Tensor structure on the Kazhdan-Lusztig category for affine gl(1|1)
aa r X i v : . [ m a t h . QA ] S e p Tensor structure on the Kazhdan-Lusztig categoryfor affine gl (1 | Thomas Creutzig, Robert McRae and Jinwei Yang
Abstract
We show that the Kazhdan-Lusztig category KL k of level- k finite-length moduleswith highest-weight composition factors for the affine Lie superalgebra \ gl (1 |
1) hasvertex algebraic braided tensor supercategory structure, and that its full subcategory O fink of objects with semisimple Cartan subalgebra actions is a tensor subcategory.We show that every simple \ gl (1 | KL k has a projective cover in O fink ,and we determine all fusion rules involving simple and projective objects in O fink .Then using Knizhnik-Zamolodchikov equations, we prove that KL k and O fink arerigid. As an application of the tensor supercategory structure on O fink , we studycertain module categories for the affine Lie superalgebra \ sl (2 |
1) at levels 1 and − .In particular, we obtain a tensor category of \ sl (2 | − that includesrelaxed highest-weight modules and their images under spectral flow. Contents d gl (1 | -modules 73 Fusion rules 194 Rigidity 275 Simple current extensions of V ( gl (1 | Affine Lie (super)algebras and their representations play meaningful roles in various areasof both mathematics and physics. Increasingly, the representation categories of interest areneither finite nor semisimple, and such categories are expected to give rise to interestingnon-semisimple topological field theories and invariants of knots and links. They are1lso expected to relate to quantum group module categories via non-semisimple Kazhdan-Lusztig correspondences.We are concerned with representation categories of the affine Lie superalgebra \ gl (1 | k , or equivalently of the affine vertex operator superalgebra V k ( gl (1 | V k ( gl (1 | GL (1 | In this paper, we study two representation categories for \ gl (1 | KL k of level- k finite-length modules with highest-weight compositionfactors. Equivalently, this is the category of finite-length grading-restricted generalized V k ( gl (1 | L . The second cate-gory we consider is the full subcategory O fink of KL k consisting of modules with semisimpleCartan subalgebra actions. Using the sufficient conditions given in [CY] for the existenceof the logarithmic tensor categories constructed in [HLZ1]-[HLZ9], we prove in Theorems2.3.1 and 2.3.2 that both KL k and O fink have the vertex and braided tensor supercategorystructures of [HLZ1]-[HLZ9] (see [CKM] for a description of the supercategory structurein the superalgebra generality).There are two classes of simple objects in KL k and O fink . The typical modules b V kn,e for n ∈ C , e/k / ∈ Z are irreducible Verma \ gl (1 | n, e ) indicates Cartansubalgebra eigenvalues on a highest-weight vector. The atypical modules b A n,ℓk for n ∈ C , ℓ ∈ Z are then the unique irreducible quotients of the corresponding reducible Vermamodules. In Section 2.4, we show that all irreducible modules have projective covers in O fink : the typical modules are their own projective covers, while each b A kn,ℓk has a length-4projective cover b P kn,ℓk .We next determine tensor products of irreducible modules in KL k and O fink in Theorems3.2.2, 3.2.3, and 3.2.4: The following are the tensor products of simple \ gl (1 | -modules:1. For n ∈ C and ℓ ∈ Z , b A kn,ℓk ⊠ b A kn ′ ,ℓ ′ k ∼ = b A kn + n ′ − ε ( ℓ,ℓ ′ ) , ( ℓ + ℓ ′ ) k , where the scalar ε ( ℓ, ℓ ′ ) is defined in (2.15) and Theorem 3.2.2 below.2. For n, n ′ ∈ C , ℓ ∈ Z , and e ′ /k / ∈ Z , b A kn,ℓk ⊠ b V kn ′ ,e ′ ∼ = b V kn + n ′ − ε ( ℓ ) ,e ′ + ℓk , where the scalar ε ( ℓ ) is defined in (2.15) below. . For n, n ′ ∈ C and e/k, e ′ /k / ∈ Z , b V kn,e ⊠ b V kn ′ ,e ′ ∼ = ( b V kn + n ′ + ,e + e ′ ⊕ b V kn + n ′ − ,e + e ′ if ( e + e ′ ) /k / ∈ Z b P kn + n ′ + ε (( e + e ′ ) /k ) ,e + e ′ if ( e + e ′ ) /k ∈ Z . These fusion rules follow from the relationship (developed in [FZ1, Li, FZ2, HY, MY],among other references) between vertex algebraic intertwining operators among \ gl (1 | gl (1 | \ gl (1 | W and W are two simple modules in KL k , the canonical tensorproduct intertwining operator of type (cid:0) W ⊠ W W W (cid:1) restricts to a gl (1 | W ⊠ W , which can then be induced to a homomor-phism from a generalized Verma \ gl (1 | W ⊠ W . On the other hand, we canget homomorphisms coming out of W ⊠ W if we can construct intertwining operatorsusing suitable gl (1 | b A kn,ℓk , ℓ ∈ Z \ { } , we also use information coming fromexplicit singular vectors in the reducible Verma modules b V k , ± k .Finally, in Theorem 4.2.3, we prove that KL k and O fink are rigid with duals given bycontragredient modules; since KL k and O fink are also braided and have a natural twistisomorphism, this means they are braided ribbon tensor supercategories. To prove thisresult, we first show that simple modules in KL k are rigid, and then we use [CMY2, Thm.4.4.1] to extend rigidity to all finite-length modules. Rigidity for the atypical irreduciblesis easy: the fusion rules of Theorem 1.1.1 show that b A kn,ℓk is a simple current, so evaluationand coevaluation morphisms are isomorphisms.To prove that each typical irreducible module b V kn,e is rigid, with contragredient dual b V k − n, − e , we need to use explicit formulas for 4-point correlation functions of the form φ ( z ) = h v , Y ( v , Y ( v , z ) v i , where Y and Y are suitable intertwining operators involving b V kn,e and its contragredient, v and v are certain lowest-conformal-weight vectors in b V kn,e , and v , v are certain lowest-conformal-weight vectors in b V k − n, − e . We use Knizhnik-Zamolodchikov equations to showin Theorem 4.1.4 that φ ( z ) satisfies the second-order regular-singular-point differentialequation z (1 − z ) φ ′′ ( z ) + (cid:2) (4∆ n,e + 1) − (8∆ n,e + 1) z (cid:3) φ ′ ( z ) + 4∆ n,e z − φ ( z )+ 2∆ n,e (2∆ n,e − − z ) − φ ( z ) + (cid:20)(cid:16) ek (cid:17) − n,e (cid:21) φ ( z ) = 0 , where ∆ n,e = ek (cid:0) n + e k (cid:1) is the lowest conformal weight of b V kn,e . This differential equationcan be solved explicitly in terms of hypergeometric functions, and we can then prove rigidityfor b V kn,e with the help of well-known formulas relating series expansions of hypergeometricfunctions on different regions of C \ { , } . 3 .2 On relaxed highest-weight modules The key result we use to establish the existence of vertex tensor category structure is [CY,Thm. 3.3.4], which shows that KL k has the vertex algebraic braided tensor (super)categorystructure of [HLZ1]-[HLZ9] provided that every C -cofinite grading-restricted generalized V k ( gl (1 | C -cofinite modules for affine vertex operator algebras atadmissible level is rather small. Especially, it misses all relaxed highest-weight modules andtheir images under spectral flow: these modules have infinite-dimensional conformal weightspaces, and conformal weights need not be lower-bounded. This means that finitenessconditions assumed in [HLZ1]-[HLZ9] to prove existence of vertex tensor category structuredo not hold, and one thus needs other strategies to obtain tensor categories that includerelaxed highest-weight modules.For example, a vertex operator (super)algebra V might have a vertex operator subal-gebra U that has a module category C U that admits vertex tensor category structure. If V is an object in a suitable completion of C U , then one can use the theory of vertex operator(super)algebra extensions [HKL, CKM] to obtain and study tensor categories of V -modulesthat lie in the completion of C U . For this reason, we have established that under reasonableassumptions, the completion of a vertex tensor category under direct limits inherits vertextensor category structure [CMY1]. In [CMY2], we applied this strategy to module cate-gories for singlet vertex operator algebras that live in the direct limit completions of the C -cofinite module categories for the corresponding Virasoro vertex operator subalgebras.These Virasoro module categories have vertex tensor category structure by [CJORY].In the present work, we apply the same idea to obtain vertex tensor categories of re-laxed highest-weight modules for the simple affine vertex operator superalgebra of sl (2 | − . The superalgebra V − ( sl (2 | V ( gl (1 | V − ( sl (2 | V ( gl (1 | V − ( sl (2 | V − ( sl (2 | O fin as inductions of simple V ( gl (1 | V − ( sl (2 | V ( gl (1 | V ( gl (1 | S induces to a local V − ( sl (2 | P S of S is the pro-jective cover of the induction of S . 4. The induction functor is monoidal [CKM, Sec. 2.7] and preserves duals [CKM, Sec.2.8], so fusion rules and rigidity are inherited from O fin .Another example of a vertex operator superalgebra extension of V ( gl (1 |
1) is a pairof βγ -ghosts tensored with two free fermions, which is just an additional order-2 simplecurrent extension of V − ( sl (2 | βγ -ghosts was the subjectof recent work by Allen and Wood [AW]; in complete analogy to V − ( sl (2 | V ( gl (1 | V ( gl (1 | V ( sl (2 | O fin in the last section of this work. In this case, all V ( sl (2 | V ( gl (1 | V ( sl (2 | GL (1 | and topological invariants In the early 1990s Rozansky and Saleur studied the Wess-Zumino-Witten theory of theLie supergroup GL (1 |
1) [RS1, RS2, RS3] in order to obtain invariants of 3-manifolds andlinks. They were motivated by Witten’s celebrated insight [Wi] that the Jones polynomialarises from SU (2) Chern-Simons theory, whose Hilbert space can be identified with thespace of conformal blocks of the WZW theory of SU (2) [Ga]. In the GL (1 |
1) analogue,Rozansky and Saleur obtained Alexander-Conway polynomials. Recall that any modulartensor category (that is, a non-degenerate semisimple finite braided ribbon tensor cate-gory) gives rise to invariants of compact 3-manifolds [Tu]. By now it is understood thatnon-semisimple non-finite categories can also give rise to 3-manifold invariants via non-semisimple topological field theories (see for example [CGP1]). It is thus natural to expectthat the invariants of Rozansky and Saleur can be reproduced from a topological fieldtheory constructed from the tensor category O fink of modules for affine gl (1 | GL (1 |
1) was actually the first example of a logarithmic conformalfield theory to be studied in detail. The term “logarithmic” here refers to logarithmicsingularities in the correlation functions. Such singularities arise from non-semisimpleaction of the Virasoro zero-mode, and by now conformal field theories associated to non-semisimple module categories for vertex operator algebras are called logarithmic conformalfield theories; see [CRi3] for an introduction. The GL (1 |
1) WZW theory has been furtherexplored in the bulk [SS, CRø] and boundary [CQS, CS]. In particular, fusion rules weresuggested by computations of correlation functions [SS, CRø], boundary states [CQS], theVerlinde formula [CQS, CRi2], and the NGK algorithm [CRi2]. Our work here shows thatthese methods indeed predicted the correct fusion rules.5 .4 Outlook
The most popular vertex operator algebras with non-semisimple representation theory areprobably the (1 , p ) singlet and triplet algebras. In the first case p = 2, the (1 ,
2) singletalgebra is the Heisenberg coset of V k ( gl (1 | ,
2) triplet algebra is a simplecurrent extension of the singlet. It is also the even subalgebra of a pair of symplecticfermions, which is the affine vertex operator superalgebra of psl (1 | sl (2). This conjecture is correct on the level of linearcategories [NT], and fusion rules for simple and projective modules also coincide [TW, KS].However the category of the quantum group is not braidable [KS], and it has only recentlybeen realized that there is a quasi-Hopf algebra modification of the quantum group yieldingbraided tensor category structure [CGR]. This quasi-Hopf modification is obtained byrelating the restricted quantum group to the category of local modules for a simple currentextension in the category of weight modules for the unrolled restricted quantum group.The latter category is conjecturally equivalent to the category of ordinary modules for thesinglet vertex operator algebra [CGP2, CMR].Thus clearly a central problem in this area is to prove the Kazhdan-Lusztig-type cor-respondences between triplet algebra categories and quasi-Hopf modifications of the re-stricted quantum groups. A variant of this conjecture is an equivalence of our tensor cat-egory O fink with a category of weight modules for U q ( gl (1 | U q ( gl (1 | q is related to the level of the affine Lie algebra, it seems realistic that similararguments can prove a correspondence between our O fink and a category of weight modulesfor U q ( gl (1 | V k ( gl (1 | ,
2) singlet and tripletalgebras, the quantum group triplet and singlet correspondences should then follow.The next natural question is whether one can understand higher-rank affine superal-gebras and W -superalgebras. As a first step, one can consider simple current extensionsof tensor products of V k ( gl (1 | V ( gl ( n | n )) as an extension of n copies of V k ( gl (1 | V k ( gl (1 | W -superalgebras for understanding braided tensor categories. Namely, Feigin-Semikhatov triality [FS] asserts that Heisenberg cosets of subregular W -algebras of sl ( n )coincide with Heisenberg cosets of principal W -superalgebras of sl ( n | sl ( n | n = 1 version of this duality asserts that the Heisenberg coset of a pair of6 γ -ghosts coincides with the U (1)-orbifold of the affine vertex superalgebra of psl (1 | W -algebra tensored with a pair of fermions is the principal W -superalgebra, and conversely,a Heisenberg coset of the principal W -superalgebra tensored with a lattice vertex superal-gebra is the subregular W -algebra [CGN]. Here, the n = 1 version is the relation between βγ -ghosts and V k ( gl (1 | n = 2 version of this duality is the Kazama-Suzuki duality between the affinevertex algebra of sl (2) and the N = 2 super Virasoro algebra; this was exploited by Feigin,Semikhatov, and Tipunin a while ago [FST] and recently has received renewed attention[CLRW, KoSa, Sa1, Sa2]. All these works use the duality for understanding super Virasoromodules. In our opinion it is easier to prove vertex tensor category structure for V k ( gl (1 | βγ -ghosts, so we hope that one can also establish and study vertex tensor categorystructure on certain module categories for the super Virasoro algebra, and then use theduality to understand dual categories of relaxed highest-weight modules of the affine vertexalgebra of sl (2). Acknowledgements
TC acknowledges support from NSERC discovery grant RES0048511. \ gl (1 | -modules In this section, we first review the basic representation theory of the Lie superalgebra gl (1 |
1) and its affinization \ gl (1 | KL k and O fink of \ gl (1 | KL k and O fink have vertex algebraic braided tensorsupercategory structure, and we construct projective covers of irreducible modules in O fink . gl (1 | The general linear superalgebra gl (1 |
1) consists of endomorphisms of the vector superspace C | . It has basis N = 12 (cid:18) − (cid:19) , E = (cid:18) (cid:19) , ψ + = (cid:18) (cid:19) , ψ − = (cid:18) (cid:19) , so that the even and odd subspaces of gl (1 |
1) are gl (1 | ¯0 = Span { N, E } and gl (1 | ¯1 =Span { ψ + , ψ − } , respectively. The non-zero Lie superbrackets of basis elements are[ N, ψ ± ] = ± ψ ± , { ψ + , ψ − } = E. There is a nondegenerate even invariant supersymmetric bilinear form κ ( · , · ) on gl (1 | κ ( N, E ) = κ ( E, N ) = 1 , κ ( ψ + , ψ − ) = − κ ( ψ − , ψ + ) = 1 , with κ vanishing on all other pairs of basis elements.7 .1.1 Remark. There is a second invariant bilinear form κ on gl (1 |
1) that vanishes onall pairs of basis elements except for κ ( N, N ) = 1. Moreover, gl (1 |
1) has automorphisms ω λ,µ for λ ∈ C , µ ∈ C \ { } defined by ω λ,µ ( N ) = N + λE, ω λ,µ ( ψ ± ) = µψ ± , ω λ,µ ( E ) = µ E, so that every non-degenerate bilinear form of gl (1 |
1) is related to κ by an automorphism.That is, κ ( ω λ,µ ( a ) , ω λ,µ ( b )) = µ κ ( a, b ) + 2 λκ ( a, b )for any a, b ∈ gl (1 | V n − ,e for n, e ∈ C be the Verma module generatedby a highest-weight vector v such that N · v = nv, E · v = ev, ψ + · v = 0 . Since ψ − squares to zero in the universal enveloping algebra of gl (1 | n is the average of the two N -eigenvalues of V n,e . The Verma module V n,e is irreducible if and only if e = 0. When e = 0, we denote the 1-dimensional irreduciblequotient of V n,e by A n + . The irreducibles with e = 0 are said to be typical and those with e = 0 are said to be atypical . For each n ∈ C , there is a non-split exact sequence0 → A n − → V n, → A n + → . For n ∈ C , we also define the induced module P n = U ( gl (1 | ⊗ U ( gl (1 | ¯0 C v n , where E · v n = 0 and N · v n = nv n (that is, C v n is the restriction of A n to gl (1 | ¯0 ). The module P n is indecomposable but reducible and satisfies the non-split exact sequence0 → V n + , → P n → V n − , → . (2.1)It has Loewy diagram A n P n : A n − A n +1 A n and has basis { v n , ψ + v n , ψ − v n , ψ + ψ − v n } .Let O be the category of finitely-generated gl (1 | gl (1 | ¯0 -actions (they have nilpotent gl (1 | ¯1 -actions automatically); see for example the expositionin [Br] for more details on this category. Every object in O has finite length, and O hasenough projective objects. The typical irreducible modules V n,e for n ∈ C , e ∈ C \ { } P n for n ∈ C is the projective cover of the atypicalirreducible module A n .We remark that O is more precisely a supercategory : every object M has a Z -grading M ¯0 ⊕ M ¯1 such that the gl (1 | ¯0 -action is even. The Z -gradings on modules induce Z -gradings on morphism spaces: a linear map f : M → N is a morphism of gl (1 | | f | if f ( M i ) ⊆ N i + | f | for i ∈ Z and a · f ( m ) = ( − | a || f | f ( a · m )for m ∈ M and homogeneous a ∈ gl (1 | V n,e , for example, have Z -gradings such that v is even and ψ − v is odd. We can reverse parities to obtain a differentobject Π( V n,e ) which is isomorphic to V n,e via an odd isomorphism. \ gl (1 | The affine Lie superalgebra \ gl (1 |
1) associated with gl (1 |
1) and the bilinear form κ ( · , · ) isthe superspace gl (1 | ⊗ C [ t, t − ] ⊕ C k , where C [ t, t − ] and k are even, with bracket definedby [ a ⊗ t r , b ⊗ t s ] = [ a, b ] ⊗ t r + s + κ ( a, b ) rδ r + s, k , [ a ⊗ t r , k ] = 0for a, b ∈ gl (1 |
1) and r, s ∈ Z . The non-vanishing brackets of basis elements for \ gl (1 |
1) are[ N r , E s ] = r k δ r + s, , [ N r , ψ ± s ] = ± ψ ± r + s , { ψ + r , ψ − s } = E r + s + r k δ r + s, , where a r denotes a ⊗ t r for a ∈ gl (1 |
1) and r ∈ Z .Given a gl (1 | M and k ∈ C , M is a gl (1 | ⊗ C [ t ] ⊕ C k -module with gl (1 | ⊗ t C [ t ] acting trivially and k acting as scalar multiplication by k . We then have the induced \ gl (1 | c M k = U ( \ gl (1 | ⊗ U ( gl (1 | ⊗ C [ t ] ⊕ C k ) M. When M is the trivial gl (1 | A = C , c M k has a vertex superalgebra structure,which we denote by V k ( gl (1 | M , the modules c M k are also V k ( gl (1 | k is called the level of c M k . The space of invariant bilinear forms of gl (1 |
1) is two-dimensional.
Apriori , this means that one should consider a two-parameter family of affine vertex super-algebras V B ( gl (1 |
1) associated to gl (1 | B . As noted inRemark 2.1.1 all non-degenerate bilinear forms are related by an automorphism of gl (1 | V B ( gl (1 | ∼ = V ( gl (1 |
1) if B is non-degenerate. It thus suffices to restrict atten-tion to V ( gl (1 | k , we willstay with that notation. 9hen k = 0, V k ( gl (1 | ω = 12 k ( N − E − + E − N − − ψ + − ψ −− + ψ −− ψ + − ) + 12 k E − . (2.2)Its associated vertex operator is given by Y ( ω, x ) = 12 k : N ( x ) E ( x ) + E ( x ) N ( x ) − ψ + ( x ) ψ − ( x ) + ψ − ( x ) ψ + ( x ) : + 12 k : E ( x ) E ( x ) : . (2.3)In particular, L − = 1 k X r ∈ Z ≥ ( N − r − E r + E − r − N r − ψ + − r − ψ − r + ψ −− r − ψ + r ) + 1 k X r ∈ Z ≥ E − r − E r , (2.4) L = 1 k X r ∈ Z > ( N − r E r + E − r N r − ψ + − r ψ − r + ψ −− r ψ + r ) + 1 k X r ∈ Z > E − r E r + 1 k ( N E − ψ +0 ψ − ) + 12 k E + 12 k E . (2.5)Now we introduce two representation categories of V k ( gl (1 |
1. The Kazhdan-Lusztig category KL k is the supercategory of finite-length grading-restricted generalized V k ( gl (1 | O fink is the full subcategory of KL k consisting of modules onwhich H and E act semisimply.We start by describing the simple objects in KL k . First, just as for affine vertexoperator algebras (see for example [LL, Sec. 6.2]), every irreducible (grading-restricted) V k ( gl (1 | gl (1 | KL k is a quo-tient of a generalized Verma \ gl (1 | b V kn,e for e = 0 or b A kn, . The structure of thesegeneralized Verma \ gl (1 | b V kn,e is irreducible if and only if e/k / ∈ Z .2. When e = 0, the unique irreducible quotient of b V kn, is b A kn + , , and there is a non-splitexact sequence 0 → b A kn − , → b V kn, → b A kn + , → . (2.6)10. When e/k ∈ Z \ { } , we use b A kn,e to denote the unique irreducible quotient of b V kn,e ,and there are non-split exact sequences0 → b A kn +1 ,e → b V kn,e → b A kn,e → e/k = 1 , , , . . . ) , → b A kn − ,e → b V kn,e → b A kn,e → e/k = − , − , − , . . . ) . (2.7)Thus KL k has two classes of simple objects: the typical irreducible modules b V kn,e for n ∈ C , e/k / ∈ Z , and the atypical irreducible modules b A n,e for n ∈ C , e/k ∈ Z . Note that as amodule for itself, V k ( gl (1 | b A k , ; in particular, V k ( gl (1 | KL k are C -cofinite, and they arealso objects of O fink .We now discuss some properties of O fink that we will use in the following sections. First,since E is central in \ gl (1 |
1) and acts semisimply on modules in O fink , we have a directsum decomposition O fink = M e ∈ C ( O fink ) e where ( O fink ) e is the full subcategory of modules in O fink on which E acts by the scalarmultiplication e . In particular, there are no non-zero morphisms between modules in( O fink ) e and ( O fink ) e when e = e .Now suppose W is a module in ( O fink ) e for some e ∈ C . If v n,e ∈ W is some highest-weight vector for \ gl (1 |
1) with N -eigenvalue n + (in particular ψ +0 · v n,e = 0), then (2.5)shows that v n,e is an L -eigenvector with conformal weight∆ n,e = ek (cid:16) n + e k (cid:17) . (2.8)That is, the minimal conformal weights of (grading-restricted) modules in ( O fink ) e havethe form ∆ n,e for n ∈ C ; more precisely, the conformal weights of such modules lie in ∪ i (∆ n i ,e + N ) for finitely many n i ∈ C . In the case e = 0, this means: For any non-zero module W in ( O fink ) :1. The unique minimal conformal weight space of W is W [0] .2. W is generated by W [0] as a \ gl (1 | -module.Proof. The first part is immediate from the e = 0 case of (2.8). For the second part, W/ \ gl (1 | · W [0] is a module in ( O fink ) with vanishing conformal weight-0 space, so thefirst part implies W/ \ gl (1 | · W [0] = 0.Now recall that the contragredient of a generalized module W = L h ∈ C W [ h ] for a vertexoperator (super)algebra V is a module structure on the graded dual W ′ = L h ∈ C W ∗ [ h ] . For11uperalgebras, W ′ has a Z -grading given by W ′ i = ( W i ) ′ for i ∈ Z , and we define thevertex operator Y W ′ by h Y W ′ ( v, x ) w ′ , w i = ( − | v || w ′ | h w ′ , Y W ( e xL ( − x − ) L v, x − ) w i , (2.9)following the convention of [CKM]. (Note that for vertex operator superalgebras, there areseveral different but equivalent definitions in the literature for the contragredient modulevertex operator; see [CKM, Rem. 3.5].) The contragredient of an irreducible V -module isirreducible (see for example [FHL, Prop. 5.3.2]), and for a V -module W , there is a naturaleven isomorphism δ W : W → W ′′ defined by h δ W ( w ) , w ′ i = ( − | w || w ′ | h w ′ , w i for w ∈ W , w ′ ∈ W ′ . Moreover, taking contragredients defines an even contravariantfunctor on V -modules, with the contragredient of a parity-homogeneous morphism f : W → W defined by h f ′ ( w ′ ) , w i = ( − | f || w ′ | h w ′ , f ( w ) i for w ∈ W and parity-homogeneous w ′ ∈ W ′ . It is straightforward to show that thecontragredient functor preserves exactness of sequences involving parity-homogeneous ho-momorphisms.For V k ( gl (1 | \ gl (1 |
1) and the Virasoro algebra on acontragredient module W ′ are given by h a ′ r w ′ , w i = − ( − | a || w ′ | h w ′ , a − r w i (2.10) h L ′ ( n ) w ′ , w i = h w ′ , L ( − n ) w i for a ∈ gl (1 | r, n ∈ Z , w ′ ∈ W ′ , and w ∈ W . In particular, the lowest conformal weightspace of the contragredient ( c M k ) ′ of a generalized Verma module is the gl (1 | M ∗ .Using this observation, we can determine the contragredients of many modules in O fink : Contragredients of V k ( gl (1 | -modules are as follows:1. ( b A kn,ℓk ) ′ ∼ = b A k − n, − ℓk for n ∈ C and ℓ ∈ Z .2. ( b V kn,e ) ′ ∼ = b V k − n, − e for n ∈ C and e/k / ∈ Z .3. ( b P kn ) ′ ∼ = b P k − n for n ∈ C .Proof. For W an irreducible V k ( gl (1 | W ′ is an irreducible V k ( gl (1 | gl (1 | W . Thus the first two cases of the proposition follow from the identities A ∗ n ∼ = A − n for n ∈ C and V ∗ n,e ∼ = V − n, − e for e = 0 (by an odd isomorphism).For the third case, we use the even isomorphism P ∗ n ∼ = P − n in category O . By theuniversal property of induced \ gl (1 | f : b P k − n → ( b P kn ) ′ which is an isomorphism on lowest conformal weight spaces.Since ker f and coker f are objects of ( O fink ) with (ker f ) [0] = 0 = (coker f ) [0] , Lemma 2.2.3implies that ker f = 0 = coker f , so that f is an isomorphism.12 .3 Tensor supercategory structure We now establish vertex and braided tensor supercategory structure on KL k and on itssubcategory O fink . The key result we use is [CY, Thm. 3.3.4], which shows that KL k has thevertex algebraic braided tensor (super)category structure of [HLZ1]-[HLZ9] provided thatevery lower-bounded C -cofinite V k ( gl (1 | The supercategory KL k has vertex algebraic braided tensor supercategorystructure.Proof. By [CY, Thm. 3.3.4], we just need to show that every lower-bounded C -cofinite V k ( gl (1 | W has a finite filtration 0 ⊂ W ⊂ W ⊂ · · · ⊂ W n ⊂ W n +1 = W such that each W i is a Z -graded submodule of W and such that each W i +1 /W i for i =1 , . . . , n is a quotient of a Verma modules b V kn,e for some n, e ∈ C (or its parity-reversedversion). As the exact sequences (2.6) and (2.7) show that each b V kn,e has length at most 2, W has finite length.Now the subcategory O fink inherits vertex and braided tensor supercategory structurefrom KL k , provided it is closed under the P ( z )-tensor products of [HLZ4]: The supercategory O fink has vertex algebraic braided tensor supercategorystructure.Proof. Suppose W and W are any two objects of O fink and ( W ⊠ P ( z ) W , ⊠ P ( z ) ) is their P ( z )-tensor product in KL k , with ⊠ P ( z ) the canonical even P ( z )-intertwining map of type (cid:0) W ⊠ P ( z ) W W W (cid:1) . It is clear that if W ⊠ P ( z ) W is an object of O fink , then ( W ⊠ P ( z ) W , ⊠ P ( z ) )also satisfies the universal property of [HLZ4, Def. 4.15] for a P ( z )-tensor product in O fink .Thus the vertex algebraic braided tensor supercategory structure on KL k will restrict tosuch structure on O fink .It remains to show that E and N act semisimply on W ⊠ P ( z ) W . By (the superalgebrageneralization of) [HLZ4, Eqn. 4.34], v m ( w ⊠ P ( z ) w ) = ( − | v || w | w ⊠ P ( z ) v m w + X i ≥ (cid:18) mi (cid:19) z m − i ( v i w ) ⊠ P ( z ) w for homogeneous v ∈ V k ( gl (1 | w ∈ W , w ∈ W , and m ∈ Z . In particular, X ( w ⊠ P ( z ) w ) = X w ⊠ P ( z ) w + w ⊠ P ( z ) X w X = E, N . Thus because W ⊠ P ( z ) W is spanned by projections of vectors w ⊠ P ( z ) w to the conformal weight spaces of W ⊠ P ( z ) W (see [HLZ4, Prop. 4.23]) and because E and N act semisimply on W and W , we see that W ⊠ P ( z ) W is an object of O fink . The proof of the preceding theorem shows more specifically that if W is an object of the subcategory ( O fink ) e for e ∈ C and W is an object of ( O fink ) e , then W ⊠ P ( z ) W is an object of ( O fink ) e + e . We shall use this fact in computing fusion rulesfor modules in O fink .In the braided tensor supercategory structure on KL k and O fink , the tensor productbifunctor is given by the P (1)-tensor product ⊠ P (1) , which for simplicity we will denote by ⊠ . For any two modules W , W in KL k or O fink , the canonical P (1)-intertwining map ⊠ corresponds to an even (logarithmic) intertwining operator Y ⊠ ( · , x ) of type (cid:0) W ⊠ W W W (cid:1) suchthat Y ⊠ ( w , w = w ⊠ w for w ∈ W and w ∈ W ; here the substitution x W ⊠ W , Y ⊠ ) then satisfies the following universalproperty: for any intertwining operator Y of type (cid:0) W W W (cid:1) with W an object of KL k , thereis a unique V k ( gl (1 | f : W ⊠ W → W such that f ◦ Y ⊠ = Y .For more details on vertex algebraic braided tensor supercategory structure, in partic-ular the unit and associativity isomorphisms which we will use briefly in Section 4.2, see[HLZ9] or the exposition in [CKM, Sec. 3.3]. In this section, we show that every irreducible V k ( gl (1 | O fink , beginning with the atypical irreducible modules.We will show below that for n ∈ C , the generalized Verma module b P kn is the projectivecover of b A kn, . Since induction is an exact functor between module categories for Liesuperalgebras (thanks to the Poincar´e-Birkhoff-Witt Theorem for superalgebras), (2.1)induces to the exact sequence0 → b V kn + , → b P kn → b V kn − , → n ∈ C . The Loewy diagram of b P kn is b A n, b P kn : b A n +1 , b A n − , . b A n, oreover, b P kn is a logarithmic module with L -block size .Proof. The Loewy diagram of b P kn follows from the exact sequences (2.6), (2.11) and theLoewy diagram of P n . To prove that b P kn is logarithmic, take a generator v n of the lowestconformal weight space ( b P kn ) [0] , which is the gl (1 | P n (see Sec. 2.1). Then fromthe expression (2.5) for L , we get L v n = − k ψ + ψ − v n = 0, while L v n = 0. So v n is ageneralized eigenvector for L with block size 2, and since v n generates b P kn as a V k ( gl (1 | b P kn is a logarithmic module with L -block size 2.To obtain the projective covers of the atypical irreducibles b A kn,ℓk , ℓ ∈ Z \ { } , we needthe spectral flow automorphisms σ ℓ of \ gl (1 | σ ℓ ( N r ) = N r , σ ℓ ( E r ) = E r − ℓ k δ r, , σ ℓ ( ψ ± r ) = ψ ± r ∓ ℓ , σ ℓ ( k ) = k . (2.12)For ℓ ∈ Z , σ ℓ extends to a vertex superalgebra automorphism of V k ( gl (1 | V k ( gl (1 | W , the spectral flow module σ ℓ ( W ) has the same underlying superspaceof W but has vertex operator Y σ ℓ ( W ) ( v, x ) = Y W ( σ − ℓ ( v ) , x )for v ∈ V k ( gl (1 | n ∈ C and ℓ ∈ − Z + , we define b P kn − ℓ − ,ℓk := σ ℓ ( b P kn ) . Since it was shown in [CRi2, Sec. 3.2] that σ ℓ ( b V kn, ) = b V kn − ℓ,ℓk , applying σ ℓ to (2.11) yieldsthe exact sequence 0 → b V kn − ℓ + ,ℓk → b P kn − ℓ − ,ℓk → b V kn − ℓ − ,ℓk → . (2.13)For ℓ ∈ Z + , we also need the conjugation automorphism of \ gl (1 |
1) defined by w ( N r ) = − N r , w ( E r ) = − E r , w ( ψ + r ) = ψ − r , w ( ψ − r ) = − ψ + r , w ( k ) = k . We then define b P k − n − ℓ + ,ℓk := σ ℓ ( w ( b P kn ))for ℓ ∈ Z + . Since σ ℓ ( w ( b V kn, )) = b V k − n − ℓ,ℓk (again see [CRi2, Sec. 3.2]), applying σ ℓ ◦ w to(2.11) yields the exact sequence0 → b V k − n − ℓ − ,ℓk → b P k − n − ℓ + ,ℓk → b V k − n − ℓ + ,ℓk → . (2.14)If we use the notation ε ( ℓ ) = if ℓ ∈ Z + ℓ = 0 − if ℓ ∈ − Z + , (2.15)15he exact sequences (2.13) and (2.14) can be written uniformly:0 → b V kn − ℓ − ε ( ℓ ) ,ℓk → b P kn − ℓ + ε ( ℓ ) ,ℓk → b V kn − ℓ + ε ( ℓ ) ,ℓk → n ∈ C , ℓ ∈ Z \ { } .We will sometimes use the notational convention b P kn, for b P kn . Now we can prove: For n ∈ C and ℓ ∈ Z , the modules b P kn − ℓ + ε ( ℓ ) ,ℓk are projective in O fink .Proof. Consider a diagram b P kn − ℓ + ε ( ℓ ) ,ℓkq (cid:15) (cid:15) A π / / B (2.17)in O fink where π is surjective and we may assume q = 0. We first consider the case ℓ = 0.Since there are no non-zero morphisms between the subcategories ( O fink ) e for different e ’s,the diagram restricts to a surjection π : A ։ B and to q : b P kn → B where A = ker A E and B = ker B E . Then the diagram further restricts to a diagram P nq [0] (cid:15) (cid:15) A π [0] / / B of gl (1 | π [0] surjective.Now since N also acts semisimply on A and B , A and B are objects of the category O . Since P n is projective in O , there is a gl (1 | e q [0] : P n → A such that q [0] = π [0] ◦ e q [0] . Then Lemma 2.2.3 implies that positive modes from \ gl (1 |
1) annihilate A ,so the universal property of generalized Verma \ gl (1 | e q [0] extends toa homomorphism e q : b P kn → A ֒ → A. Since P n generates b P kn , it follows that q = π ◦ e q , and we have shown that b P kn is projectivein O fink .Now for ℓ = 0, we apply σ − ℓ or w − ◦ σ − ℓ to the diagram (2.17) to get b P k ± nq (cid:15) (cid:15) e A π / / e B with π still surjective. Since the spectral flows e A and e B are still objects of O fink and since b P k ± n is projective in O fink , we get e q such that q = π ◦ e q . Applying σ ℓ or σ ℓ ◦ w then showsthat e q defines a homomorphism b P kn − ℓ + ε ( ℓ ) ,ℓk → A , so b P kn − ℓ + ε ( ℓ ) ,ℓk is projective in O fink .16ow we can prove that the projective modules b P kn − ℓ + ε ( ℓ ) ,ℓk are projective covers: For n ∈ C and ℓ ∈ Z , b P kn − ℓ + ε ( ℓ ) ,ℓk is a projective cover in O fink of theatypical irreducible module b A kn − ℓ + ε ( ℓ ) ,ℓk .Proof. The exact sequences (2.6), (2.7), (2.11), and (2.16) show that there is a surjectivemap π : b P kn − ℓ + ε ( ℓ ) ,ℓk → b A kn − ℓ + ε ( ℓ ) ,ℓk . Then if q : P → b A kn − ℓ + ε ( ℓ ) ,ℓk is any surjective morphismin O fink with P projective, there is a homomorphism e q : P → b P kn − ℓ + ε ( ℓ ) ,ℓk such that thediagram P q (cid:15) (cid:15) e q w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ b P kn − ℓ + ε ( ℓ ) ,ℓk π / / b A kn − ℓ + ε ( ℓ ) ,ℓk commutes. We need to show that e q is surjective.Applying σ − ℓ or w − ◦ σ − ℓ to the diagram reduces the surjectivity of e q to the ℓ = 0case. But if ℓ = 0, the Loewy diagram of b P kn from Proposition 2.4.1 shows that ker π isthe unique maximal proper submodule of b P kn . Since q = 0, the image of e q is not containedin ker π , and we conclude e q is surjective.We now turn to the typical irreducibles b V n,e for e/k / ∈ Z ; we will show that they areprojective and thus are their own projective covers in O fink . Since they are irreducible andsince every module in O fink has finite length, it is sufficient to show that when e/k / ∈ Z ,then Ext O fink ( b V kn,e , W ) = 0 (2.18)for any irreducible V k ( gl (1 | W . From the direct sum decomposition O fink , it isclear that an extension of b V kn,e by W must split unless perhaps W is an object of ( O fink ) e ;so we may assume W = b V n ′ ,e for some n ′ ∈ C . Furthermore, the commutation relationsfor N show that there is a direct sum decomposition( O fink ) e = M n ∈ C / Z ( O fink ) n,e where ( O fink ) n,e is the full subcategory consisting of modules in ( O fink ) e on which N actsby eigenvalues from the coset n . Thus we may assume W in (2.18) is isomorphic to V n ′ ,e with n ′ − n ∈ Z . We divide the proof of (2.18) into the cases n ′ = n and n ′ = n in the twofollowing lemmas: If e/k / ∈ Z and ( n, e ) = ( n ′ , e ′ ) , then Ext O fink ( b V n,e , b V n ′ ,e ′ ) = 0 .Proof. We need to prove that any exact sequence0 → b V kn ′ ,e ′ q −→ A π −→ b V kn,e → A is a module in O fink . As discussed above, we may assume e ′ = e , so that n ′ = n . Since A has a direct sum decomposition A = L h ∈ C / Z A h where A h = L h ∈ h A [ h ] ,(2.19) must split unless perhaps lowest conformal weights satisfy ∆ n,e − ∆ n ′ ,e ∈ Z . By(2.8), ∆ n,e − ∆ n ′ ,e = ek ( n − n ′ ) = 0 , so we may assume either ∆ n,e − ∆ n ′ ,e < n,e − ∆ n ′ ,e > n,e − ∆ n ′ ,e <
0, then the lowest conformal weight space of A is isomorphic to V n,e as a gl (1 | π restricts to an isomorphism on lowest conformal weightspaces. Thus the universal property of induced \ gl (1 | σ : b V kn,e → A splitting the sequence.If ∆ n,e − ∆ n ′ ,e >
0, then q restricts to a gl (1 | V n ′ ,e ′ ontothe lowest conformal weight space of A , and this restriction must be even or odd. Since V n ′ ,e ′ generates b V kn ′ ,e ′ , it follows that q is parity-homogeneous and Im q = ker π is Z -graded.Then π factors as A ։ A/ ker π ∼ = −→ b V kn,e where the first map is even and second is even or odd because A/ ker π is either b V kn,e orits parity-reversed version. Thus π is also parity-homogeneous, and we may apply thecontragredient functor to get an exact sequence0 → ( b V kn,e ) ′ π ′ −→ A ′ q ′ −→ ( b V kn ′ ,e ) ′ → . Since e/k / ∈ Z , ( b V kn,e ) ′ ∼ = b V k − n, − e and ( b V kn ′ ,e ) ′ ∼ = b V k − n ′ , − e by Proposition 2.2.4. Then the previ-ous case implies that the contragredient sequence splits, that is, we have σ ′ : ( b V kn ′ ,e ) ′ → A ′ such that q ′ ◦ σ ′ = id. Taking contragredients again and applying the natural isomorphism δ , we see that σ := δ − b V kn ′ ,e ◦ σ ′′ ◦ δ A : A → b V kn ′ ,e satisfies σ ◦ q = id. Thus im q is a direct summand of A isomorphic to b V kn ′ ,e , and there isan isomorphism b V kn,e ∼ = ker σ splitting (2.19). If e = 0 , then Ext O fink ( b V kn,e , b V kn,e ) = 0 Proof.
Assume that we have an exact sequence0 → b V kn,e → A → b V kn,e → . (2.20)where A is a module in O fink . Then there is an exact sequence of gl (1 | → V n,e → A [∆ n,e ] → V n,e → , (2.21)where A [∆ n,e ] is the lowest conformal weight space of A . Since A is a module in O fink , A [∆ n,e ] is a gl (1 | O . Then since V n,e is projective in O for e =18, (2.21) splits. By the universal property of generalized Verma modules, the splittinghomomorphism V n,e → A [∆ n,e ] then extends to a unique \ gl (1 | b V n,e → A that splits (2.20).Combining Lemmas 2.4.4 and 2.4.5 with the discussion preceding Lemma 2.4.4, weconclude: For n ∈ C and e/k / ∈ Z , the typical irreducible module b V n,e is projectivein O fink . Every irreducible V k ( gl (1 | -module has a projective cover in O fink . In this section, we compute tensor products of irreducible \ gl (1 | gl (1 |
1) case.
Here we collect some general results on determining fusion rules for affine vertex operatorsuperalgebras. For vertex operator algebras, the general theory is developed in [FZ1, Li]for the non-logarithmic case and [HY] for the logarithmic case.Let g be a finite-dimensional Lie superalgebra with corresponding affine Lie superalge-bra b g , and let V k ( g ) be the level- k affine vertex operator superalgebra. A V k ( g )-module W is N - gradable if it has an N -grading W = L i ∈ N W ( i ) compatible with the Z -gradingsuch that a r · W ( i ) ⊆ W ( i − r )for a ∈ g , r ∈ Z , and i ∈ N , where W ( i ) = 0 for i <
0. Note that an N -grading satisfyingthese properties need not be unique.Now suppose Y is an even or odd (possibly logarithmic) V k ( g )-module intertwiningoperator of type (cid:0) W W W (cid:1) . The intertwining operator Jacobi identity (see [CKM, Def. 3.7]for the proper sign factors in the superalgebra generality) implies the following commutatorand iterate formulas:( − |Y|| a | a r Y ( w , x ) − ( − | a || w | Y ( w , x ) a r = X i ≥ (cid:18) ri (cid:19) x r − i Y ( a i w , x ) (3.1)and Y ( a r w , x ) = ( − |Y|| a | X i ≥ (cid:18) ri (cid:19) ( − x ) i a r − i Y ( w , x ) − ( − | a || w | X i ≥ (cid:18) ri (cid:19) ( − x ) r − i Y ( w , x ) a i (3.2)19or homogeneous a ∈ g , w ∈ W , and r ∈ Z .Suppose W and W are generalized Verma modules c M k and c M k , respectively, andthat W is N -gradable such that each W ( i ) is the sum of finitely many generalized L -eigenspaces. This means that the substitution x Y is well defined (using the branchof logarithm log 1 = 0), and Y ( w , w is a vector in the algebraic completion Q i ∈ N W ( i ).Then the r = 0 case of the commutator formula (3.1) implies that Y induces a g -modulehomomorphism π ( Y ) : M ⊗ M → W (0)with parity |Y | defined by π ( Y )( m ⊗ m ) = π ( Y ( m , m )for m ∈ M , m ∈ M , where π denotes projection onto W (0) with respect to the N -grading of W . The following is essentially a special case of [TW, Prop. 24], which isattributed to Nahm [Na]: If Y is a surjective intertwining operator, then π ( Y ) is a surjective g -module homomorphism.Proof. Because Y is surjective, W (0) is spanned by vectors of the form π ( Y ( w , w ) for w ∈ c M k and w ∈ c M k . Thus we need to show that π ( Y ( w , w ) ∈ Span { π ( Y ( m , m ) | m ∈ M , m ∈ M } for all w ∈ c M k , w ∈ c M k . This is true by definition when w ∈ M and w ∈ M .Now assume that for some w ∈ c M k , we already know that π ( Y ( m , w ) ∈ Im π ( Y )for all m ∈ M . Then (3.1) implies that( − | a || m | π ( Y ( m , a − r w ) = ( − |Y|| a | π ( a − r Y ( m , w ) − X i ∈ N (cid:18) − ri (cid:19) π ( Y ( a i m , w )= − π ( Y ( a m , w ) ∈ Im π ( Y )for homogeneous a ∈ g , m ∈ M , and r ∈ Z + . Since c M k is generated by M under theaction of the modes a − r , this shows that π ( Y ( m , w ) ∈ Im π ( Y ) for all m ∈ M andall w ∈ c M k .Now assume that for some homogeneous w ∈ c M k , we know π ( Y ( w , w ) ∈ Im π ( Y )for all w ∈ c M k . Then (3.2) implies that π ( Y ( a − r w , w )= ( − |Y|| a | X i ≥ (cid:18) − ri (cid:19) π ( a − r − i Y ( w , w ) − ( − | a || w | X i ≥ (cid:18) − ri (cid:19) π ( Y ( w , a i w )= − ( − | a || w | X i ≥ (cid:18) − ri (cid:19) π ( Y ( w , a i w ) ∈ Im π ( Y )20or homogeneous a ∈ g r ∈ Z + , and w ∈ W . Since c M k is a generalized Verma module,this shows that π ( Y ( w , w ) ∈ Im π ( Y ) for all w ∈ c M k , w ∈ c M k , as required.For V k ( g )-modules c M k , c M k and W as above, it turns out that the even linear map π from intertwining operators of type (cid:0) W c M k c M k (cid:1) to Hom g ( M ⊗ M , W (0)) is an isomorphism if W is the contragredient ( c M k ) ′ of a generalized Verma module, in which case W (0) = M ∗ .In fact, the following theorem is the affine Lie superalgebra case of (the superalgebrageneralization of) [HY, Thm. 6.6], which generalizes [Li, Thm. 2.11] and [FZ2, Lem. 2.19]to logarithmic intertwining operators: Suppose M , M , and M are finite-dimensional g -modules and f : M ⊗ M → M ∗ is a g -module homomorphism. Then there is a unique intertwiningoperator Y of type (cid:0) ( c M k ) ′ c M k c M k (cid:1) such that π ( Y ) = f . To make the proof of [HY, Thm. 6.6] in the affine Lie superalgebra special case moreconcrete, we sketch a construction of Y from a homogeneous f . We first use (3.1) and(2.10) to define a sequence of linear maps f i : M ⊗ M → ( c M k ) ′ ( i ) = c M k ( i ) ∗ recursively. We start with f = f , and then assuming f , . . . , f i − have been defined, weset h f i ( m ⊗ m ) , a − r w i = − ( − | a | ( | m | + | m | ) h f i − r ( a m ⊗ m ) , w i for homogeneous m ∈ M , m ∈ M , a ∈ g , 1 ≤ r ≤ i , and w ∈ c M k ( i − r ). Once it isshown that the maps f i are well defined, we can then set Y ( m , x ) m = X i ∈ N x L f i ( x − L m ⊗ x − L m ) : M ⊗ M → ( c M k ) ′ [log x ] { x } . Next, we use the method of [MY] to extend Y first to M ⊗ c M k , and then to c M k ⊗ c M k .Both extensions are defined recursively: assuming we have already defined Y ( m , x ) w for m ∈ M , w ∈ c M k , we use the commutator formula (3.1) to define Y ( m , x ) a − r w = ( − | a || m | (cid:0) ( − |Y|| a | a − r Y ( w , x ) w − x − r Y ( a m , x ) w (cid:1) for homogeneous m ∈ M , a ∈ g , and r ∈ Z + . Finally, assuming we have defined Y ( w , x ) w for some homogeneous w ∈ c M k and all w ∈ c M k , we use (3.2) to define Y ( a − r w , x ) = ( − | f || a | X i ≥ (cid:18) − ri (cid:19) ( − x ) i a − r − i Y ( w , x ) − ( − | a || w | X i ≥ (cid:18) − ri (cid:19) ( − x ) − r − i Y ( w , x ) a i for homogeneous a ∈ g and r ∈ Z + . Once it is shown that these extensions are well defined,one can prove that Y is an intertwining operator, as in [MY, Thm. 6.2]. Note that Y hasthe same parity of | f | ; the uniqueness of Y follows because the construction of Y is forcedby the formulas (3.1), (3.2), and (2.9). 21 .2 Fusion rules in K L k and O f ink In this section, we compute all tensor products of irreducible V k ( gl (1 | b A k , ± k , we will need the following lemma onthe kernels of gl (1 | π ( Y ), where Y is an intertwining operator involving b A k , ± k . In the statement, we use v n,e to denote a highest-weight vector in the gl (1 | V n,e :
1. Suppose Y is an even intertwining operator of type (cid:0) W b A k ,k W (cid:1) where W and W are N -gradable V k ( gl (1 | -modules with W (0) = V n ′ ,e ′ for e ′ = 0 . Then v ,k ⊗ v n ′ ,e ′ ∈ ker π ( Y ) .2. Suppose Y is an even intertwining operator of type (cid:0) W W b A k , − k (cid:1) where W and W are N -gradable V k ( gl (1 | -modules with W (0) = V n,e for e = 0 . Then ψ − v n,e ⊗ ψ − v , − k ∈ ker π ( Y ) .Proof. We use explicit singular vectors in b V k , ± k : it is straightforward to show that ψ + − v ,k ∈ b V k ,k and ( kψ −− + E − ψ − ) v , − k ∈ b V k , − k vanish in the irreducible quotients b A k , ± k . Thus in the first case we use (3.2) to calculate0 = π (cid:0) Y ( ψ + − v ,k , ψ − v n ′ ,e ′ (cid:1) = π X i ≥ (cid:16) ψ + − − i Y ( v ,k , ψ − v n ′ ,e ′ + ( − | ψ + || v ,k | Y ( v ,k , ψ + i ψ − v n ′ ,e ′ (cid:17) = ( − | v ,k | π ( Y ( v ,k , E v n ′ ,e ′ ) = e ′ ( − | v ,k | π ( Y )( v ,k ⊗ v n ′ ,e ′ ) . Since e ′ = 0, this shows that v ,k ⊗ v n ′ ,e ′ ∈ ker π ( Y ). For the second case, we use (3.1) tocalculate0 = π (cid:0) Y ( ψ − v n,e , kψ −− + E − ψ − ) v , − k (cid:1) = k ( − | ψ − | ( | ψ − | + | v n,e | ) π (cid:18) ψ −− Y ( ψ − v n,e , v , − k − X i ≥ ( − i Y ( ψ − i ψ − v n,e , v , − k (cid:19) + ( − | E | ( | ψ − | + | v n,e | ) π (cid:18) E − Y ( ψ − v n,e , ψ − v , − k − X i ≥ ( − i Y ( E i ψ − v n,e , ψ − v , − k (cid:19) = − e π ( Y )( ψ − v n,e ⊗ ψ − v , − k ) . Since e = 0, ψ − v n,e ⊗ ψ − v , − k ∈ ker π ( Y ).First we compute the tensor products of atypical irreducible V k ( gl (1 | .2.2 Theorem. The atypical irreducible V k ( gl (1 | -modules are simple currents withfusion rules b A kn,ℓk ⊠ b A kn ′ ,ℓ ′ k ∼ = b A kn + n ′ − ε ( ℓ,ℓ ′ ) , ( ℓ + ℓ ′ ) k for n, n ′ ∈ C , ℓ, ℓ ′ ∈ Z , where ε ( ℓ, ℓ ′ ) = ε ( ℓ ) + ε ( ℓ ′ ) − ε ( ℓ + ℓ ′ ) .Proof. We will prove two special cases of the fusion rules: the ℓ = 0 case b A kn, ⊠ b A kn ′ ,ℓ ′ k ∼ = b A kn + n ′ ,ℓ ′ k , (3.3)and also b A k − ℓ + ε ( ℓ ) ,ℓk ⊠ b A k − ℓ ′ + ε ( ℓ ′ ) ,ℓ ′ k ∼ = b A k − ℓ + ℓ ′ + ε ( ℓ + ℓ ′ ) , ( ℓ + ℓ ′ ) k (3.4)for ℓ, ℓ ′ ∈ Z . The general formula then follows from these by associativity and commuta-tivity of tensor products: b A kn,ℓk ⊠ b A kn ′ ,ℓ ′ k ∼ = ( b A kn + ℓ − ε ( ℓ ) , ⊠ b A k − ℓ + ε ( ℓ ) ,ℓk ) ⊠ ( b A kn ′ + ℓ ′ − ε ( ℓ ′ ) , ⊠ b A k − ℓ ′ + ε ( ℓ ′ ) ,ℓ ′ k ) ∼ = ( b A kn + ℓ − ε ( ℓ ) , ⊠ b A kn ′ + ℓ ′ − ε ( ℓ ′ ) , ) ⊠ ( b A k − ℓ + ε ( ℓ ) ,ℓk ⊠ b A k − ℓ ′ + ε ( ℓ ′ ) ,ℓ ′ k ) ∼ = b A kn + n ′ + ℓ + ℓ ′ − ε ( ℓ ) − ε ( ℓ ′ ) , ⊠ b A k − ℓ + ℓ ′ + ε ( ℓ + ℓ ′ ) , ( ℓ + ℓ ′ ) k ∼ = b A kn + n ′ − ε ( ℓ + ℓ ′ ) , ( ℓ + ℓ ′ ) k for n, n ′ ∈ C and ℓ, ℓ ∈ Z .To prove (3.3), we first take ℓ ′ = 0. Then Proposition 3.1.1 applied to the surjectivetensor product intertwining operator yields a (non-zero) surjective gl (1 | A n ⊗ A n ′ ∼ = A n + n ′ ։ ( b A kn, ⊠ b A kn ′ , )(0) = ( b A kn, ⊠ b A kn ′ , ) [0] (recall also Lemma 2.2.3 and Remark 2.3.3). The universal property of induced \ gl (1 | b A kn + n ′ , → b A kn, ⊠ b A kn ′ , which is injectivebecause b A kn + n ′ , is simple. It is also surjective because Lemma 2.2.3 says that b A kn, ⊠ b A kn ′ , is generated by its weight-0 subspace, so we have proved the ℓ ′ = 0 case of (3.3). This alsoshows that b A kn, is a simple current for n ∈ C , with tensor inverse b A k − n, .For ℓ ′ = 0, we now know that b A kn, ⊠ b A kn ′ ,ℓ ′ k is simple because b A kn, is a simple current.Since Proposition 3.1.1 gives a surjective gl (1 | A n ⊗ V n ′ ,ℓ ′ k ∼ = V n + n ′ ,ℓ ′ k ։ ( b A kn, ⊠ b A kn ′ ,ℓ ′ k )(0) , it follows that b A kn, ⊠ b A kn ′ ,ℓ ′ k is the unique irreducible quotient b A kn + n ′ ,ℓ ′ k of b V kn + n ′ ,ℓ ′ k . Thisfinishes the proof of (3.3).For (3.4), we first take ℓ = 1 and ℓ ′ = −
1. In this case the tensor product intertwiningoperator Y of type (cid:0) b A k ,k ⊠ b A k , − k b A k ,k b A k , − k (cid:1) induces a surjective gl (1 | π ( Y ) : V ,k ⊗ V , − k ∼ = P ։ ( b A k ,k ⊠ b A k , − k )(0) = ( b A k ,k ⊠ b A k , − k ) [0] b P k → b A k ,k ⊠ b A k , − k which is surjective by Lemma 2.2.3. Itwill induce the required isomorphism b A k , ∼ = b A k ,k ⊠ b A k , − k if ker Π is the (unique) maximalproper submodule of b P k . To prove this, we just need to show that ker π ( Y ) contains theunique maximal proper submodule of P . In fact, Lemma 3.2.1 says that v ,k ⊗ v , − k , ψ − v ,k ⊗ ψ − v , − k ∈ ker π ( Y ) , and it is easy to see that these two vectors generate the maximal proper submodule of V ,k ⊗ V , − k ∼ = P . This proves the ℓ = 1, ℓ ′ = − b A k , ± k are mutually inverse simple currents.Now since b A k ,k generates a group of simple currents, (3.4) for general ℓ, ℓ ′ ∈ Z willfollow if we can show that ( b A k ,k ) ⊠ ℓ ∼ = b A k − ℓ + ε ( ℓ ) ,ℓk for ℓ ∈ Z . Since this relationship holds for ℓ = − , ,
1, it will hold in general by inductionon ℓ if we can show that b A k ,k ⊠ b A k − ℓ + ,ℓk ∼ = b A k − ℓ , ( ℓ +1) k and b A k ℓ − , − ℓk ⊠ b A k , − k ∼ = b A k ℓ , − ( ℓ +1) k (3.5)for ℓ ∈ Z + .To prove (3.5), we first note that both tensor product modules are simple because b A k , ± k are simple currents. Then Proposition 3.1.1 again shows that the minimal conformalweight spaces of the tensor product modules are gl (1 | V ,k ⊗ V − ℓ + ,ℓk ∼ = V − ℓ +1 , ( ℓ +1) k ⊕ V − ℓ , ( ℓ +1) k and V ℓ − , − ℓk ⊗ V , − k ∼ = V ℓ , − ( ℓ +1) k ⊕ V ℓ − , − ( ℓ +1) k , respectively. Thus we just need to show that the kernels of the respective gl (1 | π ( Y ) agree with V − ℓ +1 , ( ℓ +1) k and V ℓ − , − ( ℓ +1) k . For the first case, Lemma 3.2.1(1) showsthat v ,k ⊗ v − ℓ + ,ℓk ∈ ker π ( Y ); this is a highest-weight vector generating V − ℓ +1 , ( ℓ +1) k ,so the first case of (3.5) is proved. For the second case, Lemma 3.2.1(2) shows that ψ − v ℓ − , − ℓk ⊗ ψ − v , − k ∈ ker π ( Y ). As this is a lowest-weight vector generating V ℓ − , − ( ℓ +1) k ,the second case of (3.5) is proved. This completes the proof of the theorem.Next, we compute the tensor products of atypical with typical irreducible V k ( gl (1 | For n, n ′ ∈ C , ℓ ∈ Z , and e ′ /k / ∈ Z , b A kn,ℓk ⊠ b V kn ′ ,e ′ ∼ = b V kn + n ′ − ε ( ℓ ) ,e ′ + ℓk . roof. We will prove the special cases b A kn, ⊠ b V kn ′ ,e ′ ∼ = b V kn + n ′ ,e ′ (3.6)and b A k , ± k ⊠ b V kn ′ ,e ′ ∼ = b V kn ′ ∓ ,e ′ ± k . (3.7)The general case then follows from these fusion rules together with associativity of thetensor product and the fusion rules for atypical modules from Theorem 3.2.2: b A kn,ℓk ⊠ b V kn ′ ,e ′ ∼ = ( b A k − ℓ + ε ( ℓ ) ,ℓk ⊠ b A kn + ℓ − ε ( ℓ ) , ) ⊠ b V kn ′ ,e ′ ∼ = ( b A k , ± k ) ⊠ | ℓ | ⊠ b V kn + n ′ + ℓ − ε ( ℓ ) ,e ′ ∼ = b V kn + n ′ + ℓ − ε ( ℓ ) ∓ | ℓ | ,e ′ ±| ℓ | k = b V kn + n ′ − ε ( ℓ ) ,e ′ + ℓk for all n, n ′ ∈ C , ℓ ∈ Z , and e ′ /k / ∈ Z .For (3.6), Proposition 3.1.1 yields a surjective gl (1 | A n ⊗ V n ′ ,e ′ ∼ = V n + n ′ ,e ′ ։ ( b A kn, ⊠ b V kn ′ ,e ′ )(0) , which lifts to a non-zero map Π : b V kn + n ′ ,e ′ → b A kn, ⊠ b V kn ′ ,e ′ by the universal property ofinduced \ gl (1 | e ′ /k / ∈ Z and b A kn, is a simple current, both domain andcodomain of Π are simple; therefore Π is an isomorphism.For (3.7), Proposition 3.1.1 yields surjective gl (1 | π ( Y + ) : V ,k ⊗ V n ′ ,e ′ ∼ = V n ′ + ,e ′ + k ⊕ V n ′ − ,e ′ + k ։ ( b A k ,k ⊠ b V kn ′ ,e ′ )(0) π ( Y − ) : V n ′ ,e ′ ⊗ V , − k ∼ = V n ′ + ,e ′ − k ⊕ V n ′ − ,e ′ − k ։ ( b V kn ′ ,e ′ ⊠ b A k , − k )(0) . The modules b A k , ± k ⊠ b V kn ′ ,e ′ are again irreducible, so it is enough to determine the kernelsof π ( Y ± ). Since e ′ = 0, Lemma 3.2.1(1) shows that v ,k ⊗ v n ′ ,e ′ ∈ ker π ( Y + ), and thusker π ( Y + ) = V n ′ + ,e ′ + k , while Lemma 3.2.1(2) shows that ψ − v n ′ ,e ′ ⊗ ψ − v , − k ∈ ker π ( Y − ),and thus ker π ( Y − ) = V n ′ − ,e ′ − k . This completes the proof of (3.7) and of the theorem.Finally, we compute products of typical irreducible V k ( gl (1 | For n, n ′ ∈ C , b V kn,e ⊠ b V kn ′ ,e ′ ∼ = b V kn + n ′ + ,e + e ′ ⊕ b V kn + n ′ − ,e + e ′ if ( e + e ′ ) /k / ∈ Z b P kn + n ′ if e + e ′ = 0 b P kn + n ′ + ε (( e + e ′ ) /k ) ,e + e ′ if ( e + e ′ ) /k ∈ Z \ { } and e ′ /k / ∈ Z . roof. It is straightforward to compute that, as gl (1 | V n,e ⊗ V n ′ ,e ′ ∼ = (cid:26) V n + n ′ + ,e + e ′ ⊕ V n + n ′ − ,e + e ′ if e + e ′ = 0 P n + n ′ if e + e ′ = 0 . In particular, V n,e ⊗ V n ′ ,e ′ ∼ = b P k (0), where b P k is the generalized Verma module b P k = ( b V kn + n ′ + ,e + e ′ ⊕ b V kn + n ′ − ,e + e ′ if e + e ′ = 0 b P kn + n ′ if e + e ′ = 0 . When ( e + e ′ ) /k / ∈ Z \ { } , therefore, b P k is projective in O fink (by the theorems of Section2.4) and b P k is the contragredient of a generalized Verma module (by Proposition 2.2.4).This means we can apply Theorem 3.1.2 and the universal property of tensor products toget a homomorphism Π : b V kn,e ⊠ b V kn ′ ,e ′ → b P k . The map Π is surjective because b P k is generated by b P k (0), so because b P k is projective,it occurs as a direct summand of b V kn,e ⊠ b V kn ′ ,e ′ . If W is a submodule complement of b P k in b V kn,e ⊠ b V kn ′ ,e ′ , then the tensor product module has an N -grading such that( b V kn,e ⊠ b V kn ′ ,e ′ )(0) = W (0) ⊕ b P k (0) . However, Proposition 3.1.1 says that ( b V kn,e ⊠ b V kn ′ ,e ′ )(0) is a homomorphic image of b P k (0)(for any allowable choice of N -grading on the tensor product). Therefore W (0) = 0 for anypossible N -grading of W , and we conclude W = 0. This proves the ( e + e ′ ) /k / ∈ Z \ { } cases of the theorem.Now when ( e + e ′ ) /k = ℓ ∈ Z \ { } and e ′ /k / ∈ Z , we use Theorem 3.2.3 and the e + e ′ = 0 case of the present theorem to calculate b V kn,e ⊠ b V kn ′ ,e ′ ∼ = ( b A kε (( e + e ′ ) /k ) ,e + e ′ ⊠ b V kn, − e ′ ) ⊠ b V kn ′ ,e ′ ∼ = b A kε ( ℓ ) ,ℓk ⊠ ( b V kn, − e ′ ⊠ b V kn ′ ,e ′ ) ∼ = b A kε ( ℓ ) ,ℓk ⊠ b P kn + n ′ . Thus it is sufficient to show that b A kε ( ℓ ) ,ℓk ⊠ b P kn ∼ = b P kn + ε ( ℓ ) ,ℓk (3.8)for any ℓ ∈ Z \ { } , n ∈ C . Since b A kε ( ℓ ) ,ℓk is a necessarily rigid simple current, since b P kn isprojective in O fink , and since tensor products of rigid with projective objects are projective(see for example [KL5, Cor. 2, Appendix]), b A kε ( ℓ ) ,ℓk ⊠ b P kn is projective in O fink . We alsohave a surjection b A kε ( ℓ ) ,ℓk ⊠ b P kn id ⊠ q −−→ b A kε ( ℓ ) ,ℓk ⊠ b A kn, ∼ = −→ b A kn + ε ( ℓ ) ,ℓk , where q is a surjection from b P kn onto b A kn, . Since b P kn + ε ( ℓ ) ,ℓk is a projective cover of b A kn + ε ( ℓ ) ,ℓk ,it follows that b P kn + ε ( ℓ ) ,ℓk is a direct summand of b A kε ( ℓ ) ,ℓk ⊠ b P kn . Then because b A kε ( ℓ ) ,ℓk is asimple current, b A kε ( ℓ ) ,ℓk ⊠ b P kn has length 4 (see for example [CKLR, Prop. 2.5], as does b P kn + ε ( ℓ ) ,ℓk , and (3.8) follows. 26e can also compute tensor products involving projective modules using the fusionrules for simple modules together with associativity of tensor products; we record themhere: Tensor products involving projective modules in O fink are as follows:1. For n, n ′ ∈ C and ℓ, ℓ ′ ∈ Z , b A kn,ℓk ⊠ b P kn ′ ,ℓ ′ k ∼ = b P kn + n ′ − ε ( ℓ,ℓ ′ ) , ( ℓ + ℓ ′ ) k .
2. For n, n ′ ∈ C , e/k / ∈ Z , and ℓ ∈ Z , b V kn,e ⊠ b P kn ′ ,ℓ ′ k = b V kn + n ′ +1 − ε ( ℓ ′ ) ,e + ℓ ′ k ⊕ · b V kn + n ′ − ε ( ℓ ′ ) ,e + ℓ ′ k ⊕ b V kn + n ′ − − ε ( ℓ ′ ) ,e + ℓ ′ k .
3. For n, n ′ ∈ C and ℓ, ℓ ′ ∈ Z , b P kn,ℓk ⊠ b P kn ′ ,ℓ ′ k = b P kn + n ′ +1 − ε ( ℓ,ℓ ′ ) , ( ℓ + ℓ ′ ) k ⊕ · b P kn + n ′ − ε ( ℓ,ℓ ′ ) , ( ℓ + ℓ ′ ) k ⊕ b P kn + n ′ − − ε ( ℓ,ℓ ′ ) , ( ℓ + ℓ ′ ) k . In this section, we prove that the tensor supercategories KL k and O fink are rigid. Thestrategy is to first prove that all simple V k ( gl (1 | In order to prove that the typical irreducible V k ( gl (1 | b V kn,e are rigid, we willneed explicit formulas for four-point correlation functions involving highest-weight vectorsin b V kn,e and its contragredient. We will obtain these correlation functions as solutions toKnizhnik-Zamolodchikov (KZ) equations.Recall from Proposition 2.2.4 that ( b V kn,e ) ′ ∼ = b V k − n, − e , but by an odd isomorphism. Thusit is better to take ( b V kn,e ) ′ to be the parity-reversed module Π( b V k − n, − e ); that is, we takea highest-weight vector v − n, − e ∈ ( b V kn,e ) ′ to be odd. This way, intertwining operators ofinterest involving b V kn,e and its contragredient, in particular the ones induced by the homo-morphisms b V kn,e ⊠ Π( b V k − n, − e ) ∼ = −→ b P k and b V kn,e ⊠ Π( b V k − n, − e ) ∼ = −→ b P k ։ b A k , , will be even. For simplicity of notation, in what follows we will use V to denote b V kn,e and V ′ to denote Π( b V k − n, − e ). 27ow let W be some V k ( gl (1 | Y an even intertwining operator of type (cid:0) VV W (cid:1) ,and Y an even intertwining operator of type (cid:0) WV ′ V (cid:1) . Then we define the multivaluedanalytic functionsΦ( z , z )( v , v , v , v ) := h v , Y ( v , z ) Y ( v , z ) v i , | z | > | z | > v , v ∈ V ′ and v , v ∈ V are vectors of (minimal) conformalweight ∆ n,e . We also assume each v i is an N -eigenvector with eigenvalue n i ; so n , n ∈{ n ± } and n , n ∈ {− n ± } . It is then straightforward to use the expression (2.4) for L − , the L − -derivative for intertwining operators, the commutator formula (3.1), and theiterate formula (3.2) to derive the following partial differential equations (for examples ofdetailed calculations of this type see for example [HL, McR, CMY2]): (Knizhnik-Zamolodchikov equations) . The functions Φ satisfy the fol-lowing partial differential equations: ∂ z Φ( z , z )( v , v , v , v )= h − ek (cid:16) n − n + ek (cid:17) ( z − z ) − + ek (cid:16) n + n + ek (cid:17) z − i Φ( z , z )( v , v , v , v )+ ( − | v | k ( z − z ) − (cid:2) Φ( z , z )( v , ψ − v , ψ + v , v ) − Φ( z , z )( v , ψ + v , ψ − v , v ) (cid:3) + ( − | v | + | v | k z − (cid:2) Φ( z , z )( v , ψ − v , v , ψ + v ) − Φ( z , z )( v , ψ + v , v , ψ − v ) (cid:3) (4.1) ∂ z Φ( z , z )( v , v , v , v )= h ek (cid:16) n − n + ek (cid:17) ( z − z ) − + ek (cid:16) n − n − ek (cid:17) z − i Φ( z , z )( v , v , v , v )+ ( − | v | k ( z − z ) − (cid:2) Φ( z , z )( v , ψ + v , ψ − v , v ) − Φ( z , z )( v , ψ − v , ψ + v , v ) (cid:3) + ( − | v | k z − (cid:2) Φ( z , z )( v , v , ψ − v , ψ + v ) − Φ( z , z )( v , v , ψ + v , ψ − v ) (cid:3) , (4.2) For simplicity, we will sometimes drop the dependence on v , v , v , and v from the notation for Φ.To solve the KZ equations, we reduce Φ to a one-variable function using the L -conjugation formula for intertwining operators. In fact, since the lowest conformal weightof both V and V ′ is ∆ n,e , L -conjugation implies thatΦ( z , z ) = z − n,e φ ( z /z ) (4.3)where φ ( z ) := Φ(1 , z ) = h v , Y ( v , Y ( v , z ) v i .
28t is convenient to view φ ( z ) as a single-valued analytic function on the simply-connecteddomain U = { z ∈ C | | z | < } \ ( − , z = log | z | + i arg z , where − π < arg z < π .We will need some relations between the functions φ for varying v , v , and v : For any lowest-conformal-weight vector v ∈ V ′ , the following rela-tions hold: φ ( z )( v , v n,e , v − n, − e , v n,e ) = 0 , (4.4) φ ( z )( v , ψ − v n,e , v − n, − e , v n,e ) + φ ( z )( v , v n,e , ψ − v − n, − e , v n,e )= φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ) . (4.5) Proof.
For v = v = v n,e and v = v − n, − e , the KZ equations (4.1) and (4.2) become ∂ z Φ( z , z ) = h − ek (cid:16) n + ek (cid:17) ( z − z ) − + ek (cid:16) n + 1 + ek (cid:17) z − i Φ( z , z ) ∂ z Φ( z , z ) = ek (cid:16) n + ek (cid:17) (cid:2) ( z − z ) − − z − (cid:3) Φ( z , z ) . From (4.3), we also get the relations ∂ z Φ( z , z ) = − n,e z − n,e − φ ( z /z ) − z − n,e − z φ ′ ( z /z ) ,∂ z Φ( z , z ) = z − n,e − φ ′ ( z /z ) . Thus setting z = 1, z = z and using the definition (2.8) of ∆ n,e , we get − n,e φ ( z ) − zφ ′ ( z ) = (cid:16) − n,e (1 − z ) − + 2∆ n,e + ek (cid:17) φ ( z ) ,φ ′ ( z ) = 2∆ n,e (cid:0) (1 − z ) − − z − (cid:1) φ ( z ) . These two equations simplify to ek φ ( z ) = 0; since e = 0, this means φ ( z ) = 0, proving(4.4).For (4.5), we use (4.4), the contragredient formula (2.10), and the r = 0 case of thecommutator formula (3.1) to get0 = φ ( z )( ψ − v , v n,e , v − n, − e , v n,e )= − ( − | v | φ ( z )( v , ψ − v n,e , v − n, − e , v n,e ) − ( − | v | + | v n,e | φ ( z )( v , v n,e , ψ − v − n, − e , v n,e ) − ( − | v | + | v n,e | + | v − n, − e | φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ) . Since | v n,e | = 0 and | v − n, − e | = 1, (4.5) follows.29ow we derive a second-order differential equation for φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ). Webegin with two cases of the KZ equation (4.2) specialized to z = 1, z = z : φ ′ ( z )( v , ψ − v n,e , v − n, − e , v n,e )= h ek (cid:16) n − ek (cid:17) (1 − z ) − − ek (cid:16) n − ek (cid:17) z − i φ ( z )( v , ψ − v n,e , v − n, − e , v n,e ) − ek (1 − z ) − φ ( z )( v , v n,e , ψ − v − n, − e , v n,e )= 2∆ n,e (cid:2) (1 − z ) − − z − (cid:3) φ ( z )( v , ψ − v n,e , v − n, − e , v n,e ) − ek (1 − z ) − φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ) , (4.6)where the second equality uses (2.8) and (4.5), and φ ′ ( z )( v , v n,e , v − n, − e , ψ − v n,e )= h ek (cid:16) n + ek (cid:17) (1 − z ) − + ek (cid:16) n − − ek (cid:17) z − i φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ) − ek z − φ ( z )( v , v n,e , ψ − v − n, − e , v n,e )= 2∆ n,e (cid:2) (1 − z ) − − z − (cid:3) φ ( z )( v , v n,e , v − n, − e , ψ − v n,e )+ ek z − φ ( z )( v , ψ − v n,e , v − n, − e , v n,e ) . (4.7)We can solve (4.7) for φ ( z )( v , ψ − v n,e , v − n, − e , v n,e ) in terms of φ ( z )( v , v n,e , v − n, − e , ψ − v n,e )and its derivative and then plug into (4.6). The result is the following differential equationfor φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ): For any lowest-conformal-weight vector v ∈ V ′ , the analytic function φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ) is a solution to the differential equation z (1 − z ) φ ′′ ( z ) + (cid:2) (4∆ n,e + 1) − (8∆ n,e + 1) z (cid:3) φ ′ ( z ) + 4∆ n,e z − φ ( z )+ 2∆ n,e (2∆ n,e − − z ) − φ ( z ) + (cid:20)(cid:16) ek (cid:17) − n,e (cid:21) φ ( z ) = 0 (4.8) in the region U . Note that φ ( z )( v , v n,e , v − n, − e , ψ − v n,e ), as a series in z , is the series expansion of asolution to (4.8) about the regular singular point 0. Since (4.8) is a second-order differ-ential equation with regular singular points at 0, 1, and ∞ , it can be transformed into ahypergeometric differential equation. Indeed, if f ( z ) = z n,e (1 − z ) n,e φ ( z )where φ ( z ) is a solution to (4.8), then f ( z ) satisfies the hypergeometric equation z (1 − z ) f ′′ ( z ) + (1 − z ) f ′ ( z ) + (cid:16) ek (cid:17) f ( z ) = 0 . (4.9)30he solutions of (4.9) are well known (see for example [AS, Chap. 15] or [DLMF, Sec.15.10]), and it follows that two linearly independent solutions of the original equation(4.8) on U are φ (1) ( z ) = z − n,e (1 − z ) − n,e F (cid:16) ek , − ek ; 1; z (cid:17) φ (2) ( z ) = z − n,e (1 − z ) − n,e (cid:16) F (cid:16) ek , − ek ; 1; z (cid:17) log z + G ( z ) (cid:17) , where G ( z ) is a power series that converges in the region U (by adding a multiple of φ (1) ( z )if necessary, we may assume G ( z ) has no constant term, but this is not important for us).We also need some analytic properties of iterates of intertwining operators involving V and V ′ , but we will not need explicit formulas. Let M be some V k ( gl (1 | Y an even intertwining operator of type (cid:0) MV V ′ (cid:1) , and Y an even intertwining operator of type (cid:0) VM V (cid:1) . The seriesΨ( z , z )( v , v , v , v ) := h v , Y ( Y ( v , z ) v , z ) v i , | z | > | z | > z and z converges to a multivalued analytic function in the indicated region. We thendefine the single-variable function ψ ( z ) := Ψ(1 − z, z ) = h v , Y ( Y ( v , − z ) v , z ) v i . Using the same branch of logarithm for log z and log(1 − z ) as we used for φ ( z ), ψ ( z )defines a single-valued analytic function on the simply-connected region e U = { z ∈ C | | z | > | − z | > } \ [1 , ∞ ) = (cid:26) z ∈ C (cid:12)(cid:12) Re z > (cid:27) \ [1 , ∞ ) . Also, by the L -conjugation property, ψ ( z ) = z − n,e (cid:28) v , Y (cid:18) Y (cid:18) v , − zz (cid:19) v , (cid:19) v (cid:29) = (cid:18) − zz (cid:19) n,e (cid:28) v , Y (cid:18) Y (cid:18) v , − zz (cid:19) v , (cid:19) v (cid:29) (4.10)for lowest-conformal-weight vectors v , v ∈ V ′ and v , v ∈ V . Thus the iterate of inter-twining operators gives the expansion of the analytic function ψ ( z ) as a series in − zz onthe region e U . Using the associativity of intertwining operators from [HLZ7], ψ ( z ) is theanalytic continuation to the region e U of a corresponding product of intertwining operators φ ( z ). Thus the functions ψ ( z ) satisfy the same differential equations as the functions φ ( z ).31 .2 Rigidity for K L k and O f ink In this section, we will prove that KL k and O fink are rigid, beginning with the simpleobjects. To simplify the proof, we first discuss some results on rigidity in general tensor(super)categories.Recall that a (left) dual of an object W in a tensor (super)category with unit object is an object W ′ equipped with an even evaluation morphism e W : W ′ ⊠ W → and evencoevaluation i W : → W ⊠ W ′ such that the compositions W l − W −−→ ⊠ W i W ⊠ id W −−−−−→ ( W ⊠ W ′ ) ⊠ W A − W,W ′ ,W −−−−−→ W ⊠ ( W ′ ⊠ W ) id W ⊠ e W −−−−−→ W ⊠ r W −−→ W and W ′ r − W ′ −−→ W ′ ⊠ id W ′ ⊠ i W −−−−−→ W ′ ⊠ ( W ⊠ W ′ ) A W ′ ,W,W ′ −−−−−−→ ( W ′ ⊠ W ) ⊠ W ′ e W ⊠ id W ′ −−−−−→ ⊠ W ′ l W ′ −−→ W ′ are identities. The object W is rigid if it also has a right dual, defined analogously; fortensor supercategories of modules for a vertex operator superalgebra, left duals are alsoright duals due to braiding and ribbon structure. In the following we will denote the abovetwo rigidity compositions by R W and R W ′ , respectively.In tensor supercategories of modules for a self-contragredient vertex operator superal-gebra, the contragredient W ′ of a module W satisfies the following universal propertydue to symmetries of intertwining operators [HLZ3, Xu]: there is an even evaluation e W : W ′ ⊠ W → (where the unit object is the superalgebra itself) such that forany morphism f : X ⊠ W → , there is a unique ϕ : X → W ′ such that the diagram X ⊠ W ϕ ⊠ id W (cid:15) (cid:15) f $ $ ■■■■■■■■■■ W ′ ⊠ W e W / / commutes. In a general tensor supercategory, we say that a pair ( W ′ , e W ) is a contragredient of W if it satisfies this universal property. If two objects W and X have contragredients,then a morphism f : W → X has a contragredient f ′ : X ′ → W ′ defined to be the uniquemorphism such that the diagram X ′ ⊠ W id X ′ ⊠ f / / f ′ ⊠ id W (cid:15) (cid:15) X ′ ⊠ X e X (cid:15) (cid:15) W ′ ⊠ W e W / / commutes. Let W be an object of a tensor supercategory with contragredient ( W ′ , e W ) ,and let i W : → W ⊠ W ′ be a morphism. Then the rigidity compositions with respect to e W and i W satisfy R W ′ = R ′ W . roof. We need to show that e W ◦ ( R W ′ ⊠ id W ) = e W ◦ (id W ′ ⊠ R W ). The left side is thecomposition W ′ ⊠ W r − W ′ ⊠ id W −−−−−→ ( W ′ ⊠ ) ⊠ W (id W ′ ⊠ i W ) ⊠ id W −−−−−−−−−→ ( W ′ ⊠ ( W ⊠ W ′ )) ⊠ W A W ′ ,W,W ′ ⊠ id W −−−−−−−−−→ (( W ′ ⊠ W ) ⊠ W ′ ) ⊠ W ( e W ⊠ id W ′ ) ⊠ id W −−−−−−−−−→ ( ⊠ W ′ ) ⊠ W l W ′ ⊠ id W −−−−−→ W ′ ⊠ W e W −−→ . By properties of the unit isomorphisms and naturality of associativity isomorphisms, thisbecomes W ′ ⊠ W id W ′ ⊠ l − W −−−−−→ W ′ ⊠ ( ⊠ W ) id W ′ ⊠ ( i 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 A − W ′ ⊠ W,W ′ ,W −−−−−−−−→ ( W ′ ⊠ W ) ⊠ ( W ′ ⊠ W ) e W ⊠ id W ′ ⊠ W −−−−−−−→ ⊠ ( W ′ ⊠ W ) l W ′ ⊠ W −−−−→ W ′ ⊠ W e W −−→ . We then apply the pentagon axiom to the associativity isomorphisms and naturality ofunit and associativity isomorphisms to the second e W to obtain W ′ ⊠ W id W ′ ⊠ l − W −−−−−→ W ′ ⊠ ( ⊠ W ) id W ′ ⊠ ( i W ⊠ id W ) −−−−−−−−−→ W ′ ⊠ (( W ⊠ W ′ ) ⊠ W ) id W ′ ⊠ A − W,W ′ ,W −−−−−−−−−→ W ′ ⊠ ( W ⊠ ( W ′ ⊠ W ) id W ′ ⊠ (id W ⊠ e W ) −−−−−−−−−→ W ′ ⊠ ( W ⊠ ) A W ′ ,W, −−−−−→ ( W ′ ⊠ W ) ⊠ e W ⊠ id −−−−→ ⊠ l −→ . But since l = r , the last three arrows become r ◦ ( e W ⊠ id ) ◦ A W ′ ,W, = e W ◦ r W ′ ⊠ W ◦ A W ′ ,W, = e W ◦ (id W ′ ⊠ r W ) , so the entire composition is e W ◦ (id W ′ ◦ R W ). Let W be a simple object of a tensor supercategory with contragredient ( W ′ , e W ) , and let i W : → W ⊠ W ′ be an even morphism. If the rigidity composition R W with respect to e W and i W is non-zero, then W is left rigid.Proof. Since W is simple and R W is even and non-zero, we have R W = c · id W for somenon-zero scalar c . Then by the lemma, R W ′ = R ′ W = ( c · id W ) ′ = c · id W ′ , so ( W ′ , e W , c − · i W ) is a left dual of W .Now we can prove that the simple objects of KL k and O fink are rigid. For the atypicalsimple modules b A kn,ℓk , ℓ ∈ Z , this is easy: By Proposition 2.2.4, ( b A kn,ℓk ) ′ ∼ = b A k − n, − ℓk , and33heorem 3.2.2 shows that b A kn,ℓk ⊠ b A k − n, − ℓk ∼ = b A k , ∼ = V k ( gl (1 | b A kn,ℓk is rigid. (Actually, for ℓ = 0, we should take ( b A kn,ℓk ) ′ to be the parity-reversed moduleΠ( b A k − n, − ℓk ) to ensure that evaluation and coevaluation are both even.)Now for the typical irreducible modules, Proposition 2.2.4 shows that we may take( b V kn,e ) ′ to be Π( b V k − n, − e ). We first fix a choice of evaluation and coevaluation. For theevaluation, let E denote the intertwining operator of type (cid:0) V k ( gl (1 | b V k − n, − e ) b V kn,e (cid:1) induced by thenondegenerate bilinear form h· , ·i : Π( b V k − n, − e ) × b V kn,e → C such that h ψ − v − n, − e , v n,e i = h v − n, − e , ψ − v n,e i = 1. In particular, for lowest-conformal-weight vectors v ′ ∈ Π( V − n, − e ), v ∈ V n,e , we have E ( v ′ , x ) v ∈ x − n,e (cid:0) h v ′ , v i + x V k ( gl (1 | x ]] (cid:1) . We then define the evaluation ε : Π( b V k − n, − e ) ⊠ b V kn,e → V k ( gl (1 | ε ◦ Y ⊠ = E , where Y ⊠ denotes the canonical even tensor productintertwining operator of type (cid:0) Π( b V k − n, − e ) ⊠ b V kn,e Π( b V k − n, − e ) b V kn,e (cid:1) .For the coevaluation, we first note that there is an even gl (1 | A → V n,e ⊗ Π( V − n, − e ) defined by ψ − v n,e ⊗ v − n, − e + v n,e ⊗ ψ − v − n, − e . We can compose this with the gl (1 | π ( Y ⊠ ) : V n,e ⊗ Π( V − n, − e ) → ( b V kn,e ⊠ Π( b V k − n, − e ))(0)of Section 3.1, and then use the universal property of induced \ gl (1 | i : V k ( gl (1 | → b V kn,e ⊠ Π( b V k − n, − e ) . By definition, i ( ) = π (cid:0) Y ⊠ ( ψ − v n,e , v − n, − e + Y ⊠ ( v n,e , ψ − v − n, − e (cid:1) . Equivalently, i ( ) is the coefficient of x − n,e in Y ⊠ ( ψ − v n,e , x ) v − n, − e + Y ⊠ ( v n,e , x ) ψ − v − n, − e . Now Corollary 4.2.2 implies that b V kn,e will be rigid if the rigidity composition R = r ◦ (id ⊠ ε ) ◦ A − ◦ ( i ⊠ id) ◦ l − is non-zero. We shall prove this by showing h ψ − v − n, − e , R ( v n,e ) i 6 = 0. From the definitions,( i ⊠ id) ◦ l − ( v n,e ) is the coefficient of (cid:0) − xx (cid:1) − n,e in the series Y ⊠ (cid:18) Y ⊠ (cid:18) ψ − v n,e , − xx (cid:19) v − n, − e , (cid:19) v n,e + Y ⊠ (cid:18) Y ⊠ (cid:18) v n,e , − xx (cid:19) ψ − v − n, − e , (cid:19) v n,e . (cid:0) − xx (cid:1) − n,e is the lowest power of − xx in this series since 0 is the lowestconformal weight of b V kn,e ⊠ Π( b V k − n, − e ) (recall Lemma 2.2.3 and Remark 2.3.3). We take x to be a real number in the interval ( ,
1) = U ∩ e U ∩ R , and then recalling (4.10), we findthat h ψ − v − n, − e , R ( v n,e ) i is the coefficient of (cid:0) − xx (cid:1) − n,e in the expansion of the followinganalytic function as a series in − xx and ln (cid:0) − xx (cid:1) on ( , D ψ − v − n, − e , r ◦ (id ⊠ ε ) ◦ A − ◦ Y ⊠ ( Y ⊠ ( ψ − v n,e , − x ) v − n, − e , x ) v n,e E + D ψ − v − n, − e , r ◦ (id ⊠ ε ) ◦ A − ◦ Y ⊠ ( Y ⊠ ( v n,e , − x ) ψ − v − n, − e , x ) v n,e E = D ψ − v − n, − e , r ◦ (id ⊠ ε ) ◦ Y ⊠ ( ψ − v n,e , Y ⊠ ( v − n, − e , x ) v n,e E + D ψ − v − n, − e , r ◦ (id ⊠ ε ) ◦ Y ⊠ ( v n,e , Y ⊠ ( ψ − v − n, − e , x ) v n,e E = D ψ − v − n, − e , Ω( Y b V kn,e )( v n,e , E ( v − n, − e , x ) ψ − v n,e E , (4.11)where we have used (4.5) for the last equality, and Ω( Y b V kn,e ) is the intertwining operator oftype (cid:0) b V n,e b V n,e V k ( gl (1 | (cid:1) obtained from the vertex operator by skew-symmetry.By Theorem 4.1.4, (4.11) is a solution to the differential equation (4.8). As a series in x , it is non-logarithmic with lowest-degree term h ψ − v − n, − e , v n,e ih v − n, − e , ψ − v n,e i x − n,e = x − n,e , so (4.11) is the fundamental basis solution φ (1) ( x ) = x − n,e (1 − x ) − n,e F (cid:16) ek , − ek ; 1; x (cid:17) . Thus we are reduced to showing that in the expansion of F (cid:0) ek , − ek ; 1; x (cid:1) as a series in − xx and ln (cid:0) − xx (cid:1) on the interval ( , (cid:2) Γ (cid:0) ek (cid:1) Γ (cid:0) − ek (cid:1)(cid:3) − = sin( πe/k ) πe/k = 0since e/k / ∈ Z . We conclude that b V kn,e is rigid.We now extend rigidity from simple objects to the full supercategories KL k and O fink .First recall that all modules in both categories have finite length (in the sense that everymodule has a finite filtration of Z -graded submodules such that the consecutive quotientsare Z -graded simple V k ( gl (1 | KL k is closed under Z -graded submodules and quotients, and the same is also clear for O fink . Moreover, sincetaking contragredients is an exact functor, KL k is closed under contragredients; O fink is alsoclosed under contragredients by the a r = E , N cases of (2.10). These are the conditionsneeded to apply the (straightforward superalgebra generalization of) [CMY2, Thm 4.4.1],which states that if simple objects are rigid, then so are finite-length objects: The tensor supercategories KL k and O fink are rigid; moreover, they arebraided ribbon tensor supercategories with even natural twist isomorphism θ = e πiL . Simple current extensions of V ( gl (1 | There are a few families of vertex operator superalgebras extending V k ( gl (1 | V ( gl (1 | W n + ,ℓ = b A , ⊕ M m ≥ ( b A ⊠ mn + ,ℓ ⊕ b A ⊠ m − n − , − ℓ ) (5.1)for ℓ ∈ Z and n ∈ R such that | ℓ | ≤ n + ,ℓ and 2 nℓ ∈ Z . In particular, it is shown thatas vertex superalgebras W , − ∼ = V − ( sl (2 | , W , ∼ = V ( sl (2 | , where V − ( sl (2 | V ( sl (2 | sl (2 | − and 1, respectively. These superalgebras are thus objects of the direct limitcompletion Ind( O fin ) of the category O fin of V ( gl (1 | V ( gl (1 | O fin ; the existence of vertex and braided tensor category structures on Ind( O fin ) followsafter verifying the conditions of [CMY1, Thm. 1.1]:1. The vertex operator superalgebra V ( gl (1 | O fin .2. The category O fin is closed under submodules, quotients, and finite direct sums.3. Every module in O fin is finitely generated (since every module in O fin has finitelength).4. The category O fin admits vertex and braided tensor category structures by Theorem2.3.2.5. For any intertwining operator Y of type (cid:0) XW W (cid:1) where W , W are modules in O fin and X is a generalized V ( gl (1 | O fin ), the submodule Im Y ⊆ X isan object of O fin . (By [CMY1, Cor. 2.13], Im Y is C -cofinite, that is, an object of KL . Then Im Y is an object of O fin by the a = E, N , r = 0 cases of (3.1).)For more details on the braided tensor category structure on Ind( O fin ), see [CMY1].We can now study V − ( sl (2 | F : O fin → Rep V − ( sl (2 | V − ( sl (2 | V − ( sl (2 | O fin ). By [CKM, Lem. 2.65], a V ( gl (1 | − ( sl (2 | V − ( sl (2 | V − ( sl (2 | W , − = b A , ⊕ M m ≥ ( b A m − , − m ⊕ b A − m + , m ) = M m ∈ Z b A m − ε ( m ) , − m . So by the fusion rules in Theorem 3.2.2 and the balancing equation with twist θ = e πiL ,the monodromy of b A n,ℓ with b A m − ε ( m ) , − m for n ∈ C , ℓ ∈ Z , m ∈ Z is M b A n,ℓ , b A m − ε ( m ) , − m = θ b A n,ℓ ⊠ b A m − ε ( m ) , − m ◦ ( θ − b A n,ℓ ⊠ θ − b A m − ε ( m ) , − m )= θ b A n + m − ε ( ℓ )+ ε ( ℓ − m ) ,ℓ − m ◦ ( θ − b A n,ℓ ⊠ θ − b A m − ε ( m ) , − m )= e πi (∆ n + m − ε ( ℓ )+ ε ( ℓ − m ) ,ℓ − m − ∆ n,ℓ − ∆ m − ε ( m ) , − m ) . From this, we see that b A n,ℓ induces to a local module if and only if 2 n ∈ Z . Similarly by thefusion rules in Theorem 3.2.3, the monodromy of the typical module b V n,e for n ∈ C , e / ∈ Z with b A m − ε ( m ) , − m is M b V n,e , b A m − ε ( m ) , − m = θ b V n,e ⊠ b A m − ε ( m ) , − m ◦ ( θ − b V n,e ⊠ θ − b A m − ε ( m ) , − m )= θ b V n + m,e − m ◦ ( θ − b V n,e ⊠ θ − b A m − ε ( m ) , − m )= e πi (∆ n + m,e − m − ∆ n,e − ∆ m − ε ( m ) , − m ) , From this, we see that b V n,e induces to a local module if and only if 2 n + e ∈ Z . Moreover,we can determine the simple objects of Rep V − ( sl (2 | The simple objects of
Rep V − ( sl (2 | are of the form F ( S ) , where S is either b A n,ℓ for n ∈ Z , ℓ ∈ Z or b V n,e for n ∈ C , e / ∈ Z such that n + e ∈ Z . Moreover, F ( S ) ∼ = F ( S ′ ) if and only if there exists m ∈ Z such that S ′ ∼ = S ⊠ b A m − ε ( m ) , − m . (5.2) Proof.
Let X be a simple object of Rep V − ( sl (2 | G be the restriction functorRep V − ( sl (2 | → Ind( O fin ). As an object of Ind( O fin ), G ( X ) is the union of O fin -submodules; thus because every non-zero object of O fin contains an irreducible submodule, G ( X ) contains an irreducible submodule S . By Frobenius reciprocity,Hom( F ( S ) , X ) ∼ = Hom( S, G ( X )) = 0 , so if F ( S ) is simple, then F ( S ) ∼ = X . Indeed, by examining the E -eigenvalues of b A m − ε ( m ) , − m ⊠ S , b A m − ε ( m ) , − m ⊠ S ≇ b A m ′ − ε ( m ′ ) , − m ′ ⊠ S m = m ′ , and then [CKM, Prop. 4.4] shows F ( S ) is simple. Moreover, we have seenthat F ( S ) is an object in Rep V − ( sl (2 | S = b A n,ℓ for n ∈ Z , ℓ ∈ Z or b V n,e for n ∈ C , e / ∈ Z such that 2 n + e ∈ Z .The condition (5.2) follows from Frobenius reciprocity. As V ( gl (1 | F ( b A n,ℓ ) = M m ∈ Z b A n + m − ε ( ℓ )+ ε ( ℓ − m ) ,ℓ − m , and F ( b V n,e ) = M m ∈ Z b V n + m,e − m . Since the lowest conformal weights of the summands b A n + m − ε ( ℓ )+ ε ( n − m ) ,ℓ − m and b V n + m,e − m are both linear functions of m , most of the induced simple objects F ( b A n,ℓ ) and F ( b V n,e ) arenot lowest-weight modules for V − ( sl (2 | b V n,e isch[ b V n,e ]( z, y ; q ) = q ∆ n,e y e z n ∞ Y i =0 (1 + zq i +1 )(1 + z − q i )(1 − q i +1 ) so that we get ch[ F ( b V n,e )]( z, y ; q ) = ch[ b V n,e ]( z, y ; q ) X m ∈ Z q − m (2 n + e ) y − m z m . (5.3)Thus F ( b V n,e ) has an infinite-dimensional lowest-conformal-weight space if 2 n + e = 0 andhas unbounded conformal weights otherwise. These are examples of relaxed highest-weightmodules and their images under spectral flow.Now that we have determined the simple modules in Rep V − ( sl (2 | If P is projective in O fin , then F ( P ) is projective in Rep V − ( sl (2 | .Proof. We first verify that P is projective in Ind( O fin ). Thus suppose we have a diagram P q (cid:15) (cid:15) X p / / Y in Ind( O fin ) with p surjective. Since O fin is closed under quotients, Im q ⊆ Y is a (finitely-generated) object of O fin . Then since X , as an object of Ind( O fin ), is the union of its38 fin -submodules, there are finitely many O fin -submodules { W i } which contain preimagesof a generating set for Im q . Because O fin is closed under finite direct sums and quotients,the submodule W = P W i is an object of O fin , and p ( W ) ⊆ Y is an O fin -submodule thatcontains Im q . Now because P is projective in O fin , there is a map f : P → W such that q = p | W ◦ f . Interpreting f as a map into X , we conclude that P is projective in Ind( O fin ).Now, as in [ACKR, Thm. 17], Frobenius reciprocity implies that F ( P ) is projective inRep V − ( sl (2 | Rep V − ( sl (2 | ( F ( P ) , · ) is exact since it is the composi-tion of two exact functors: the restriction functor G : Rep V − ( sl (2 | → Ind( O fin ) andthe Hom functor Hom Ind( O fin ) ( P, · ).Now we can prove: Suppose S is a simple V ( gl (1 | -module with projective cover P S in O fin such that F ( S ) is an object of Rep V − ( sl (2 | . Then F ( P S ) is a projective coverof F ( S ) in Rep V − ( sl (2 | .Proof. By [CKL, Thm. 1.4(1)], the conformal weight criterion for F ( P S ) to be local is thesame as that for F ( S ), so F ( P S ) is an object of Rep V − ( sl (2 | F ( S ) is, and then F ( P S )is projective in Rep V − ( sl (2 | p : P S → S induces to a surjection F ( p ) : F ( P S ) → F ( S ).Now if S is a typical simple V ( gl (1 | P S = S and it is clear that F ( P S )is a projective cover of F ( S ). If S = b A n,ℓ is atypical with projective cover b P n,ℓ in O fin ,we need to show that for any surjection q : P → F ( b A n,ℓ ) in Rep V − ( sl (2 | P projective, there is a surjection f : P → F ( b P n,ℓ ) such that q = F ( p ) ◦ f . Indeed, because P and F ( b P n,ℓ ) are both projective, we have maps f : P → F ( b P n,ℓ ) and g : F ( b P n,ℓ ) → P such that the diagrams P f y y rrrrrrrrrr q (cid:15) (cid:15) F ( b P n,ℓ ) F ( p ) / / F ( b A n,ℓ ) P q (cid:15) (cid:15) F ( b P n,ℓ ) g rrrrrrrrrr F ( p ) / / F ( b A n,ℓ )commute. Since induction is exact, F ( b P n,ℓ ) has finite length; so if we can show that F ( b P n,ℓ )is also indecomposable, then Fitting’s Lemma implies that f ◦ g is either nilpotent or anisomorphism. It cannot be nilpotent because F ( p ) ◦ ( f ◦ g ) N = F ( p ) = 0for all N ∈ N , so it is an isomorphism. It follows that f is surjective (and g is injective).It remains to show that F ( b P n,ℓ ) is indecomposable. It is enough to show that its socleis F ( b A n,ℓ ) (and in particular is simple), because then the intersection of any two non-zero39ubmodules in F ( b P n,ℓ ) will contain the socle. By Frobenius reciprocity and Corollary 3.2.5,for any simple V ( gl (1 | S such that F ( S ) is local,dim Hom Rep V − ( sl (2 | ( F ( S ) , F ( b P n,ℓ )) = X m ∈ Z dim Hom O fin ( S, b A m − ε ( m ) , − m ⊠ b P n,ℓ )= (cid:26) S ∼ = b A m − ε ( m ) , − m ⊠ b A n,ℓ for some m ∈ Z . Thus F ( S ) occurs as a submodule of F ( b P n,ℓ ), with multiplicity one, if and only if F ( S ) ∼ = F ( b A n,ℓ ) (recall (5.2)). Consequently, Soc F ( b P n,ℓ ) ∼ = F ( b A n,ℓ ) as required.For the vertex operator superalgebra V ( sl (2 | W , = b A , ⊕ M m ∈ Z ≥ ( b A ,m ⊕ b A − , − m ) = M m ∈ Z b A ε ( m ) ,m . (5.4)Then similar analysis as above gives the simple objects in Rep V ( sl (2 | The simple objects of
Rep V ( sl (2 | are of the form F ( S ) , where S is either b A n,ℓ for n ∈ + Z , ℓ ∈ Z \ { } or n ∈ Z , ℓ = 0 , or b V n,e for n ∈ C , e / ∈ Z such that n + e ∈ + Z . For any simple V ( gl (1 | -module S such that F ( S ) is an objectof Rep V ( sl (2 | , F ( P S ) is a projective cover of F ( S ) in Rep V ( sl (2 | , where P S is aprojective cover of S in O fin . As V ( gl (1 | F ( b A n,ℓ ) = M m ∈ Z b A n − ε ( ℓ )+ ε ( ℓ + m ) ,ℓ + m , and F ( b V n,e ) = M m ∈ Z b V n,e + m . Note that the lowest conformal weights of the summands b A n − ε ( ℓ )+ ε ( ℓ + m ) ,ℓ + m and b V n,e + m areboth quadratic functions of m with leading term m . Thus the induced simple objects F ( b A n,ℓ ) and F ( b V n,e ) are all lowest-weight modules for V ( sl (2 | References [AS] M. Abramowitz and I. Stegun,
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J. Algebra (1995), no. 1, 241–273.Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton,Alberta T6G 2R3, Canada
E-mail: [email protected], [email protected]
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
E-mail: [email protected]@tsinghua.edu.cn