simple modules in C M ( p ) are rigid.(3) Finally, use rigidity of simple modules and induction on length to prove that allmodules in C M ( p ) are rigid. Here, our proof differs from that of [TW], which useda lemma from the Appendix of [KL5] whose proof is valid only in categories withenough rigid projective objects.We begin with a discussion of the BPZ equations satisfied by compositions of intertwiningoperators involving the module M , . Let Y and Y be M ( p )-module intertwining operators of types (cid:0) M , M , M (cid:1) and (cid:0) M M , M , (cid:1) ,respectively, for some M ( p )-module M . Also, let Y and Y be M ( p )-module intertwiningoperators of types (cid:0) M , N M , (cid:1) and (cid:0) N M , M , (cid:1) , respectively, for some M ( p )-module N . Wethen define ϕ ( x ) = h v ′ , Y ( v, Y ( v, x ) v i ψ ( x ) = h v ′ , Y ( Y ( v, − x ) v, x ) v i , where v is a highest-weight vector in M , , of conformal weight h , , and v ′ is a highest-weight vector in the contragredient module M ′ , . (It turns out that M ′ , ∼ = M , , so laterwe will identify h· , ·i with a non-degenerate invariant bilinear form on M , .) We can view ϕ ( x ) and ψ ( x ) either as formal series in x and 1 − x , respectively, or, replacing x with z ,as analytic functions on the (simply-connected) regions U = { z ∈ C | | z | < } \ ( − , U = { z ∈ C | | − z | < | z |} \ [1 , ∞ ) = { z ∈ C | Re z > / } \ [1 , ∞ ) , respectively.It is well known that singular vectors in the Virasoro Verma module of conformal weight h , lead to a differential equation for ϕ ( x ) and ψ ( x ) [BPZ, Hu1]. To derive this equation,we need the following consequences of the intertwining operator Jacobi identity: Let Y be a logarithmic intertwining operator among V -modules of type (cid:0) W W W (cid:1) . Then for u ∈ V , w ∈ W , and n ∈ Z , [ u n , Y ( w , x )] = X k ≥ (cid:18) nk (cid:19) x n − k Y ( u k w , x ) , (4.1)26 nd Y ( u n w, z ) = X k ≥ (cid:18) nk (cid:19) ( − x ) k u n − k Y ( w , x ) − X k ≥ (cid:18) nk (cid:19) ( − x ) n − k Y ( w , x ) u k . (4.2) The series ϕ ( x ) and ψ ( x ) satisfy the differential equation px (1 − x ) f ′′ ( x ) + (1 − x ) f ′ ( x ) − h , x (1 − x ) f ( x ) = 0 . (4.3) Proof.
We first derive a partial differential equation for the formal seriesΦ( x , x ) = h v ′ , Y ( v, x ) Y ( v, x ) v i using the fact that (cid:18) L ( − − p L ( − (cid:19) v = 0in the irreducible Virasoro module L ( c p , h , ) ⊆ M , . Thus by the L ( − p ∂ x Φ( x , x ) = h v ′ , Y ( v, x ) Y ( L ( − v, x ) v i = X k ≥ x k h v ′ , Y ( v, x ) ω − k − Y ( v, x ) v i + X k ≥ x − k − h v ′ , Y ( v, x ) Y ( v, x ) ω k v i = X k ≥ x k h v ′ , ω − k − Y ( v, x ) Y ( v, x ) v i− X k ≥ X i ≥ (cid:18) − k − i (cid:19) x − k − i − x k h v ′ , Y ( ω i v, x ) Y ( v, x ) v i + (cid:0) x − h v ′ , Y ( v, x ) Y ( v, x ) L ( − v i + x − h v ′ , Y ( v, x ) Y ( v, x ) L (0) v i (cid:1) = − X k ≥ x − k − x k h v ′ , Y ( L ( − v, x ) Y ( v, x ) v i + X k ≥ ( k + 1) x − k − x k h v ′ , Y ( L (0) v, x ) Y ( v, x ) v i− x − ( h v ′ , Y ( v, x ) Y ( L ( − v, x ) v i + h v ′ , Y ( L ( − v, x ) Y ( v, x ) v i )+ h , x − h v ′ , Y ( v, x ) Y ( v, x ) v i = − (cid:0) ( x − x ) − + x − (cid:1) ∂ x Φ( x , x ) − x − ∂ x Φ( x , x )+ h , (cid:0) − ∂ x ( x − x ) − + x − (cid:1) Φ( x , x )= − (cid:18) x − x + 1 x (cid:19) ∂ x Φ( x , x ) − x ∂ x Φ( x , x ) + h , (cid:18) x − x ) + 1 x (cid:19) Φ( x , x ) . Now, the L (0)-conjugation formula for intertwining operators implies thatΦ( x , x ) = x − h , ϕ ( x /x ) . , x ) = ϕ ( x ) , ∂ x Φ( x , x ) | x =1 ,x = x = ϕ ′ ( x ) , ∂ x Φ( x , x ) | x =1 ,x = x = ϕ ′′ ( x ) , and ∂ x Φ( x , x ) | x =1 ,x = x = (cid:16) − h , x − h , − ϕ ( x /x ) − x − h , − x ϕ ′ ( x /x ) (cid:17)(cid:12)(cid:12)(cid:12) x =1 ,x = x = − h , ϕ ( x ) − xϕ ′ ( x ) . Plugging these relations into the partial differential equation for Φ( x , x ), we get p ϕ ′′ ( x ) = (cid:0) (1 − x ) − + x − (cid:1) (2 h , ϕ ( x ) + xϕ ′ ( x )) − x − ϕ ′ ( x ) + h , (cid:0) (1 − x ) − + x − (cid:1) ϕ ( x )= (cid:18) x − x + 1 − x (cid:19) ϕ ′ ( x ) + h , (cid:18) − x + 2 x + 1(1 − x ) + 1 x (cid:19) ϕ ( x ) , which means px (1 − x ) ϕ ′′ s ( x ) = (2 x − ϕ ′ s ( x ) + h , x (1 − x ) ϕ s ( x )as required.For ψ ( x ), we consider the formal seriesΨ( x , x ) = h v ′ , Y ( Y ( v, x ) v, x − x ) v i . So using the L ( − p ∂ x Ψ( x , x )= p ∂ x (cid:0) h v ′ , Y ( Y ( L ( − v, x ) v, x − x ) v i − h v ′ , Y ( L ( − Y ( v, x ) v, x − x ) v i (cid:1) = − p ∂ x h v ′ , Y ( Y ( v, x ) L ( − v, x − x ) v i = p h v ′ , Y ( Y ( v, x ) L ( − v, x − x ) v i = h v ′ , Y ( Y ( v, y ) L ( − v, y ) v i| y = x ,y = x − x . Then a calculation using (4.1) and (4.2) as before leads to p ∂ x Ψ( x , x ) = − y − ∂ y Ψ( y , y + y ) | y = x ,y = x − x − x − ∂ y Ψ( y , y + y ) | y = x ,y = x − x + h , ( y − + y − )Ψ( y , y + y ) | y = x ,y = x − x = − x − ∂ x Ψ( x , x ) − (cid:0) x − + ( x − x ) − (cid:1) ∂ x Ψ( x , x )+ h , (cid:0) x − + ( x − x ) − (cid:1) Ψ( x , x ) . Now defining e ψ ( x ) = ψ (1 − x ), the L (0)-conjugation property for intertwining operatorsimplies Ψ( x , x ) = x − h , e ψ ( x /x ) . x,
1) = e ψ ( x ) , ∂ x Ψ( x , x ) | x = x,x =1 = e ψ ′ ( x ) , ∂ x Ψ( x , x ) | x = x,x =1 = e ψ ′′ ( x ) , and ∂ x Ψ s ( x , x ) | x = x,x =1 = − h , e ψ ( x ) − x e ψ ′ ( x ) . Plugging these relations into the partial differential equation for e Ψ( x , x ), we get p e ψ ′′ ( x ) = − x − e ψ ′ ( x ) − ( x − + (1 − x ) − )( − h , e ψ ( x ) − x f ψ s ′ ( x ))+ h , ( x − + (1 − x ) − ) e ψ ( x )= (cid:18) x − x + 1 − x (cid:19) e ψ ′ ( x ) + h , (cid:18) − x + 2 x + 1(1 − x ) + 1 x (cid:19) e ψ ( x ) . This simplifies to p x (1 − x ) e ψ ′′ ( x ) = (2 x − e ψ ′ ( x ) + h , x (1 − x ) e ψ ( x ) , so substituting x − x shows that ψ ( x ) satisfies (4.3).To solve the differential equation (4.3), set g ( x ) = x − / p (1 − x ) − / p f ( x ) . Then a straightforward calculation shows that g ( x ) satisfies the hypergeometric differentialequation p x (1 − x ) g ′′ ( x ) + 2(1 − x ) g ′ ( x ) + (1 − /p ) g ( x ) = 0 , (4.4)as in [TW, Equation 4.83], whose solutions on U and U can be obtained by the methodof Frobenius and are well known.For p ≥
3, the indicial equation of (4.4) has two distinct roots, and it follows that thesolutions to (4.3) on U have the basis: ϕ ( x ) = x / p (1 − x ) / p F (cid:18) p , p − , p ; x (cid:19) ∈ x / p C [[ x ]] ,ϕ ( x ) = x − / p (1 − x ) / p F (cid:18) − p , p , − p ; x (cid:19) ∈ x − / p C [[ x ]] . On U , (4.3) has basis of solutions ψ ( x ) = x / p (1 − x ) / p F (cid:18) p , p − , p ; 1 − x (cid:19) ∈ (1 − x ) / p C [[1 − x ]] ,ψ ( x ) = x / p (1 − x ) − / p F (cid:18) − p , p , − p ; 1 − x (cid:19) ∈ (1 − x ) − / p C [[1 − x ]] .
29s in [TW, Equation 4.85], these two bases of solutions are related on the intersection U ∩ U = { z ∈ C | > | z | > | − z | > } by the hypergeometric function connection formulas: ϕ ( z ) = 12 cos πp ψ ( z ) + 3 − p − p Γ( p ) Γ( p )Γ( p ) ψ ( z ) (4.5) ψ ( z ) = 12 cos πp ϕ ( z ) + 3 − p − p Γ( p ) Γ( p )Γ( p ) ϕ ( z ) (4.6)when 1 > | z | > | − z | > p = 2, on the other hand, the indicial equation of (4.4) has a repeated root, so oneof the basis solutions for (4.3) on U is logarithmic: ϕ ( x ) = x / (1 − x ) / F (cid:18) , , x (cid:19) ,ϕ ( x ) = ϕ ( x ) log x + x / (1 − x ) / G ( x )for a power series G ( x ) with constant term 0. As in the p ≥ U can be obtained via the substitution x − x , that is, ψ i ( x ) = ϕ i (1 − x ) for i = 1 , ϕ ( z ) = ln 4 π ψ ( z ) − π ψ ( z ) (4.7)for 1 > | z | > | − z | > M , In this section, we prove that the simple M ( p )-module M , is rigid. Since (3.9) showsthat the identity of M , ⊠ M , depends on p , the proof is divided into two cases. p ≥ p ≥
3, in which case (3.9) shows M , ⊠ M , ∼ = M , ⊕ M , . This direct sum decomposition means that for s = 1 ,
3, we have homomorphisms i s : M ,s → M , ⊠ M , , p s : M , ⊠ M , → M ,s such that p s ◦ i s = id M ,s s = 1 , i ◦ p + i ◦ p = id M , ⊠ M , . If we take Y ⊠ to be the tensor product intertwining operator of type (cid:0) M , ⊠ M , M , M , (cid:1) , then Y s = p s ◦ Y ⊠ for s = 1 , (cid:0) M ,s M , M , (cid:1) .Moreover, Y ⊠ = i ◦ Y + i ◦ Y . (4.8)We may take i and p to be preliminary candidates for the coevaluation and evalua-tion, respectively, for M , . Then to show M , is rigid, it is sufficient to show that thehomomorphisms f, g : M , → M , defined by the commutative diagrams M , f (cid:15) (cid:15) r − M , / / M , ⊠ M , M , ⊠ i / / M , ⊠ ( M , ⊠ M , ) A M , , M , , M , (cid:15) (cid:15) M , M , ⊠ M , l M , o o ( M , ⊠ M , ) ⊠ M , p ⊠ id M , o o (4.9)and M , g (cid:15) (cid:15) l − M , / / M , ⊠ M , i ⊠ id M , / / ( M , ⊠ M , ) ⊠ M , A − M , , M , , M , (cid:15) (cid:15) M , M , ⊠ M , r M , o o M , ⊠ ( M , ⊠ M , ) id M , ⊠ p o o (4.10)are non-zero multiples of the identity. In fact, since M , is simple, it is sufficient to showthat f and g are non-zero.To prove f = 0, we need to show that the intertwining operator Y = l M , ◦ ( p ⊠ id M , ) ◦ A M , , M , , M , ◦ (id M , ⊠ i ) ◦ Y ⊠ = l M , ◦ ( p ⊠ id M , ) ◦ A M , , M , , M , ◦ Y ⊠ (2 ⊠ ◦ (id M , ⊗ i )of type (cid:0) M , M , M , (cid:1) is non-zero. For this, it is sufficient to show that h v ′ , Y ( v, Y ( v, x ) v i 6 = 0for non-zero highest weight vectors v ∈ M , , v ′ ∈ M ′ , ∼ = M , , and for some x ∈ R such that 1 > x > − x >
0. We define the intertwining operator Y of type (cid:0) M , M , M , (cid:1) i instead of i . Then we use (4.8) to calculate h v ′ , Y ( v, Y ( v, x ) v i + h v ′ , Y ( v, Y ( v, x ) v i = D v ′ , l M , ◦ ( p ⊠ id M , ) ◦ A M , , M , , M , (cid:0) Y ⊠ (2 ⊠ ( v, Y ⊠ ( v, x ) v (cid:1)E = D v ′ , l M , ◦ ( p ⊠ id M , ) (cid:0) Y (2 ⊠ ⊠ ( Y ⊠ ( v, − x ) v, x ) v (cid:1)E = (cid:10) v ′ , l M , (cid:0) Y ⊠ ( Y ( v, − x ) v, x ) v (cid:1)(cid:11) = h v ′ , Y M , ( Y ( v, − x ) v, x ) v i . (4.11)Now by Proposition 4.1.2, both sides of (4.11) are solutions to (4.3). Moreover, since Y s ( v, x ) v ∈ x h ,s − h , M , [[ x ]] = (cid:26) x − / p M , [[ x ]] if s = 1 x / p M , [[ x ]] if s = 3 , we have h v ′ , Y ( v, Y ( v, x ) v i = c ϕ ( x ) h v ′ , Y ( v, Y ( v, x ) v i = c ϕ ( x )for c , c ∈ C , using the notation of the previous subsection.We can be more precise about the series expansion of Y : since M , is simple andself-contragredient, Y is necessarily related to Y M , by the skew-symmetry and adjointintertwining operators of [HLZ2]. In particular, Y ( v, x ) v ∈ x − h , (cid:0) h e (2 r +1) πiL (0) v, v i + x C [[ x ]] (cid:1) for some r ∈ Z , where h· , ·i is a non-degenerate invariant bilinear form on M , . Then itfollows that h v ′ , Y M , ( Y ( v, − x ) v, x ) v i ∈ (1 − x ) − h , (cid:0) h v ′ , v ih e (2 r +1) πiL (0) v, v i + (1 − x ) C [[1 − x ]] (cid:1) , which means that if we choose v and v ′ to be non-zero, then the right side of (4.11) is anon-zero multiple dψ ( x ). Now in the case p ≥
4, we can solve (4.5) and (4.6) to show that c = − d π/p ) = 0, which means Y = 0, as required. For p = 3, (4.5) and (4.6) showthat ϕ ( x ) = ψ ( x ) for 1 > x > − x >
0, so that (4.11) amounts to h v ′ , Y ( v, Y ( v, x ) v i = − c ψ ( x ) + dψ ( x ) . Since ψ ( x ), ψ ( x ) are linearly independent on the interval ( ,
1) and since d = 0, we againconclude Y = 0. This proves that f in (4.9) is non-zero. The proof that g in (4.10) isnon-zero is similar, so M , is rigid when p ≥ .2.2 The case p = 2Now for p = 2, M , = F α , is an irreducible Fock module, and we have a surjectiveintertwining operator of type (cid:0) F α , M , M , (cid:1) , which induces a surjective homomorphism p : M , ⊠ M , → F α , → M , . Since M , is irreducible, this means M , is self-contragredient, and the intertwiningoperator Y = p ◦ Y ⊠ must be related to Y M , by skew-symmetry and adjoints. Inparticular, if v ∈ M , is a lowest-conformal-weight vector, then Y ( v, x ) v = e − (2 r +1) h , h v, v i x − h , + . . . (4.12)for some r ∈ Z and non-degenerate invariant bilinear form h· , ·i on M , . We can rescale v and/or p so that the coefficient of x − h , in Y ( v, x ) v is . Then let e denote thecoefficient of x − h , in Y ⊠ ( v, x ) v , so p ( e ) = . Since conformal weights of modules in C M (2) are bounded below by − p ( p − = − , the lowest power of x from the coset − h , + Z occurring in Y ⊠ ( v, x ) v is x − h , .We have not yet shown that M , ⊠ M , ∼ = P , as in (3.9), but we will use P , toget a coevaluation i : M , → M , ⊠ M , . From the structure of P , illustrated in(3.12), we have an injection i : M , → P , and a surjection p : P , → M , . Themodule P , has a two-dimensional lowest conformal weight space of weight 0, and since P , is logarithmic, ( P , ) [0] has a basis { , L (0) } such that p ( ) = and (rescaling i ifnecessary) i ( ) = L (0) . Recall from Corollary 3.3.4 that P , is projective in C M (2) .Now let M be the M (2)-submodule of M , ⊠ M , generated by e . Since F W (2) ( M ) ֒ → F W (2) ( M , ⊠ M , ) ∼ = W , ⊠ W (2) W , and F W (2) ( M , ) ∼ = W , are objects of C W (2) , p | M : M → M , is a (surjective) homo-morphism in C M (2) . Thus because P , is projective in C M (2) , we get a homomorphism q : P , → M ֒ → M , ⊠ M , , and then i : M , → M , ⊠ M , , such that the diagram M , i / / i ' ' PPPPPPPPPPPPP P , q (cid:15) (cid:15) p ' ' PPPPPPPPPPPPPP M , ⊠ M , p / / M , commutes. We want to show that M , is rigid self-dual with evaluation (a multiple of) p and coevaluation i . As in the p ≥ f and g definedby the diagrams (4.9) and (4.10) are non-zero. Here we will focus on showing g = 0, sincethe proof for f is similar.We first show that L (0) e = i ( ). Note that since q ( ) = e + m for some m ∈ ker p | M [0] ,we have L (0) e = q ( L (0) ) − L (0) m = i ( ) − L (0) m.
33o it is sufficient to show that L (0) ker p | M [0] = 0. Note that because conformal weightsof modules in C M (2) are bounded below by − , M [0] is contained in the top level of M andthus is a finite-dimensional module for the Zhu algebra A ( M (2)) computed in [Ad1]. Asan A ( M (2))-module, M [0] is generated by e , and it is spanned by monomials H (0) i L (0) j e since M (2) is generated by H and ω , and since [ H ], [ ω ] commute in A ( M (2)). Moreover,since [ H ] in A ( M (2)) equals a polynomial in [ ω ], and since L (0) is nilpotent on M [0] , wemay take i ≤ j ≤ N for some N ∈ N . Note that ker p | M [0] is spanned by monomials H (0) i L (0) j e with i + j >
0, since the kernel has codimension 1 and since L (0), H (0) acttrivially on ( M , ) [0] = C .Now we can write q ( ) = X i =0 N X j =0 a i,j H (0) i L (0) j e (4.13)with a , = 1. By the relations L (0) = 0 = H (0) L (0) and a , = 1, successive applications of H (0) L (0) N , L (0) N , H (0) L (0) N − , . . . , L (0) , H (0) L (0)to both sides of (4.13) show that H (0) i L (0) j e = 0 for i + j >
1. This means thatker p | M [0] = span { L (0) e , H (0) e } and L (0) ker p | M [0] = 0, as required.Now we begin applying the composition in (4.10) to a lowest-conformal-weight vector v ∈ M , :( i ⊠ id M , ) ◦ l − M , ( v ) = ( i ⊠ id M , )( Y ⊠ ( , v ) = Y (2 ⊠ ⊠ ( i ( ) , v, where Y ⊠ and Y (2 ⊠ ⊠ are the relevant tensor product intertwining operators. But since i ( ) = L (0) e is the coefficient of x − h , in L (0) Y ⊠ ( v, x ) v , and since x − h , is the lowestpower of x from the coset − h , + Z occurring in Y ⊠ ( v, x ) v , we see that ( i ⊠ id M , ) ◦ l − M , ( v ) is the coefficient of (1 − x ) − h , in the expansion of Y (2 ⊠ ⊠ ( L (0) Y ⊠ ( v, − x ) v, x ) v = Y (2 ⊠ ⊠ ( L (0) Y ⊠ ( v, − x ) v, − (1 − x )) v as a series in 1 − x . Defining the intertwining operator Y ⊠ , = r M , ◦ (id M , ⊠ p ) ◦ A − M , , M , , M , ◦ Y (2 ⊠ ⊠ of type (cid:0) M , M , ⊠ M , M , (cid:1) , we will thus get g = 0 provided the coefficient of (1 − x ) − h , inthe series expansion of (cid:10) v ′ , Y ⊠ , ( L (0) Y ⊠ ( v, − x ) v, x ) v (cid:11) (4.14)34s non-zero.To analyze (4.14), we first replace 1 and x with z and z , respectively, and use the L (0)-commutator formula L (0) Y ⊠ ( v, z ) v = 2 h , Y ⊠ ( v, z ) v + z Y ⊠ ( L ( − v, z ) (4.15)(see [HLZ2, Equation 3.28]) to calculate, for | z | > | z | > | z − z | > (cid:10) v, Y ⊠ , ( L (0) Y ⊠ ( v, z − z ) v, z ) v (cid:11) = 2 h , D v, r M , ◦ (id M , ⊠ p ) ◦ A − M , , M , , M , (cid:0) Y (2 ⊠ ⊠ ( Y ⊠ ( v, z − z ) v, z ) v (cid:1)E + ( z − z ) ·· D v, r M , ◦ (id M , ⊠ p ) ◦ A − M , , M , , M , (cid:0) Y (2 ⊠ ⊠ ( Y ⊠ ( L ( − v, z − z ) v, z ) v (cid:1)E = 2 h , D v, r M , ◦ (id M , ⊠ p ) (cid:0) Y ⊠ (2 ⊠ ( v, z ) Y ⊠ ( v, z ) v (cid:1)E + ( z − z ) D v, r M , ◦ (id M , ⊠ p ) (cid:0) Y ⊠ (2 ⊠ ( L ( − v, z ) Y ⊠ ( v, z ) v (cid:1)E = (2 h , + ( z − z ) ∂ z ) (cid:10) v, r M , (cid:0) Y ⊠ ( v, z ) Y ( v, z ) v (cid:1)(cid:11) = (2 h , + ( z − z ) ∂ z ) (cid:10) v, Ω( Y M , )( v, z ) Y ( v, z ) v (cid:11) , where Ω represents the skew-symmetry operation on vertex operators. Now, by the L (0)-conjugation formula for intertwining operators and the fact that Y is not a logarithmicintertwining operator, (cid:10) v, Ω( Y M , )( v, z ) Y ( v, z ) v (cid:11) = dz − h , ϕ ( z /z )for some d ∈ C , using the notation of the previous subsection. It follows from (4.12) that d = 0. So now (cid:10) v, Y ⊠ , ( L (0) Y ⊠ ( v, z − z ) v, z ) v (cid:11) = (2 h , + ( z − z ) ∂ z ) (cid:16) dz − h , ϕ ( z /z ) (cid:17) = d z (cid:16) h , z − h , − ϕ ( z /z ) − ( z − z ) z − h , − ϕ ′ ( z /z ) (cid:17) . Setting z = 1 and z = x for x ∈ R , 1 > x > − x > , and using (4.7), we have (cid:10) v, Y ⊠ , ( L (0) Y ⊠ ( v, − x ) v, x ) v (cid:11) = d x (cid:18) h , − (1 − x ) ddx (cid:19) ϕ ( x )= x (cid:18) h , − (1 − x ) ddx (cid:19) ( c ψ ( x ) + c ψ ( x ))= x (cid:18) h , − (1 − x ) ddx (cid:19) (cid:0) ( c + c log(1 − x )) ϕ (1 − x ) + c (1 − x ) − h , x − h , G (1 − x ) (cid:1) where c = d ln 4 /π and c = − d/π = 0. We need to expand the right side as a series in1 − x and show that the coefficient of (1 − x ) − h , is non-zero.35et us introduce the notation(1 − x ) − h , e ϕ (1 − x ) = ϕ (1 − x )(1 − x ) − h , e G (1 − x ) = (1 − x ) − h , x − h , G (1 − x ) , so that e ϕ (1 − x ), e G (1 − x ) are power series in 1 − x . Thus we have (cid:10) v, Y ⊠ , ( L (0) Y ⊠ ( v, − x ) v, x ) v (cid:11) = x (cid:18) h , − (1 − x ) ddx (cid:19) (1 − x ) − h , (cid:16) ( c + c log(1 − x )) e ϕ (1 − x ) + c e G (1 − x ) (cid:17) = − x (1 − x ) − h , +1 ddx (cid:16) ( c + c log(1 − x )) e ϕ (1 − x ) + c e G (1 − x ) (cid:17) = x (1 − x ) − h , +1 (cid:16) c (1 − x ) − e ϕ (1 − x ) + ( c + c log(1 − x )) e ϕ ′ (1 − x ) + c e G ′ (1 − x ) (cid:17) ∈ (1 − x ) − h , (cid:0) c e ϕ (1 − x ) + (1 − x ) C [[1 − x ]][log(1 − x )] (cid:1) . Since c = 0 and e ϕ (1 − x ) = x − h , F (cid:0) , ,
1; 1 − x (cid:1) is a power series in 1 − x withconstant term 1, it follows that the coefficient of (1 − x ) − h , here is indeed non-zero. Thiscompletes the proof that g = 0. As mentioned previously, the proof that f = 0 is similar:it uses (4.7) with ψ and ϕ solutions exchanged, which is valid because the basis solutionsto (4.3) in U ∩ R and U ∩ R are related by the substitution x − x . Thus M , isrigid and self-dual when p = 2. s = p case of (3.9)Now that we know M , is rigid, we use the projectivity of P r,p − and M r,p in C M ( p ) (recallCorollary 3.3.4) to prove the fusion rule (3.9) for s = p . Since the tensor product of arigid object with a projective object in any tensor category is projective (see for exampleCorollary 2 in the Appendix of [KL5]), M , ⊠ M r,p is projective in C M ( p ) . We also have anintertwining operator of type (cid:0) M r,p − M , M r,p (cid:1) obtained by restricting and projecting a non-zerointertwining operator among Fock modules. This intertwining operator is surjective atleast as long as r ≤
0, so we get a surjective homomorphism p : M , ⊠ M r,p → M r,p − .Letting p : P r,p − → M r,p − denote a surjective homomorphism, projectivity of M , ⊠ M r,p in C M ( p ) implies the existence of a homomorphism q : M , ⊠ M r,p → P r,p − such that the diagram P r,p − p ' ' ❖❖❖❖❖❖❖❖❖❖❖ M , ⊠ M r,pq O O p / / M r,p − P r,p − , p ) is a projective cover of M r,p − by Proposition 3.3.5, q is surjec-tive. Then projectivity of P r,p − means that P r,p − is a direct summand of M , ⊠ M r,p : M , ⊠ M r,p ∼ = P r,p − ⊕ Q for some submodule Q . But if we induce, we get R ¯ r,p − ∼ = W , ⊠ W ( p ) W ¯ r,p ∼ = F W ( p ) ( M , ) ⊠ W ( p ) F W ( p ) ( M r,p ) ∼ = F W ( p ) ( M , ⊠ M r,p ) ∼ = F W ( p ) ( P r,p − ) ⊕ F W ( p ) ( Q ) ∼ = R ¯ r,p − ⊕ F W ( p ) ( Q ) . Since R ¯ r,p − is indecomposable and non-zero, it follows that F W ( p ) ( Q ) = 0, and then Q = 0as well. So M , ⊠ M r,p ∼ = P r,p − . This proves the s = p , r ≤ r > We now prove that all simple modules M r,s in the tensor category C M ( p ) are rigid, withduals given by the contragredient modules of [FHL].Recall that for a vertex operator algebra V and a grading-restricted generalized V -module W = L h ∈ C W [ h ] , the contragredient is a module structure on the graded dual W ′ = L h ∈ C W ∗ [ h ] . If C is a category of grading-restricted V -modules closed under contragredients,then contragredients define a contravariant functor, with the contragredient of a morphism f : W → W defined by h f ′ ( w ′ ) , w i = h w ′ , f ( w ) i for w ∈ W , w ′ ∈ W ′ . Moreover, for any module W in C , we have a natural isomorphism δ W : W → W ′′ defined by h δ W ( w ) , w ′ i = h w ′ , w i for w ∈ W , w ′ ∈ W ′ . It is straightforward to prove: The contragredient functor on C is exact, that is, if a sequence of V -modules W f −→ W g −→ W is exact at W , then W ′ f ′ −→ W ′ g ′ −→ W ′ is exact at W ′ . We use this proposition to prove:
The category C M ( p ) of M ( p ) -modules is closed under contragredients.Proof. Since every module in C M ( p ) has finite length, we can use induction on length. Forthe base case, we need to show that the contragredient of M r,s for r ∈ Z , 1 ≤ s ≤ p isisomorphic to some M r ′ ,s ′ . Since M ′ r,s is an irreducible M ( p )-module by [FHL, Proposition5.3.2], and since the lowest conformal weight of M ′ r,s agrees with that of M r,s , the onlyother possibility is that M ′ r,s ∼ = F λ for some λ / ∈ L ◦ such that h λ = h α r,s or h λ = h α − r,s r ≥ r ≤
0. However, we can see from (2.2) and (2.3) that h λ = h r,s forsome r ≥
1, 1 ≤ s ≤ p only if λ = (( p − ± ( pr − s )) α p ∈ L ◦ . So in fact F λ ∼ = M ′ r,s for λ / ∈ L ◦ is impossible, proving the base case of the induction.For the inductive step, suppose M is a module in C M ( p ) with length ℓ ( M ) ≥
2. Thenwe have an exact sequence 0 → N → M → N → N , N modules in C M ( p ) such that ℓ ( N ) , ℓ ( N ) < ℓ ( M ). By Proposition 4.3.2, thesequence of M ( p )-modules 0 → N ′ → M ′ → N ′ → N ′ , N ′ are modules in C M ( p ) . Thus M ′ , asan extension of two modules in C M ( p ) , is also a module in C M ( p ) . By analyzing conformal weights using (2.3) again, we can see that h ,s = h r ′ ,s ′ for r ′ ≥
1, 1 ≤ s ′ ≤ p if and only if r ′ = 1, s ′ = s . This forces M ′ ,s ∼ = M ,s for all1 ≤ s ≤ p ; in particular, M ( p ) = M , is a self-contragredient vertex operator algebra.When a category C of grading-restricted generalized modules for a self-contragredientvertex operator algebra V is closed under contragredients and has vertex algebraic braidedtensor category structure as in [HLZ1]-[HLZ8], we have an alternative characterization ofthe contragredient functor (see for example [CKM2]). In particular, the isomorphisms ofintertwining operator spaces V WV,W → V
WW,V → V V ′ W,W ′ → V VW ′ ,W → Hom V ( W ′ ⊠ W, V )for a V -module W in C allow us to identify the vertex operator Y W with an evaluationhomomorphism e W : W ′ ⊠ W → V . The pair ( W ′ , e W ) satisfies the following universalproperty: for any homomorphism f : X ⊠ W → V in C , there is a unique homomorphism ϕ : X → W ′ such that the diagram X ⊠ W f ' ' PPPPPPPPPPPPPP ϕ ⊠ id W (cid:15) (cid:15) W ′ ⊠ W e W / / V (4.16)commutes. This universal property allows us to characterize contragredient homomor-phisms and the ribbon isomorphisms δ W in terms of commutative diagrams: Suppose C is a braided tensor category of modules for a self-contragredientvertex operator algebra that is closed under contragredients. Then we can choose the eval-uations e W : W ′ ⊠ W → V for modules W in C in such a way that:
1) For any homomorphism f : W → W in C , f ′ : W ′ → W ′ is the unique homomor-phism such that the diagram W ′ ⊠ W W ′ ⊠ f / / f ′ ⊠ id W (cid:15) (cid:15) W ′ ⊠ W e W (cid:15) (cid:15) W ′ ⊠ W e W / / V (4.17) commutes.(2) For any module W in C , δ W : W → W ′′ is the unique homomorphism such that thediagram W ⊠ W ′ δ W ⊠ id W ′ (cid:15) (cid:15) R W,W ′ ◦ ( θ W ⊠ id W ′ ) / / W ′ ⊠ W e W (cid:15) (cid:15) W ′′ ⊠ W ′ e W ′ / / V (4.18) commutes, where θ W = e πiL (0) is the twist on W .Proof. Fix an isomorphism κ : V ′ → V and for a module W in C , let E W ∈ V VW ′ ,W denotethe intertwining operator corresponding to a choice of e W , that is, E W = e W ◦ Y W ′ ,W where Y W ′ ,W ∈ V W ′ ⊠ WW ′ ,W is the tensor product intertwining operator. Let us choose E W = κ ◦ Ω ( A (Ω( Y W ))) , using the notation of [HLZ2], for all modules W in C . A calculation shows that this means E W satisfies h v ′ , E W ( w ′ , x ) w i = h e − x − L (1) w ′ , Y W ( e xL (1) e κ ( v ′ ) , x − ) e − xL (1) e − πiL (0) x − L (0) w i (4.19)for v ′ ∈ V ′ , w ′ ∈ W ′ , and w ∈ W , where e κ = δ − V ◦ κ ′ : V ′ → V .Now for a V -homomorphism f : W → W in C , we use the definitions, (4.19), andproperties of V -module homomorphisms to calculate (cid:10) v ′ , ( e W ◦ ( f ′ ⊠ id W )) Y W ′ ,W ( w ′ , x ) w (cid:11) = h v ′ , E W ( f ′ ( w ′ ) , x ) w i = D e − x − L (1) f ′ ( w ′ ) , Y W ( e xL (1) e κ ( v ′ ) , x − ) e − xL (1) e − πiL (0) x − L (0) w E = D e − x − L (1) w ′ , Y W ( e xL (1) e κ ( v ′ ) , x − ) e − xL (1) e − πiL (0) x − L (0) f ( w ) E = h v ′ , E W ( w ′ , x ) f ( w ) i = (cid:10) v ′ , (cid:0) e W ◦ (id W ′ ⊠ f ) (cid:1) Y W ′ ,W ( w ′ , x ) w (cid:11) v ′ ∈ V ′ , w ′ ∈ W ′ , and w ∈ W . Since the tensor product intertwining operator Y W ′ ,W is surjective, this shows that (4.17) commutes.Now for a module W in C , we use the definitions and (4.19) to calculate h v ′ , ( e W ◦ R W,W ′ ◦ ( θ W ⊠ id W ′ )) Y W,W ′ ( w, x ) w ′ i = (cid:10) v ′ , e W (cid:0) e xL ( − Y W ′ ,W ( w ′ , e πi x ) e πiL (0) w (cid:1)(cid:11) = (cid:10) e xL (1) v ′ , E W ( w ′ , e πi x ) e πiL (0) w (cid:11) = D e x − L (1) w ′ , Y W ( e κ ( v ′ ) , − x − ) e xL (1) e − πiL (0) x − L (0) w E for v ′ ∈ V ′ , w ∈ W , and w ′ ∈ W ′ . On the other hand, the definitions and conjugationformulas from [FHL, Section 5] involving L (0) and L (1) imply h v ′ , ( e W ′ ◦ ( δ W ⊠ id W ′ )) Y W,W ′ ( w, x ) w ′ i = h v ′ , E W ′ ( δ W ( w ) , x ) w ′ i = D e − x − δ W ( w ) , Y W ′ ( e xL (1) e κ ( v ′ ) , x − ) e − xL (1) e − πiL (0) x − L (0) w ′ E = D Y W ′ ( e xL (1) e κ ( v ′ ) , x − ) e − xL (1) e − πiL (0) x − L (0) w ′ , e − x − L (1) w E = D e − xL (1) e − πiL (0) x − L (0) w ′ , Y W ( e x − L (1) ( − x ) L (0) e xL (1) e κ ( v ′ ) , x ) e − x − L (1) w E = D e − πiL (0) x − L (0) e x − L (1) w ′ , Y W (( − x ) L (0) e κ ( v ′ ) , x ) e − x − L (1) w E = D e x − L (1) w ′ , Y W ( e κ ( v ′ ) , − x − ) e xL (1) e − πiL (0) x − L (0) w E . Since Y W,W ′ is surjective, this shows that (4.18) commutes.From now on, we assume that we have chosen the evaluations so that the conclusion ofthe above lemma holds.If W is a rigid object in a tensor category, then its dual W ∗ satisfies the same universalproperty of the contragredient, with the morphism ϕ of (4.16) given by the composition X r − X −−→ X ⊠ V id X ⊠ i W −−−−→ X ⊠ ( W ⊠ W ′ ) A X,W,W ′ −−−−−→ ( X ⊠ W ) ⊠ W ′ f ⊠ id W ′ −−−−→ V ⊠ W ′ l W ′ −−→ W ′ . Thus if C is a braided tensor category of modules for a self-contragredient vertex operatoralgebra V that is closed under contragredients, the dual of a rigid object in C is necessarilyits contragredient. The following two lemmas are standard, and we will use them to provethat simple modules in the category C M ( p ) of M ( p )-modules are rigid: If W and W are rigid objects in a tensor category, then W ⊠ W is alsorigid (with dual W ∗ ⊠ W ∗ ). If C is a braided tensor category of modules for a self-contragredient vertexoperator algebra V that is closed under contragredients, then any direct summand of a rigidmodule in C is rigid. roof. Suppose W is a rigid module in C and we have homomorphisms q : M → W , p : W → M such that p ◦ q = id M for some module M in C . Then it is straightforwardto show using (4.17) that M is rigid with dual M ′ , evaluation e M = e W ◦ ( p ′ ⊠ q ), andcoevaluation i M = ( p ⊠ q ′ ) ◦ i W .We can now prove: Every simple module M r,s in C M ( p ) is rigid.Proof. Since the modules M n +1 , , n ∈ Z , and M , are simple currents, they are rigid.Then for any n ∈ Z , the module M n, ∼ = M , ⊠ M n − , (from (3.8)) is rigid by Lemma4.3.5. Next, the rigidity of M , from the previous subsection, the fusion rules (3.9) for2 ≤ s ≤ p −
1, and Lemma 4.3.6 imply that M ,s is rigid for each 1 ≤ s ≤ p . Finally, thefusion rule M r,s ∼ = M r, ⊠ M ,s from (3.2) and (3.8) together with Lemma 4.3.5 implythat every M r,s is rigid. In this subsection, we extend rigidity of simple modules in C M ( p ) to finite-length modules.This will be a consequence of the following more general theorem that we expect will alsobe useful for future examples: Assume that V is a self-contragredient vertex operator algebra and C isa category of grading-restricted generalized V -modules such that:(1) The category C is closed under submodules, quotients, and contragredients, and everymodule in C has finite length.(2) The category C has braided tensor category structure as in [HLZ8].(3) Every simple module in C is rigid.Then C is a rigid tensor category. As we have verified the conditions of the theorem in Theorem 3.3.2, Proposition 4.3.2,and Theorem 4.3.7, we get:
The tensor category C M ( p ) is rigid; moreover, it is a braided ribbon tensorcategory with natural twist isomorphism θ = e πiL (0) . The category C M ( p ) is closed under contragredients; in particular, C M ( p ) is a braided ribbon tensor category.Proof. Suppose that M is a module in C M ( p ) , so that F W ( p ) ( M ) is in C W ( p ) . Since M ′ is adual of M in C M ( p ) , it is easy to see that F W ( p ) ( M ′ ) is a dual of F W ( p ) ( M ) in Rep W ( p )(see for example [CKM1, Proposition 2.77]). But since C W ( p ) is a rigid tensor category, F W ( p ) ( M ) ′ is already a dual of F W ( p ) ( M ). Since duals are unique, F W ( p ) ( M ′ ) ∼ = F W ( p ) ( M ) ′ is an object of C W ( p ) , and therefore M ′ is an object of C M ( p ) .41o prove Theorem 4.4.1, we will use the V -homomorphismsΦ W ,W : W ′ ⊠ W ′ → ( W ⊠ W ) ′ for modules W , W in C induced by the universal property of contragredients such thatthe diagram W ′ ⊠ ( W ′ ⊠ ( W ⊠ W )) id W ′ ⊠ A W ′ ,W ,W / / W ′ ⊠ (( W ′ ⊠ W ) ⊠ W ) id W ′ ⊠ ( e W ⊠ id W ) + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ( W ′ ⊠ W ′ ) ⊠ ( W ⊠ W ) A − W ′ ,W ′ ,W ⊠ W O O Φ W ,W ⊠ id W ⊠ W (cid:15) (cid:15) W ′ ⊠ ( V ⊠ W ) id W ′ ⊠ l W (cid:15) (cid:15) ( W ⊠ W ) ′ ⊠ ( W ⊠ W ) e W ⊠ W / / V W ′ ⊠ W e W o o commutes. These homomorphisms determine a natural transformation: For any homomorphisms f : W → X and f : W → X in C , thediagram X ′ ⊠ X ′ f ′ ⊠ f ′ / / Φ X ,X (cid:15) (cid:15) W ′ ⊠ W ′ W ,W (cid:15) (cid:15) ( X ⊠ X ) ′ ( f ⊠ f ) ′ / / ( W ⊠ W ) ′ commutes.Proof. By the universal property of (( W ⊠ W ) ′ , e W ⊠ W ), it is enough to show that e W ⊠ W ◦ ([( f ⊠ f ) ′ ◦ Φ X ,X ] ⊠ id W ⊠ W ) = e W ⊠ W ◦ ([Φ W ,W ◦ ( f ′ ⊠ f ′ )] ⊠ id W ⊠ W ) . By (4.17), the left side of this equation is e X ⊠ X ◦ (Φ X ,X ⊠ ( f ⊠ f )) . (4.20)The right side is the composition( X ′ ⊠ X ′ ) ⊠ ( W ⊠ W ) ( f ′ ⊠ f ′ ) ⊠ id W ⊠ W −−−−−−−−−−→ ( W ′ ⊠ W ′ ) ⊠ ( W ⊠ W ) A − W ′ ,W ′ ,W ⊠ W −−−−−−−−−→ W ′ ⊠ ( W ′ ⊠ ( W ⊠ W )) id W ′ ⊠ A W ′ ,W ,W −−−−−−−−−−→ W ′ ⊠ (( W ′ ⊠ W ) ⊠ W ) id W ′ ⊠ ( e W ⊠ id W ) −−−−−−−−−−→ W ′ ⊠ ( V ⊠ W ) id W ′ ⊠ l W −−−−−−→ W ′ ⊠ W e W −−→ V. It is then straightforward to use the naturality of the associativity and unit isomorphismsapplied to f ′ and f ′ , the diagram (4.17) for f ′ and f ′ , naturality of associativity and unitapplied to f and f , and the definition of Φ X ,X to reduce this to (4.20).42f W and W are rigid objects of C , then Φ W ,W is an isomorphism with inverse givenby the composition( W ⊠ W ) ′ r − W ⊠ W ′ −−−−−−→ ( W ⊠ W ) ′ ⊠ V id ( W ⊠ W ′ ⊠ i W −−−−−−−−−→ ( W ⊠ W ) ′ ⊠ ( W ⊠ W ′ ) id ( W ⊠ W ′ ⊠ ( r − W ⊠ id W ′ ) −−−−−−−−−−−−−−→ ( W ⊠ W ) ′ ⊠ (( W ⊠ V ) ⊠ W ′ ) id ( W ⊠ W ′ ⊠ ((id W ⊠ i W ) ⊠ id W ′ ) −−−−−−−−−−−−−−−−−−−→ ( W ⊠ W ) ′ ⊠ (( W ⊠ ( W ⊠ W ′ )) ⊠ W ′ ) assoc. −−−→ (( W ⊠ W ) ′ ⊠ ( W ⊠ W )) ⊠ ( W ′ ⊠ W ′ ) e W ⊠ W ⊠ id W ′ ⊠ W ′ −−−−−−−−−−−→ V ⊠ ( W ′ ⊠ W ′ ) l W ′ ⊠ W ′ −−−−→ W ′ ⊠ W ′ , where the arrow marked assoc. represents a suitable composition of associativity isomor-phisms. Conversely, we have: If Φ W,W ′ is an isomorphism, then W is rigid.Proof. Recall the ribbon isomorphism δ W : W → W ′′ characterized by the commutativediagram (4.18). Let us use e e W to denote e W ′ ◦ ( δ W ⊠ id W ′ ) : W ⊠ W ′ → V. We also define a homomorphism ϕ : V → V ′ such that the diagram V ⊠ V l V = r V ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ ϕ ⊠ id V (cid:15) (cid:15) V ′ ⊠ V e V / / V commutes. Now assuming Φ W,W ′ is an isomorphism, we define a coevaluation i W : V → W ⊠ W ′ by the composition V ϕ −→ V ′ e e ′ W −−→ ( W ⊠ W ′ ) ′ Φ − W,W ′ −−−−→ W ′′ ⊠ W ′ δ − W ⊠ id W ′ −−−−−→ W ⊠ W ′ . To prove rigidity, we need to show that the endomorphism R W of W defined by thecomposition W l − W −−→ V ⊠ W i W ⊠ id W −−−−−→ ( W ⊠ W ′ ) ⊠ W A − W,W ′ ,W −−−−−→ W ⊠ ( W ′ ⊠ W ) id W ⊠ e W −−−−−→ W ⊠ V r W −−→ W and the endomorphism R W ′ of W ′ defined by the composition W ′ r − W ′ −−→ W ′ ⊠ V id W ′ ⊠ i W −−−−−→ W ′ ⊠ ( W ⊠ W ′ ) A W ′ ,W,W ′ −−−−−−→ ( W ′ ⊠ W ) ⊠ W ′ e W ⊠ id W ′ −−−−−→ V ⊠ W ′ l W ′ −−→ W ′ are identities. Since δ W is an isomorphism, ( W, e e W ) satisfies the universal property of thecontragredient of W ′ . So R W = id W is equivalent to e e W ◦ ( R W ⊠ id W ′ ) = e e W . (4.21)43imilarly, R W ′ = id W ′ is equivalent to e W ◦ ( R W ′ ⊠ id W ) = e W . (4.22)Here, we give the details for (4.22) only, since the proof of (4.21) is similar but simpler.We first insert R V,W ′ ◦ R − V,W ′ into R W ′ and apply naturality of the braiding isomor-phisms: W ′ r − W ′ −−→ W ′ ⊠ V R − V,W ′ −−−→ V ⊠ W ′ (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ′ −−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ W ′ ( δ − W ⊠ id W ′ ) ⊠ id W ′ −−−−−−−−−−→ ( W ⊠ W ′ ) ⊠ W ′ R W ⊠ W ′ ,W ′ −−−−−−→ W ′ ⊠ ( W ⊠ W ′ ) A W ′ ,W,W ′ −−−−−−→ ( W ′ ⊠ W ) ⊠ W ′ e W ⊠ id W ′ −−−−−→ V ⊠ W ′ l W ′ −−→ W ′ . Next we use R − V,W ◦ r − W ′ = l − W ′ , the hexagon axiom, and the naturality of braiding andassociativity: W ′ l − W ′ −−→ V ⊠ W ′ (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ′ −−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ W ′ A − W ′′ ,W ′ ,W ′ −−−−−−−→ W ′′ ⊠ ( W ′ ⊠ W ′ ) id W ′′ ⊠ R W ′ ,W ′ −−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ W ′ ) A W ′′ ,W ′ ,W ′ −−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ W ′R W ′′ ,W ′ ⊠ id W ′ −−−−−−−−→ ( W ′ ⊠ W ′′ ) ⊠ W ′ (id W ′ ⊠ δ − W ) ⊠ id W ′ −−−−−−−−−−→ ( W ′ ⊠ W ) ⊠ W ′ e W ⊠ id W ′ −−−−−→ V ⊠ W ′ l W ′ −−→ W ′ . Using (4.18) and the definitions of θ and contragredient homomorphisms, we then calculate e W ◦ (id W ′ ⊠ δ − W ) ◦ R W ′′ ,W ′ = e W ′ ◦ ( θ − W ′′ ⊠ id W ′ )= e W ′ ◦ (( θ − W ′ ) ′ ⊠ id W ′ ) = e W ′ ◦ (id W ′′ ⊠ θ − W ′ ) . Returning this to the composition and using naturality, we see that R W ′ is the composition W ′ l − W ′ ◦ θ − W ′ −−−−−→ V ⊠ W ′ (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ′ −−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ W ′ A − W ′′ ,W ′ ,W ′ −−−−−−−→ W ′′ ⊠ ( W ′ ⊠ W ′ ) id W ′′ ⊠ R W ′ ,W ′ −−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ W ′ ) A W ′′ ,W ′ ,W ′ −−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ W ′ e W ′ ⊠ id W ′ −−−−−−→ V ⊠ W ′ l W ′ −−→ W ′ . Now we begin to analyze e W ◦ ( R W ′ ⊠ id W ). We first use l − W ′ ⊠ id W = A V,W ′ ,W ◦ ( l − W ′ ⊠ W )and naturality of associativity to get W ′ ⊠ W θ − W ′ ⊠ id W −−−−−→ W ′ ⊠ W l − W ′ ⊠ W −−−−→ V ⊠ ( W ′ ⊠ W ) (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ′ ⊠ W −−−−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ ( W ′ ⊠ W ) A W ′′ ⊠ W ′ ,W ′ ,W −−−−−−−−−→ (( W ′′ ⊠ W ′ ) ⊠ W ′ ) ⊠ W A − W ′′ ,W ′ ,W ′ ⊠ id W −−−−−−−−−−→ ( W ′′ ⊠ ( W ′ ⊠ W ′ )) ⊠ W (id W ′′ ⊠ R W ′ ,W ′ ) ⊠ id W −−−−−−−−−−−−−→ ( W ′′ ⊠ ( W ′ ⊠ W ′ )) ⊠ W A W ′′ ,W ′ ,W ′ ⊠ id W −−−−−−−−−−→ (( W ′′ ⊠ W ′ ) ⊠ W ′ ) ⊠ W ( e W ′ ⊠ id W ′ ) ⊠ id W −−−−−−−−−−→ ( V ⊠ W ′ ) ⊠ W l W ′ ⊠ id W −−−−−→ W ′ ⊠ W e W −−→ V. R W ′ ,W ′ : W ′ ⊠ W θ − W ′ ⊠ id W −−−−−→ W ′ ⊠ W l − W ′ ⊠ W −−−−→ V ⊠ ( W ′ ⊠ W ) (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ′ ⊠ W −−−−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ ( W ′ ⊠ W ) A − W ′′ ,W ′ ,W ′ ⊠ W −−−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ ( W ′ ⊠ W )) id W ′′ ⊠ A W ′ ,W ′ ,W −−−−−−−−−−→ W ′′ ⊠ (( W ′ ⊠ W ′ ) ⊠ W ) id W ′′ ⊠ ( R W ′ ,W ′ ⊠ id W ) −−−−−−−−−−−−−→ W ′′ ⊠ (( W ′ ⊠ W ′ ) ⊠ W ) A W ′′ ,W ′ ⊠ W ′ ,W −−−−−−−−−→ ( W ′′ ⊠ ( W ′ ⊠ W ′ )) ⊠ W A W ′′ ,W ′ ,W ′ ⊠ id W −−−−−−−−−−→ (( W ′′ ⊠ W ′ ) ⊠ W ′ ) ⊠ W ( e W ′ ⊠ id W ′ ) ⊠ id W −−−−−−−−−−→ ( V ⊠ W ′ ) ⊠ W l W ′ ⊠ id W −−−−−→ W ′ ⊠ W e W −−→ V. Now we insert R W,W ′ ◦ R − W,W ′ between the first two associativity isomorphisms and applyvarious naturalities to the inverse braiding: W ′ ⊠ W R − W,W ′ −−−−→ W ⊠ W ′ id W ⊠ θ − W ′ −−−−−→ W ⊠ W ′ l − W ⊠ W ′ −−−−→ V ⊠ ( W ⊠ W ′ ) (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ⊠ W ′ −−−−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ ( W ⊠ W ′ ) A − W ′′ ,W ′ ,W ⊠ W ′ −−−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ ( W ⊠ W ′ )) id W ′′ ⊠ (id W ′ ⊠ R W,W ′ ) −−−−−−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ ( W ′ ⊠ W )) id W ′′ ⊠ A W ′ ,W ′ ,W −−−−−−−−−−→ W ′′ ⊠ (( W ′ ⊠ W ′ ) ⊠ W ) id W ′′ ⊠ ( R W ′ ,W ′ ⊠ id W ) −−−−−−−−−−−−−→ W ′′ ⊠ (( W ′ ⊠ W ′ ) ⊠ W ) A W ′′ ,W ′ ⊠ W ′ ,W −−−−−−−−−→ ( W ′′ ⊠ ( W ′ ⊠ W ′ )) ⊠ W A W ′′ ,W ′ ,W ′ ⊠ id W −−−−−−−−−−→ (( W ′′ ⊠ W ′ ) ⊠ W ′ ) ⊠ W ( e W ′ ⊠ id W ′ ) ⊠ id W −−−−−−−−−−→ ( V ⊠ W ′ ) ⊠ W l W ′ ⊠ id W −−−−−→ W ′ ⊠ W e W −−→ V. We apply the hexagon axiom and then the pentagon axiom to the braiding and associativityisomorphisms in rows three, four, and five: W ′ ⊠ W R − W,W ′ −−−−→ W ⊠ W ′ id W ⊠ θ − W ′ −−−−−→ W ⊠ W ′ l − W ⊠ W ′ −−−−→ V ⊠ ( W ⊠ W ′ ) (Φ − W,W ′ ◦ e e ′ W ◦ ϕ ) ⊠ id W ⊠ W ′ −−−−−−−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ ( W ⊠ W ′ ) A − W ′′ ,W ′ ,W ⊠ W ′ −−−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ ( W ⊠ W ′ )) id W ′′ ⊠ A W ′ ,W,W ′ −−−−−−−−−−→ W ′′ ⊠ (( W ′ ⊠ W ) ⊠ W ′ ) id W ′′ ⊠ R W ′ ⊠ W,W ′ −−−−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ ( W ′ ⊠ W )) A W ′′ ,W ′ ,W ′ ⊠ W −−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ ( W ′ ⊠ W ) A W ′′ ⊠ W ′ ,W ′ ,W −−−−−−−−−→ (( W ′′ ⊠ W ′ ) ⊠ W ′ ) ⊠ W ( e W ′ ⊠ id W ′ ) ⊠ id W −−−−−−−−−−→ ( V ⊠ W ′ ) ⊠ W l W ′ ⊠ id W −−−−−→ W ′ ⊠ W e W −−→ V. (4.23)Now by naturality of associativity and properties of the left unit isomorphisms, the lastsix arrows reduce to W ′′ ⊠ (( W ′ ⊠ W ) ⊠ W ′ ) id W ′′ ⊠ R W ′ ⊠ W,W ′ −−−−−−−−−−→ W ′′ ⊠ ( W ′ ⊠ ( W ′ ⊠ W )) A W ′′ ,W ′ ,W ′ ⊠ W −−−−−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ ( W ′ ⊠ W ) e W ′ ⊠ id W ′ ⊠ W −−−−−−−−→ V ⊠ ( W ′ ⊠ W ) l W ′ ⊠ W −−−−→ W ′ ⊠ W e W −−→ V. e W , we get W ′′ ⊠ (( W ′ ⊠ W ) ⊠ W ′ ) id W ′′ ⊠ ( e W ⊠ id W ′ ) −−−−−−−−−−→ W ′′ ⊠ ( V ⊠ W ′ ) id W ′′ ⊠ R V,W ′ −−−−−−−→ W ′′ ⊠ ( W ′ ⊠ V ) A W ′′ ,W ′ ,V −−−−−−→ ( W ′′ ⊠ W ′ ) ⊠ V e W ′ ⊠ id V −−−−−→ V ⊠ V l V = r V −−−−→ V. Using naturality of the right unit isomorphism, and then using the relations between unit,associativity, and braiding isomorphisms, we get W ′′ ⊠ (( W ′ ⊠ W ) ⊠ W ′ ) id W ′′ ⊠ ( e W ⊠ id W ′ ) −−−−−−−−−−→ W ′′ ⊠ ( V ⊠ W ′ ) id W ′′ ⊠ l W ′ −−−−−−→ W ′′ ⊠ W ′ e W ′ −−→ V. Returning this to (4.23) and using the definition of Φ
W,W ′ , we now find that e W ◦ ( R W ′ ⊠ id W )equals the composition W ′ ⊠ W R − W,W ′ −−−−→ W ⊠ W ′ id W ⊠ θ − W ′ −−−−−→ W ⊠ W ′ l − W ⊠ W ′ −−−−→ V ⊠ ( W ⊠ W ′ ) ( e e ′ W ◦ ϕ ) ⊠ id W ⊠ W ′ −−−−−−−−−−→ ( W ⊠ W ′ ) ′ ⊠ ( W ⊠ W ′ ) e W ⊠ W ′ −−−−→ V. The diagram (4.17) applied to e e W then yields W ′ ⊠ W R − W,W ′ −−−−→ W ⊠ W ′ id W ⊠ θ − W ′ −−−−−→ W ⊠ W ′ l − W ⊠ W ′ −−−−→ V ⊠ ( W ⊠ W ′ ) ϕ ⊠ id W ⊠ W ′ −−−−−−→ V ′ ⊠ ( W ⊠ W ′ ) id V ′ ⊠ e e W −−−−−→ V ′ ⊠ V e V −→ V. Then by the definition of ϕ and naturality of the left unit isomorphisms, this compositionsimplifies to W ′ ⊠ W R − W,W ′ −−−−→ W ⊠ W ′ id W ⊠ θ − W ′ −−−−−→ W ⊠ W ′ e e W −−→ V. Finally, we find e e W ◦ (id W ⊠ θ − W ′ ) ◦ R − W,W ′ = e W ′ ◦ ( δ W ⊠ id W ′ ) ◦ (id W ⊠ θ − W ′ ) ◦ R − W,W ′ = e W ◦ ( θ − W ′ ⊠ θ W ) = e W using (4.18) and θ − W ′ = ( θ − W ) ′ . This completes the proof that R W ′ = id W ′ .Now we finish the proof of Theorem 4.4.1: Proof.
By Proposition 4.4.5, it is sufficient to prove that Φ W ,W is an isomorphism forall modules W and W in C . Since every module W in C has finite length ℓ ( W ), wecan prove Φ W ,W is an isomorphism by induction on ℓ ( W ) + ℓ ( W ). For the base case ℓ ( W ) = ℓ ( W ) = 1, W and W are both simple and thus rigid by assumption. So Φ W ,W is an isomorphism by the discussion preceding Proposition 4.4.5.Now assume ℓ ( W ) + ℓ ( W ) = N > f W , f W is an isomorphism whenever ℓ ( f W ) + ℓ ( f W ) < N . Without loss of generality, assume that ℓ ( W ) ≥
2, so that there is an exactsequence 0 → M g −→ W f −→ N → M, N modules in C (because C is closed under submodules and quotients) and ℓ ( M ) , ℓ ( N ) < ℓ ( W ). The diagram W ′ ⊠ N ′ id W ′ ⊠ f ′ / / Φ N,W (cid:15) (cid:15) W ′ ⊠ W ′ W ′ ⊠ g ′ / / Φ W ,W (cid:15) (cid:15) W ′ ⊠ M ′ / / Φ M,W (cid:15) (cid:15) / / ( N ⊠ W ) ′ ( f ⊠ id W ) ′ / / ( W ⊠ W ) ′ ( g ⊠ id W ) ′ / / ( M ⊠ W ) ′ has exact rows because W ′ ⊠ • , • ⊠ W are right exact and because the contragredientfunctor is exact. The diagram also commutes by Proposition 4.4.4. Since Φ N,W and Φ M,W are isomorphisms by the inductive hypothesis, the short five lemma diagram chase showsthat Φ W ,W is also an isomorphism. Using Theorem 4.4.1, we can also get rigidity for the category O of C -cofinite grading-restricted generalized modules for the simple Virasoro vertex operatoralgebra L (1 ,
0) of central charge 1. Conditions (1) and (2) of the theorem were proved forthis category in [CJORY], while it was shown in [McR] that all simple modules in O arerigid. Using the natural isomorphism Φ from the proof of Theorem 4.4.1, wecan determine the contragredients of all simple modules in C M ( p ) . We have already seen inRemark 4.3.3 that each M ,s is self-contragredient, and the fusion rule M r, ⊠ M − r, ∼ = M , for r ∈ Z shows that M ′ r, ∼ = M − r, . Then we get M ′ r,s ∼ = ( M r, ⊠ M ,s ) ′ ∼ = M ′ ,s ⊠ M ′ r, ∼ = M ,s ⊠ M − r, ∼ = M − r,s for r ∈ Z , 1 ≤ s ≤ p . In this final section, we construct projective covers of the remaining irreducible modulesin C M ( p ) , and we determine all fusion products involving irreducible modules and theirprojective covers. In Corollary 3.3.4 and Proposition 3.3.5, we proved that for r ∈ Z , • The module P r,p = F α r,p = M r,p is projective and is its own projective cover. • The module P r,p − ∼ = M , ⊠ M r,p is projective and is a projective cover of M r,p − .47n the proof of Proposition 3.3.5, we also showed that P r,p − has a unique maximal propersubmodule Z r,p − , and we have the following non-split exact sequences:0 → M r,p − → Z r,p − → M r − , ⊕ M r +1 , → , (5.1)and 0 → Z r,p − → P r,p − → M r,p − → . (5.2)Moreover, the Loewy diagrams of P r,p − and Z r,p − are M r,p − P r,p − : M r − , M r +1 , M r,p − , M r,p − . Z r,p − : M r − , M r +1 , We will also need the following lemma:
For r ∈ Z , M r,p is injective in C M ( p ) .Proof. Because M − r,p is projective in C M ( p ) , M r,p ∼ = M ′ − r,p is injective.Now for p ≥
3, we begin constructing more projective modules using the fusion rules(3.9). Since M , is rigid, the functor M , ⊠ • is exact. Applying M , ⊠ • to (5.1) andusing (3.9), 0 → M r,p − ⊕ M r,p → M , ⊠ Z r,p − → M r − , ⊕ M r +1 , → M r,p is injective in C M ( p ) , it is a direct summand of M , ⊠ Z r,p − . Let Z r,p − be a submodule complement of M r,p in M , ⊠ Z r,p − , that is, M , ⊠ Z r,p − = M r,p ⊕ Z r,p − . (5.4)It is easy to see that there is an exact sequence0 → M r,p − → Z r,p − → M r − , ⊕ M r +1 , → . (5.5)We claim that Soc( Z r,p − ) = M r,p − . Otherwise either M r − , or M r +1 , is a submoduleof Z r,p − , and hence also a submodule of M , ⊠ Z r,p − . But in fact the rigidity of M , ,the fusion rules (3.9), and the Loewy diagram of Z r,p − implyHom M ( p ) ( M r ± , , M , ⊠ Z r,p − ) ∼ = Hom M ( p ) ( M , ⊠ M r ± , , Z r,p − ) ∼ = Hom M ( p ) ( M r ± , ⊕ M r, ± , , Z r,p − ) = 0 , Z r,p − is M r,p − . Z r,p − : M r − , M r +1 , Now we apply M , ⊠ • to (5.2) and use (3.9) and the decomposition (5.4) of Z r,p − toget 0 → M r,p ⊕ Z r,p − → M , ⊠ P r,p − → M r,p − ⊕ M r,p → . (5.6)Because M r,p is both projective and injective in C M ( p ) , 2 · M r,p is a direct summand of M , ⊠ P r,p − . Defining P r,p − to be a direct summand of M , ⊠ P r,p − complementary to2 · M r,p , we get an exact sequence0 → Z r,p − → P r,p − → M r,p − → . (5.7)We claim that Soc( P r,p − ) = M r,p − . Otherwise Soc( P r,p − ) = 2 · M r,p − and thendim Hom M ( p ) ( M , ⊠ M r,p − , P r,p − ) = dim Hom M ( p ) ( M r,p − , M , ⊠ P r,p − ) = 2 , whereas in factdim Hom M ( p ) ( M , ⊠ M r,p − , P r,p − ) = dim Hom M ( p ) ( M r,p − ⊕ M r,p − , P r,p − ) = 1 , a contradiction. The claim follows.The exact sequence (5.7) gives0 → Z r,p − / M r,p − → P r,p − / M r,p − → M r,p − → . (5.8)We claim that P r,p − / M r,p − is indecomposable, so that in particular (5.8) does not splitand Soc( P r,p − / M r,p − ) = Z r,p − / M r,p − = M r +1 , ⊕ M r − , . Now if P r,p − / M r,p − were decomposable, then because it has length 3 and contains M r − , ⊕ M r +1 , as a submodule, it would have to contain either M r − , or M r +1 , as asummand. But using the rigidity of M , , this would implyHom M ( p ) ( P r,p − / M r,p − , M , ⊠ M r ± , ) ∼ = Hom M ( p ) ( M , ⊠ ( P r,p − / M r,p − ) , M r ± , ) ∼ = Hom M ( p ) (cid:0) ( M , ⊠ P r,p − ) / ( M , ⊠ M r,p − ) , M r ± , (cid:1) ∼ = Hom M ( p ) (cid:0) ( P r,p − ⊕ · M r,p ) / ( M r,p ⊕ M r,p − ) , M r ± , (cid:1) ∼ = Hom M ( p ) (cid:0) ( P r,p − / M r,p − ) ⊕ M r,p , M r ± , (cid:1) = 0 , whereas in fact the Loewy diagram for P r,p − showsHom M ( p ) ( P r,p − / M r,p − , M , ⊠ M r ± , ) ∼ = Hom M ( p ) ( P r,p − / M r,p − , M r ± , ⊕ M r ± , ) = 0 . This proves the claim, and it also finishes the verification of the Loewy diagram for P r,p − in the next lemma: 49 .1.2 Lemma. For r ∈ Z , P r,p − is a projective cover of M r,p − in C M ( p ) , and it hasLoewy diagram M r,p − . P r,p − : M r − , M r +1 , M r,p − Proof.
It remains to show that P r,p − is a projective cover of M r,p − in C M ( p ) . Since M , is rigid and P r,p − is projective in C M ( p ) , M , ⊠ P r,p − is also projective (see Corollary2 from the Appendix of [KL5]). Thus as a direct summand of M , ⊠ P r,p − , P r,p − isprojective in C M ( p ) , and we have a surjective map p : P r,p − → M r,p − . Now suppose P isprojective and q : P → M r,p − is a surjective map. Then there exists f : P → P r,p − suchthat P f x x ♣♣♣♣♣♣♣♣♣♣♣♣ q (cid:15) (cid:15) P r,p − p / / M r,p − commutes. We need to show that f is surjective. The Loewy diagram for P r,p − showsthat Z r,p − is the unique maximal submodule of P r,p − , so if f is not surjective, thenim f ⊆ Z r,p − = ker p . But this contradicts q = 0, so in fact f is surjective.Now that we have projective covers P r,p − , P r,p − , and P r,p for r ∈ Z , we proceed toconstruct modules P r,s for 1 ≤ s ≤ p − s ∈ { , , . . . , p − } and assumewe have P r,σ for all s ≤ σ ≤ p − • The module P r,σ is a projective cover of M r,σ in C M ( p ) , and • The Loewy diagram of P r,σ is M r,σ P r,σ : M r − ,p − σ M r +1 ,p − σ M r,σ We now define P r,s − as follows. We have a surjection M , ⊠ P r,s → M , ⊠ M r,s ∼ = M r,s − ⊕ M r,s +1 → M r,s +1 . M , is rigid and P r,s is projective, M , ⊠ P r,s is also projective. So because P r,s +1 is the projective cover of M r,s +1 , we get a a surjective map M , ⊠ P r,s → P r,s +1 . Since P r,s +1 is projective, this surjection splits and P r,s +1 is a direct summand of M , ⊠ P r,s .We now define P r,s − to be a complement of P r,s +1 : M , ⊠ P r,s = P r,s +1 ⊕ P r,s − . The module P r,s − is in C M ( p ) because this category is closed under tensor products andsubmodules, and it is projective in C M ( p ) because it is a summand of a projective module.We can now prove: The module P r,s − is a projective cover of M r,s − in C M ( p ) with thefollowing Loewy diagram: M r,s − P r,s − : M r − ,p − s +1 M r +1 ,p − s +1 M r,s − Proof.
From the Loewy diagram for P r,s , we see that P r,s has a unique maximal propersubmodule Z r,s with Loewy diagram M r,s Z r,s : M r − ,p − s M r +1 ,p − s Then the following non-split exact sequences are clear:0 −→ M r,s −→ Z r,s −→ M r − ,p − s ⊕ M r +1 ,p − s −→ , (5.9)and 0 −→ Z r,s −→ P r,s −→ M r,s −→ . (5.10)Applying M , ⊠ • to (5.9) and using the fusion rules (3.9) yields the exact sequence0 −→ M r,s − ⊕ M r,s +1 −→ M , ⊠ Z r,s −→ M r − ,p − s − ⊕ M r − ,p − s +1 ⊕ M r +1 ,p − s − ⊕ M r +1 ,p − s +1 −→ . M , ⊠ Z r,s are contained in the two distinctcosets h r,s ± + Z , and thus M , ⊠ Z r,s decomposes as a direct sum of two modules, say e Z r,s +1 and Z r,s − , satisfying the exact sequences0 → M r,s − → Z r,s − → M r − ,p − s +1 ⊕ M r +1 ,p − s +1 → , and 0 → M r,s +1 → e Z r,s +1 → M r − ,p − s − ⊕ M r +1 ,p − s − → . If either M r − ,p − s +1 or M r +1 ,p − s +1 were a submodule of Z r,s − , and thus also a submoduleof M , ⊠ Z r,s , then rigidity of M , would implyHom M ( p ) ( M , ⊠ M r ± ,p − s +1 , Z r,s ) ∼ = Hom M ( p ) ( M r ± ,p − s +1 , M , ⊠ Z r,s ) = 0 . However, by the fusion rules (3.9) and the Loewy diagram of Z r,s , there is no non-zerohomomorphism M , ⊠ M r ± ,p − s +1 ∼ = M r ± ,p − s ⊕ M r ± ,p − s +2 → Z r,s . As a result, Soc( Z r,s − ) = M r,s − .Now we apply M , ⊠ • to (5.10) to get an exact sequence0 → Z r,s − ⊕ e Z r,s +1 → M , ⊠ P r,s → M r,s − ⊕ M r,s +1 → . Thus conformal weight considerations again show that P r,s − satisfies the exact sequence0 → Z r,s − → P r,s − → M r,s − → . (5.11)We claim that Soc( P r,s − ) = M r,s − . If not, then Soc( P r,s − ) = 2 · M r,s − and rigidity of M , would implydim Hom M ( p ) ( M , ⊠ M r,s − , P r,s ) = dim Hom M ( p ) ( M r,s − , M , ⊠ P r,s ) = 2 . However, this would contradictdim Hom M ( p ) ( M , ⊠ M r,s − , P r,s ) = dim Hom M ( p ) (cid:0) [ M r,s − ⊕ ] M r,s , P r,s (cid:1) = 1 , where the summand in brackets occurs for s ≥ → Z r,s − / M r,s − → P r,s − / M r,s − → M r,s − → P r,s − / M r,s − is indecomposable, so that in particularSoc( P r,s − / M r,s − ) = Z r,s − / M r,s − = M r − ,p − s +1 ⊕ M r +1 ,p − s +1 . P r,s − / M r,s − were decomposable, then because it contains M r − ,p − s +1 ⊕ M r +1 ,p − s +1 asa submodule and has length 3, it would have to contain either M r − ,p − s +1 or M r +1 ,p − s +1 as a direct summand. Then rigidity of M , would implyHom M ( p ) ( P r,s / M r,s , M , ⊠ M r ± ,p − s +1 ) ∼ = Hom M ( p ) ( M , ⊠ ( P r,s / M r,s ) , M r ± ,p − s +1 ) ∼ = Hom M ( p ) (cid:0) ( M , ⊠ P r,s ) / ( M , ⊠ M r,s ) , M r ± ,p − s +1 (cid:1) ∼ = Hom M ( p ) (cid:0) ( P r,s − ⊕ P r,s +1 ) / ( M r,s − ⊕ M r,s +1 ) , M r ± ,p − s +1 (cid:1) ∼ = Hom M ( p ) (cid:0) ( P r,s − / M r,s − ) ⊕ ( P r,s +1 / M r,s +1 ) , M r ± ,p − s +1 (cid:1) = 0 . However, in factHom M ( p ) ( P r,s / M r,s , M , ⊠ M r ± ,p − s +1 )= Hom M ( p ) ( P r,s / M r,s , M r ± ,p − s ⊕ M r ± ,p − s +2 ) = 0 . Thus P r,s − / M r,s − is indecomposable, and we have verified the Loewy diagram for P r,s − .Now we show that the projective module P r,s − is a projective cover of M r,s − in C M ( p ) .The Loewy diagram shows that Z r,s − is the unique maximal proper submodule of P r,s − and that there is a surjective map p : P r,s − → M r,s − with kernel Z r,s − . Thus for anysurjective map q : P → M r,s − in C M ( p ) with P projective, a map f : P → P r,s − suchthat the diagram P q (cid:15) (cid:15) f x x ♣♣♣♣♣♣♣♣♣♣♣♣ P r,s − p / / M r,s − commutes is necessarily surjective, just as in the proof of Lemma 5.1.2The projective covers P r,s for r ∈ Z , 1 ≤ s ≤ p have the following fusion rules: (1) For r ∈ Z , ≤ s ≤ p , M , ⊠ P r,s = P r +1 ,s . (5.12) (2) For p ≥ and r ∈ Z , ≤ s ≤ p − , M , ⊠ P r,s = P r, ⊕ P r +1 ,p ⊕ P r − ,p if s = 1 P r,s − ⊕ P r,s +1 , if < s < p − P r,p − ⊕ · P r,p , if s = p − . (5.13) (3) For p = 2 and r ∈ Z , M , ⊠ P r, = P r +1 , ⊕ · P r, ⊕ P r − , . (5.14)53 roof. It remains to prove (5.12), the s = 1 case of (5.13), and (5.14).Using the fusion rules (3.8), M , ⊠ P r,s is projective with a surjective map M , ⊠ P r,s →M r +1 ,s . Because P r +1 ,s is a projective cover of M r +1 ,s , it is a direct summand of M , ⊠ P r,s .However, M , ⊠ P r,s is indecomposable because M , is a simple current and P r,s isindecomposable. So in fact M , ⊠ P r,s = P r +1 ,s , proving (5.12).Now for the s = 1 case of (5.13), the unique maximal proper submodule Z r, of P r, satisfies the exact sequence0 → M r, → Z r, → M r − ,p − ⊕ M r +1 ,p − → . Applying M , ⊠ • and using the fusion rules (3.9), we have0 → M r, → M , ⊠ Z r, → M r − ,p − ⊕ M r +1 ,p − ⊕ M r − ,p ⊕ M r +1 ,p → . (5.15)Since both of M r ± ,p are projective, M r − ,p ⊕ M r +1 ,p is a direct summand of M , ⊠ Z r, .Then the complement e Z r, of M r − ,p ⊕ M r +1 ,p satisfies the exact sequence0 → M r, → e Z r, → M r − ,p − ⊕ M r +1 ,p − → . (5.16)Now consider the exact sequence0 → Z r, → P r, → M r, → . Applying M , ⊠ • and using the fusion rules (3.9), we have0 → e Z r, ⊕ M r − ,p ⊕ M r +1 ,p → M , ⊠ P r, → M r, → . Since both of M r ± ,p are injective, there exists a direct summand e P r, of M , ⊠ P r, complementary to M r − ,p ⊕ M r +1 ,p satisfying the exact sequence0 → e Z r, → e P r, → M r, → . (5.17)The module e P r, is projective in C M ( p ) since it is a summand of a projective module. Since P r, is a projective cover of M r, , there is thus a surjection e P r, → P r, ; but since (5.16) and(5.17) show that these two modules have the same length, we get e P r, ∼ = P r, . Therefore M , ⊠ P r, = P r, ⊕ M r − ,p ⊕ M r +1 ,p , proving (5.13) for s = 1.Now when p = 2, we need to replace the exact sequence (5.15) with0 −→ M r, −→ M , ⊠ Z r, −→ M r − , ⊕ M r +1 , −→ . Since both M r ± , = P r ± , are projective, this exact sequence splits. The exact sequence0 −→ M , ⊠ Z r, −→ M , ⊠ P r, −→ M , ⊠ M r, −→ M , ⊠ M r, ∼ = M r, is projective. Then these two split exact sequencestogether imply (5.14). 54 .2 General fusion rules Finally, here are all fusion rules involving the simple modules M r,s and their projectivecovers in C M ( p ) : All fusion products of the M ( p ) -modules M r,s and P r,s are as follows,with sums taken to be empty if the lower bound exceeds the upper bound:(1) For r, r ′ ∈ Z and ≤ s, s ′ ≤ p , M r,s ⊠ M r ′ ,s ′ = (cid:18) min { s + s ′ − , p − − s − s ′ } M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ M r + r ′ − ,ℓ (cid:19) ⊕ (cid:18) p M ℓ =2 p +1 − s − s ′ ℓ + s + s ′ ≡ P r + r ′ − ,ℓ (cid:19) . (5.18) (2) For r, r ′ ∈ Z , ≤ s ≤ p − and ≤ s ′ ≤ p , P r,s ⊠ M r ′ ,s ′ = (cid:18) min { s + s ′ − ,p } M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ P r + r ′ − ,ℓ (cid:19) ⊕ (cid:18) p M ℓ =2 p +1 − s − s ′ ℓ + s + s ′ ≡ P r + r ′ − ,ℓ (cid:19) ⊕ p M ℓ = p + s − s ′ +1 ℓ + p + s + s ′ ≡ (cid:0) P r + r ′ ,ℓ ⊕ P r + r ′ − ,ℓ (cid:1) . (5.19) (3) For r, r ′ ∈ Z and ≤ s, s ′ ≤ p − , P r,s ⊠ P r ′ ,s ′ = (cid:18) · min { s + s ′ − ,p } M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ P r + r ′ − ,ℓ (cid:19) ⊕ (cid:18) · p M ℓ =2 p +1 − s − s ′ ℓ + s + s ′ ≡ P r + r ′ − ,ℓ (cid:19) ⊕ · p M ℓ = p + s − s ′ +1 ℓ + p + s + s ′ ≡ (cid:0) P r + r ′ ,ℓ ⊕ P r + r ′ − ,ℓ (cid:1) ⊕ min { s − s ′ + p − ,p } M ℓ = | s + s ′ − p | +1 ℓ + p + s + s ′ ≡ (cid:0) P r + r ′ ,ℓ ⊕ P r + r ′ − ,ℓ (cid:1) ⊕ p M ℓ = p − s + s ′ +1 ℓ + p + s + s ′ ≡ (cid:0) P r + r ′ ,ℓ ⊕ P r + r ′ − ,ℓ (cid:1) ⊕ p M ℓ = s + s ′ +1 ℓ + s + s ′ ≡ (cid:0) P r + r ′ +1 ,ℓ ⊕ · P r + r ′ − ,ℓ ⊕ P r + r ′ − ,ℓ (cid:1) . (5.20)55 roof. We will see that the fusion rules (5.18) and (5.19) are completely determined byrepeated applications of (3.8), (3.9), (5.12), (5.13), and (5.14). As it can be seen thatthese recursion formulas agree with those for the unrolled quantum group of sl given in[CGP2], the general fusion rules must therefore agree with those in [CGP2, Lemma 8.1,Proposition 8.2, Corollary 8.3, Proposition 8.4]. Thus we shall give a detailed proof for(5.18) only; the fusion rules (5.19) can be proved similarly, and then (5.20) will follow from(5.19) combined with rigidity and the projectivity of P r,s in C M ( p ) .We first use induction on s to prove M ,s ⊠ M r ′ ,s ′ = (cid:18) min { s + s ′ − , p − − s − s ′ } M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ M r ′ ,ℓ (cid:19) ⊕ (cid:18) p M ℓ =2 p +1 − s − s ′ ℓ + s + s ′ ≡ P r ′ ,ℓ (cid:19) . (5.21)From the fusion rules (3.9), (5.21) is true for s = 1 ,
2. Now assume (5.21) holds for somefixed s ∈ { , , . . . p − } ; we shall prove it holds for s + 1. We first tensor the left side of(5.21) with M , to get M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) = ( M , ⊠ M ,s ) ⊠ M r ′ ,s ′ = ( M ,s − ⊕ M ,s +1 ) ⊠ M r ′ ,s ′ = ( M ,s − ⊠ M r ′ ,s ′ ) ⊕ ( M ,s +1 ⊠ M r ′ ,s ′ ) . Then because all tensor product modules here have finite length, the Krull-Schmidt The-orem guarantees that we can determine the indecomposable summands of M ,s +1 ⊠ M r ′ ,s ′ by subtracting those of M ,s − ⊠ M r ′ ,s ′ from those of M , ⊠ ( M ,s ⊠ M r ′ ,r ′ ). We considerthe following three cases: • If s + s ′ ≤ p , by induction M ,s − ⊠ M r ′ ,s ′ = s + s ′ − M ℓ = | s − s ′ − | +1 ℓ + s − s ′ ≡ M r ′ ,ℓ , and by induction and the fusion rules (3.9) M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) = s + s ′ − M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ (cid:0) M r ′ ,ℓ +1 ⊕ M r ′ ,ℓ − (cid:1) , where we define M r ′ , = 0 in case s = s ′ . Thus we can verify that M ,s +1 ⊠ M r ′ ,s ′ = M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) M ,s − ⊠ M r ′ ,s ′ = s + s ′ M ℓ = | s − s ′ +1 | +1 ℓ + s +1+ s ′ ≡ M r ′ ,ℓ , which is the same as (5.21) in the case s + 1. (This is trivial if s + s ′ < p , and weuse P r ′ ,p = M r ′ ,p if s + s ′ = p .) 56 If s + s ′ = p + 1, by induction M ,s − ⊠ M r ′ ,s ′ = p − M ℓ = | s − s ′ − | +1 ℓ + p ≡ M r ′ ,ℓ , and by induction and the fusion rules (3.9) M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) = (cid:18) p − M ℓ = | s − s ′ | +1 ℓ + p +1 ≡ (cid:0) M r ′ ,ℓ +1 ⊕ M r ′ ,ℓ − (cid:1)(cid:19) ⊕ P r ′ ,p − . Thus M ,s +1 ⊠ M r ′ ,s ′ = M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) M ,s − ⊠ M r ′ ,s ′ = (cid:18) p − M ℓ = | s − s ′ +1 | +1 ℓ + p +2 ≡ M r ′ ,ℓ (cid:19) ⊕ P r ′ ,p − , which is the same as (5.21) for s + 1. • If p + 2 ≤ s + s ′ < p , by induction, M ,s − ⊠ M r ′ ,s ′ = (cid:18) p − s − s ′ M ℓ = | s − s ′ − | +1 ℓ + s − s ′ ≡ M r ′ ,ℓ (cid:19) ⊕ (cid:18) p M ℓ =2 p +2 − s − s ′ ℓ + s − s ′ ≡ P r ′ ,ℓ (cid:19) , and by induction and the fusion rules (3.9), (5.13), M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) = p − − s − s ′ M ℓ = | s − s ′ | +1 ℓ + s + s ′ ≡ (cid:0) M r ′ ,ℓ +1 ⊕ M r ′ ,ℓ − (cid:1) ⊕ (cid:18) p M ℓ =2 p +1 − s − s ′ ℓ + s + s ′ ≡ (cid:0) P r ′ ,ℓ − ⊕ P r ′ ,ℓ +1 (cid:1)(cid:19) ⊕ A , where A = ( P r ′ ,p if p + s + s ′ ≡ , P r ′ ,p +1 = 0. From these, we can see that M ,s +1 ⊠ M r ′ ,s ′ = M , ⊠ ( M ,s ⊠ M r ′ ,s ′ ) M ,s − ⊠ M r ′ ,s ′ = (cid:18) p − − s − s ′ M ℓ = | s − s ′ +1 | +1 ℓ + s +1+ s ′ ≡ M r ′ ,ℓ (cid:19) ⊕ (cid:18) p − M ℓ =2 p − s − s ′ ℓ + s +1+ s ′ ≡ P r ′ ,ℓ (cid:19) ⊕ A = (cid:18) p − − s − s ′ M ℓ = | s − s ′ +1 | +1 ℓ + s +1+ s ′ ≡ M r ′ ,ℓ (cid:19) ⊕ (cid:18) p M ℓ = p − s − s ′ ℓ + s +1+ s ′ ≡ P r ′ ,ℓ (cid:19) , which is the same as the fusion rule given by (5.21) for s + 1.Finally, the fusion rule (5.18) for an arbitrary r ∈ Z follows from (5.21) together with thefusion rules (3.8) and (5.12) involving M , .Now the fusion rules (5.19) can be proved similarly by induction on s ′ , using the fusionrules of Theorem 5.1.4; we omit the details since they are tedious. As for (5.20), we firstnote that rigidity of C M ( p ) and projectivity of P r,s in C M ( p ) imply that the sequence0 −→ P r,s ⊠ M r ′ ,s ′ −→ P r,s ⊠ Z r ′ ,s ′ −→ P r,s ⊠ ( M r ′ +1 ,p − s ′ ⊕ M r ′ − ,p − s ′ ) −→ r ′ ∈ Z , 1 ≤ s ′ ≤ p −
1. Similarly, the sequence0 −→ P r,s ⊠ Z r ′ ,s ′ −→ P r,s ⊠ P r ′ ,s ′ −→ P r,s ⊠ M r ′ ,s ′ −→ P r,s ⊠ P r ′ ,s ′ = 2 · ( P r,s ⊠ M r ′ ,s ′ ) ⊕ ( P r,s ⊠ M r ′ +1 ,p − s ′ ) ⊕ ( P r,s ⊠ M r ′ − ,p − s ′ ) , and then (5.20) follows from (5.19). References [Ad1] D. Adamovi´c, Classification of irreducible modules of certain subalgebras of freeboson vertex algebra,
J. Algebra (2003), no. 1, 115–132.[Ad2] D. Adamovi´c, A construction of admissible A (1)1 -modules of level − , J. PureAppl. Algebra , (2005), no. 2-3, 119–134.[ACGY] D. Adamovi´c, T. Creutzig, N. Genra and J. Yang, The vertex algebras R ( p ) and V ( p ) , arXiv:2001.08048.[AM1] D. Adamovi´c and A. Milas, Vertex operator algebras associated to modularinvariant representations for A (1)1 , Math. Res. Lett. (1995), no. 5, 563–575.58AM2] D. Adamovi´c and A. Milas, Logarithmic intertwining operators and W (2 , p − J. Math. Phys. (2007), no. 7, 073503, 20 pp.[AM3] D. Adamovi´c and A. Milas, On the triplet vertex algebra W ( p ), Adv. Math. (2008), no. 6, 2664–2699.[AM4] D. Adamovi´c and A. Milas, Lattice construction of logarithmic modules forcertain vertex algebras,
Selecta Math. (N.S.) (2009), no. 4, 535–561.[AMW] D. Adamovi´c, A. Milas and Q. Wang, On parafermion vertex algebras of sl (2) − / and sl (3) − / , arXiv:2005.02631.[ACKR] J. Auger, T. Creutzig, S. Kanade and M. Rupert, Braided tensor categoriesrelated to B p vertex algebras, Comm. Math. Phys. (2020), no. 1, 219–260.[BPZ] A. Belavin, A. Polyakov and A. Zamolodchikov, Infinite conformal symmetryin two-dimensional quantum field theory,
Nuclear Phys. B (1984), no. 2,333–380.[BMM] K. Bringmann, K. Mahlburg and A. Milas, Quantum modular forms and plumb-ing graphs of 3-manifolds,
J. Combin. Theory Ser. A (2020), 105145, 32 pp.[BN] M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglastheories,
J. Phys. A (2016), no. 1, 015401, 33 pp.[CM] S. Carnahan and M. Miyamoto, Regularity of fixed-point vertex operator sub-algebras, arXiv:1603.05645.[CF] N. Carqueville and M. Flohr, Nonmeromorphic operator product expansion and C -cofiniteness for a family of W -algebras, J. Phys. A (2006), no. 4, 951–966.[CCFGH] M. Cheng, S. Chun, F. Ferrari, S. Gukov and S. Harrison, 3d modularity, J.High Energy Phys.
J. High Energy Phys.
J.Topol. (2014), no. 4, 1005–1053.[CGP2] F. Costantino, N. Geer and B. Patureau-Mirand, Some remarks on the unrolledquantum group of sl (2), J. Pure Appl. Algebra (2015), no. 8, 3238–3262.[Cr1] T. Creutzig, W -algebras for Argyres-Douglas theories, Eur. J. Math. (2017),no. 3, 659–690. 59Cr2] T. Creutzig, Logarithmic W -algebras and Argyres-Douglas theories at higherrank, J. High Energy Phys.
Selecta Math. (N.S.) (2019), no. 2, Paper No. 27, 21 pp.[CG] T. Creutzig and T. Gannon, Logarithmic conformal field theory, log-modulartensor categories and modular forms, J. Phys. A (2017), no. 40, 404004, 37pp.[CGR] T. Creutzig, A. Gainutdinov and I. Runkel, A quasi-Hopf algebra for the tripletvertex operator algebra, Commun. Contemp. Math. (2020), no. 3, 1950024,71 pp.[CGN] T. Creutzig, N. Genra and S. Nakatsuka, Duality of subregular W -algebras andprincipal W -superalgebras, arXiv:2005.10713.[CHY] T. Creutzig, Y.-Z. Huang and J. Yang, Braided tensor categories of admissiblemodules for affine Lie algebras, Comm. Math. Phys. (2018) no. 3, 827–854.[CJORY] T. Creutzig, C. Jiang, F. Orosz Hunziker, D. Ridout and J. Yang, Tensor cate-gories arising from Virasoro algebras, arXiv:2002.03180.[CKLR] T. Creutzig, S. Kanade, A. Linshaw and D. Ridout, Schur-Weyl duality forHeisenberg cosets,
Transform. Groups (2019), no. 2, 301–354.[CKM1] T. Creutzig, S. Kanade and R. McRae, Tensor categories for vertex operatorsuperalgebra extensions, arXiv:1705.05017.[CKM2] T. Creutzig, S. Kanade and R. McRae, Gluing vertex algebras, arXiv:1906.00119.[CMY] T. Creutzig, R. McRae and J. Yang, Direct limit completions of vertex tensorcategories, arXiv:2006.09711.[CM1] T. Creutzig and A. Milas, False theta functions and the Verlinde formula, Adv.Math. (2014), 520–545.[CM2] T. Creutzig and A. Milas, Higher rank partial and false theta functions andrepresentation theory,
Adv. Math. (2017), 203–227.[CMR] T. Creutzig, A. Milas and M. Rupert, Logarithmic link invariants of U Hq ( sl )and asymptotic dimensions of singlet vertex algebras, J. Pure Appl. Algebra (2018), no. 10, 3224–3247.[CRi1] T. Creutzig and D. Ridout, Relating the archetypes of logarithmic conformalfield theory,
Nuclear Phys. B (2013), no. 3, 348–391.60CRi2] T. Creutzig and D. Ridout, Modular data and Verlinde formulae for fractionallevel WZW models II,
Nuclear Phys. B (2013), no. 2, 423–458.[CRW] T. Creutzig, D. Ridout and S. Wood, Coset constructions of logarithmic (1 , p )-models,
Lett. Math. Phys. (2014), no. 5, 553–583.[CRu] T. Creutzig and M. Rupert, Uprolling unrolled quantum groups,arXiv:2005.12445.[CY] T. Creutzig and J. Yang, Tensor categories of affine Lie algebras beyond admis-sible level, arXiv:2002.05686.[DeR] M. De Renzi, Non-semisimple extended topological quantum field theories,arXiv:1703.07573.[DGP] M. De Renzi, N. Geer and B. Patureau-Mirand, Non-semisimple quantum invari-ants and TQFTs from small and unrolled quantum groups, arXiv:1812.10685.[DL] C. Dong and J. Lepowsky,
Generalized Vertex Algebras and Relative Vertex Op-erators , Progress in Math., Vol. 112, Birkh¨auser, Boston, 1993.[DLM] C. Dong, H. Li and G. Mason, Compact automorphism groups of vertex operatoralgebras,
Internat. Math. Res. Notices
Comm. Math. Phys. (2006), no. 1, 47–93.[FGST2] B. Feigin, A. Gainutdinov, A. Semikhatov and I. Tipunin, Logarithmic exten-sions of minimal models: characters and modular transformations,
Nuclear Phys.B (2006), no. 3, 303–343.[FT] B. Feigin and I. Tipunin, Logarithmic CFTs connected with simple Lie algebras,arXiv:1002.5047.[FL] I. Flandoli and S. Lentner, Logarithmic conformal field theories of type B n , ℓ = 4and symplectic fermions, J. Math. Phys. (2018), no. 7, 071701, 35 pp.[FHL] I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertexoperator algebras and modules, Mem. Amer. Math. Soc. (1993), no. 494,viii+64 pp.[FHST] J. Fuchs, S. Hwang, A. Semikhatov and I. Tipunin, Nonsemisimple fusion alge-bras and the Verlinde formula,
Comm. Math. Phys. (2004), no. 3, 713–742.[GPPV] S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants,
J. Knot Theory Ramifications (2020), no. 2, 2040003, 85 pp.61Hu1] Y.-Z. Huang, Virasoro vertex operator algebras, the (nonmeromorphic) operatorproduct expansion and the tensor product theory, J. Algebra (1996), no. 1,201–234.[Hu2] Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture,
Commun.Contemp. Math. (2008), no. 1, 103–154.[Hu3] Y.-Z. Huang, Rigidity and modularity of vertex tensor categories, Commun.Contemp. Math. (2008), suppl. 1, 871–911.[Hu4] Y.-Z. Huang, Cofiniteness conditions, projective covers and the logarithmic ten-sor product theory, J. Pure Appl. Algebra , , no. 4, (2009), 458–475.[HKL] Y.-Z. Huang, A. Kirillov, Jr. and J. Lepowsky, Braided tensor categories andextensions of vertex operator algebras, Comm. Math. Phys. (2015), no. 3,1143–1159.[HLZ1] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory forgeneralized modules for a conformal vertex algebra, I: Introduction and stronglygraded algebras and their generalized modules,
Conformal Field Theories andTensor Categories, Proceedings of a Workshop Held at Beijing InternationalCenter for Mathematics Research , ed. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong,I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University,Vol. 2, Springer, New York, 2014, 169–248.[HLZ2] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theoryfor generalized modules for a conformal vertex algebra, II: Logarithmic formalcalculus and properties of logarithmic intertwining operators, arXiv:1012.4196.[HLZ3] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theoryfor generalized modules for a conformal vertex algebra, III: Intertwining mapsand tensor product bifunctors , arXiv:1012.4197.[HLZ4] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory forgeneralized modules for a conformal vertex algebra, IV: Constructions of tensorproduct bifunctors and the compatibility conditions , arXiv:1012.4198.[HLZ5] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theoryfor generalized modules for a conformal vertex algebra, V: Convergence con-dition for intertwining maps and the corresponding compatibility condition,arXiv:1012.4199.[HLZ6] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theoryfor generalized modules for a conformal vertex algebra, VI: Expansion condi-tion, associativity of logarithmic intertwining operators, and the associativityisomorphisms, arXiv:1012.4202. 62HLZ7] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category the-ory for generalized modules for a conformal vertex algebra, VII: Convergenceand extension properties and applications to expansion for intertwining maps,arXiv:1110.1929.[HLZ8] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theoryfor generalized modules for a conformal vertex algebra, VIII: Braided tensorcategory structure on categories of generalized modules for a conformal vertexalgebra, arXiv:1110.1931.[Ka] H. Kausch, Extended conformal algebras generated by a multiplet of primaryfields,
Phys. Lett. B 259 (1991), no. 4, 448–455.[KL1] D. Kazhdan and G. Lusztig, Affine Lie algebras and quatum groups,
Interna-tional Mathematics Research Notices (in
Duke Math. J. ) (1991), 21–29.[KL2] D. Kazhdan and G. Lusztig, Tensor structure arising from affine Lie algebras, I, J. Amer. Math. Soc. (1993), 905–947.[KL3] D. Kazhdan and G. Lusztig, Tensor structure arising from affine Lie algebras,II, J. Amer. Math. Soc. (1993), 949–1011.[KL4] D. Kazhdan and G. Lusztig, Tensor structure arising from affine Lie algebras,III, J. Amer. Math. Soc. (1994), 335–381.[KL5] D. Kazhdan and G. Lusztig, Tensor structure arising from affine Lie algebras,IV, J. Amer. Math. Soc. (1994), 383–453.[KO] A. Kirillov, Jr. and V. Ostrik, On a q -analogue of the McKay correspondence andthe ADE classification of sl conformal field theories, Adv. Math. (2002),no. 2, 183–227.[KS] H. Kondo and Y. Saito, Indecomposable decomposition of tensor products ofmodules over the restricted quantum universal enveloping algebra associated to sl , J. Algebra (2011), 103–129.[Le] S. Lentner, Quantum groups and Nichols algebras acting on conformal fieldtheories, arXiv:1702.06431.[Li1] H. Li, Symmetric invariant bilinear forms on vertex operator algebras,
J. PureAppl. Algebra (1994), no. 3, 279–297.[Li2] The physics superselection principle in vertex operator algebra theory, J. Algebra (1997), no. 2, 436–457.[McR] R. McRae, On the tensor structure of modules for compact orbifold vertex op-erator algebras,
Math. Z. (2019), https://doi.org/10.1007/s00209-019-02445-z.63Mi1] M. Miyamoto, Flatness and semi-rigidity of vertex operator algebras,arXiv:1104.4675.[Mi2] M. Miyamoto, C -cofiniteness and fusion products of vertex operator algebras, Conformal Field Theories and Tensor Categories, Proceedings of a WorkshopHeld at Beijing International Center for Mathematics Research , ed. C. Bai, J.Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lec-tures from Beijing University, Vol. 2, Springer, New York, 2014, 271–279.[NT] K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W ( p ) andthe restricted quantum group U q ( sl ) at q = e πip , Exploring new structures andnatural constructions in mathematical physics , 1–49, Adv. Stud. Pure Math.,Vol. 61,
Math. Soc. Japan, Tokyo (2011).[Pa] S. Park, Higher rank ˆ Z and F K , SIGMA Symmetry Integrability Geom. MethodsAppl. (2020), Paper No. 044, 17 pages.[Ru] M. Rupert, Categories of weight modules for unrolled restricted quantum groupsat roots of unity, arXiv:1910.05922.[Su] S. Sugimoto, On the Feigin-Tipunin conjecture, arXiv:2004.05769.[TW] A. Tsuchiya and S. Wood, The tensor structure on the representation categoryof the W p triplet algebra, J. Phys. A (2013), no. 44, 445203, 40 pp.[Ve] E. Verlinde, Fusion rules and modular transformations in 2D conformal fieldtheory, Nuclear Phys. B300