Logarithmic Link Invariants of U ¯ ¯ ¯ ¯ H q ( sl 2 ) and Asymptotic Dimensions of Singlet Vertex Algebras
aa r X i v : . [ m a t h . QA ] M a y LOGARITHMIC LINK INVARIANTS OF U Hq ( sl ) ANDASYMPTOTIC DIMENSIONS OF SINGLET VERTEX ALGEBRAS
THOMAS CREUTZIG, ANTUN MILAS AND MATT RUPERT
Abstract.
We study relationships between the restricted unrolled quantum group U Hq ( sl )at 2 r -th root of unity q = e πi/r , r ≥
2, and the singlet vertex operator algebra M ( r ). We usedeformable families of modules to efficiently compute (1 , U Hq ( sl ). These relate to the colored Alexander tangle invariantsstudied in [ADO, Mu1]. It follows that the regularized asymptotic dimensions of charactersof M ( r ) coincide with the corresponding modified traces of open Hopf link invariants. Wealso discuss various categorical properties of M ( r )-mod in connection to braided tensorcategories. Introduction
This work is concerned with the restricted ”unrolled” quantum group U Hq ( sl ) at 2 r -throot of unity q = e πi/r , r ≥
2, and the singlet vertex operator algebra M ( r ). Representationcategories of both are neither semi-simple nor do they have finitely many simple objects.While this quantum group has been used to construct link and 3-manifold invariants [BCGP,CGP1, CGP2, CMu, GP, GPT1, GPT2], categorical properties of the singlet vertex operatoralgebra (and more generally, vertex algebras with infinitely many simple objects) are stillpoorly understood. Previously, it was realized in examples that modular-like properties ofcharacters [AC, CM1, CR1, CR2, CR3], as well as their asymptotic dimensions (often alsocalled quantum dimensions) [CM1, CMW], relate to the fusion ring of the singlet vertexalgebras and other vertex (super)algebras. This begs for a categorical interpretation andin this work the relation of M ( r ) and U Hq ( sl ) gives such an interpretation via open Hopflink invariants. In particular, this shows that the Jacobi variable introduced in [CM1] asa regularization parameter for the classical false theta functions has a novel interpretationfrom the point of view of quantum topology.1.1. Tensor Categories and Vertex Algebras.
Vertex operator algebras are importantsources of braided and modular tensor categories. If a vertex operator algebra V is regular(i.e. C -cofinite and rational), together with some additional mild conditions, then it iswell-understood [H1, H2] that its representation category is modular and especially ribbon.Moreover in this case, there are three actions of the modular group: on the linear span oftorus one-point functions, a categorical one given by twists and Hopf links and the one thatdiagonalizes the fusion rules (this is only a S -matrix). All three coincide in the appropriatesense. T.C. was supported by an NSERC Research Grant (RES0020460).A.M. was supported by a Simons Foundation Collaboration Grant ( f V is not rational but still C -cofinite (satisfying a few additional assumptions), thenthere is still a modular group action on the space of one-point functions on the torus [Miy2].Moreover, there is also a modular group action in the category provided it is ribbon [Ly1, Ly2]and a relation between the character S -matrix and logarithmic Hopf link invariants has beengiven in [CG1, CG2] for the triplet vertex algebras. Here we would like to extend theseobservations beyond categories with finitely many simple objects. From the vertex algebrapoint of view, vertex algebras which are not C -cofinite are considerably difficult to study.One issue with these algebras is that the category of weak modules is way too big and onlyafter restriction to a subcategory we hope to have good categorical properties. In an impor-tant series of papers, Huang, Lepowsky and Zhang [HLZ] obtained sufficient conditions ona subcategory C to posses a braided (vertex) tensor category structure. Roughly speaking,they proved that if all objects in an abelian subcategory C of generalized V -modules sat-isfy the C -cofiniteness condition, the category is closed under P ( z )-tensor products, and afew additional conditions, then the convergence and extension properties for products anditerates hold in C , and thus the category can be endowed with a braided tensor categorystructure. More precisely, Assumptions 10.1, 12.1 and 12.2 [HLZ] have to be satisfied. Themost difficult part in the verification of these axioms is that a suitable subcategory is closedunder the P ( z )-tensor product. We should mention that Miyamoto recently obtained a suf-ficient condition on the closure of the tensor product in a C -cofinite category of modules[Miy1]. But his result alone does not give a braided tensor product.The singlet vertex algebra M ( r ), r ∈ N ≥ , is a prominent example among irrational non C -cofinite vertex algebras and it was studied by many authors [AD, AM1, AM2, AM3, CM1,CMW]. This vertex algebra (subalgebra of the triplet vertex algebra) contains both atypicaland typical representations so it serves as the best testing ground for categorical explorationbeyond C -cofiniteness. We have already understood in previous works that asymptoticdimensions of characters relate to representations of the fusion ring [CM1, CMW] and ourconjecture was that this has a precise categorical meaning. Here, we will give such aninterpretation. It is believed that categories of U Hq ( sl )-modules and the singlet VOA M ( r )-modules are equivalent as monoidal categories and maybe even as braided tensor categories,after restriction to suitable sub-categories. This restriction is indeed needed in order to havea braiding on the quantum group side - one needs the category of weight modules introducedin [CGP1].1.2. Summary of the present work.
Representations of the tensor ring in a ribbon cate-gory are directly given by open logarithmic Hopf link invariants [T] (for a proof in non-strictcategories see [CG1]). Here, we first successfully compare them with the asymptotic dimen-sions of characters and secondly we find a novel way of computing them. Previously, theyhave been computed using the known tensor ring [CGP1], which from our perspective is notideal as we are seeking ways to better understand the still inaccessible fusion ring of VOAs.Our computation is a deformation argument analogous to ideas of Murakami and Nagatomo[Mu2, MN] in the case of the restricted quantum group U q ( sl ) but also motivated by theidea of deformable families of VOAs [CL]. The idea is that if one has a continuous set ofmodules, all but a discrete set of them semi-simple, then one can construct a deformable Regretfully, there is no consistent notation for the singlet algebra in the literature; previously it was alsodenoted by W (2 , r −
1) in [CM1] as well as by M (1) in [AD, AM1]. amily of modules M ( x ) which specializes to specific modules if specializing the variable x and especially at non-generic position also specializing to the indecomposable but reduciblemodules of particular interest in logarithmic conformal field theory. Moreover, this pro-cess commutes with the computation of invariants of interest such as Hopf links and twists.Since these are next to trivial to compute on simple modules our process gives a nice way ofobtaining them also for the complicated cases of indecomposable but reducible modules.Our strategy of computation actually extends straight forwardly to any open (1 ,
1) tan-gle invariants. It turns out that the results very nicely compare to the colored Alexanderinvariants introduced by Akutsu, Deguchi and Ohtsuki in [ADO] and further developed byMurakami [Mu1, MN, Mu2]. This has already been observed earlier if only simple projec-tive modules were used as colors [GPT1]. We would like to stress one important differencebetween U Hq ( sl ) and U q ( sl ). The latter is not braidable and therefore the category of finite-dimensional U q ( sl )-modules does not have a universal R -matrix [KS]. On the other handthe category of (suitably defined) weight modules for U Hq ( sl ) is braided. We also would liketo announce that this problem for U q ( sl ) can be cured by finding a suitable non-trivial asso-ciator in the module category of U q ( sl ) [GR, CGR]. This associator is found in identifyingthe module category of U q ( sl ) as the representation category of local modules of an algebrain U Hq ( sl ) [CGR]. It remains however to be proven that the resulting module category givesrise to the colored Alexander invariants studied by Murakami.1.3. Results.
It is believed that the representation categories of U Hq ( sl ) and M ( r ) areequivalent as monoidal and hopefully as braided tensor categories (we will have a moreprecise conjecture below). The fusion ring of M ( r ) is not known, its Grothendieck ring hasbeen conjectured in [CM1]. Comparison does work though (see Section 5): Theorem 1.
Irreducible representations of U Hq ( sl ) and M ( r ) are related as follows:(1) Assume that the Grothendieck ring of M ( r ) is as conjectured (see Section 2 ). Let α ∈ ( C \ Z ) ∪ r Z . Then the map ϕ : V α F α + r − √ r , S i ⊗ C Hkr M − k,i +1 betweensimple objects for U Hq ( sl ) and M ( r ) is bijective and induces a ring isomorphism.(2) Let k ∈ Z and j ∈ { , . . . , r − } and let ǫ ∈ C satisfy Re ( ǫ ) < B rǫ as well as ǫ in S ( k, j + 1 + r ( k + 1)) (for the precise definitions of these sets, see Section 2) then qdim[ ϕ ( X ) ǫ ] = t P j ⊗ C Hkr (Φ X,P j ⊗ C Hkr ◦ x j,k ) t P j ⊗ C Hkr (Φ S ,P j ⊗ C Hkr ◦ x j,k ) , and if Re ( ǫ ) > B rǫ then for α = i √ rǫ we have qdim[ ϕ ( X ) − iα √ r ] = t V α (Φ X,V α ) t V α (Φ S ,V α ) . Here t X is the modified trace on the ideal of negligible (projective) objects X . If the representation categories of U Hq ( sl ) and M ( r ) are braided equivalent then thismeans that the asymptotic dimensions of regularized characters of the VOA M ( r ) verynicely capture the modified traces of the logarithmic Hopf link invariants. In the continuousregularization regime these are the ordinary modified traces while the stripwise constant egime corresponds to traces of logarithmic Hopf link invariants weighted with the nilpotentendomorphisms.In [CM1], the regularized quantum dimensions of modules were used to at least conjecturethe Grothendieck ring of M ( r ) and in [CGP1] the Hopf link invariants of U Hq ( sl ) werecomputed using the known tensor ring of U Hq ( sl ). Our picture is that one should use theHopf link invariants to compute the tensor ring of U Hq ( sl ) and we indeed can find a strategythat works well. The key Lemma (see Theorem 4) is the construction of a deformable familyof modules X ǫ for ǫ in ( − , ) such that X ǫ = (cid:26) V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ if ǫ = 0 P i ⊗ C Hℓr if ǫ = 0This deformable family can be used to compute open Hopf link invariants and more generallyopen tangle invariants of (1 ,
1) tangles T . For this we recall Section 3 of [GPT1]. Let L be a C -colored ribbon graph, such that at least one of the colors is a simple V λ . Let T λ bethe colored (1 , V λ . Then the re-normalizedReshetikhin-Turaev link invariant is F ′ ( L ) = t V λ ( T λ ) , where t V λ is the modified trace on V λ . These invariants where shown in [GPT1] to coincidewith Murakami’s Alexander invariants [Mu1]. We can now extend these results to anyprojective module. For this let L be as above but with at least one of the colors the module P i ⊗ C Hℓr . Let T P i ⊗ C Hℓr be the colored (1 , P i ⊗ C Hℓr . Let T λ be the colored (1 , P i ⊗ C Hℓr ) with V λ . Let d ( X ) = t X ( Id X ) be the modified dimension ofa projective module X and let x i,ℓr be the nilpotent endomorphism of P i ⊗ C Hℓr normalizedas in section 3.2, then (Theorem 8):
Theorem 2.
The colored (1 , -ribbon graph T P i ⊗ C Hℓr satisfies t P i ⊗ C Hℓr (cid:16) T P i ⊗ C Hℓr (cid:17) = lim ǫ → (cid:0) t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) + t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) (cid:1) and T P i ⊗ C Hℓr = a Id P i ⊗ C Hℓr + bx i,ℓr with coefficients a = lim ǫ → t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( T − − i + r + ℓr + ǫ ) = lim ǫ → t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) and b = r πi { i } (cid:18) ddλ t V λ ( T λ ) d ( T λ ) (cid:12)(cid:12)(cid:12) λ = ℓr + r − i − − ddλ t V λ ( T λ ) d ( V λ ) (cid:12)(cid:12)(cid:12) λ = i + i − r + ℓr (cid:19) . The construction of the deformable family of modules X ǫ is the only key ingredient inproving this theorem. Murakami and Nagatomo [MN] also constructed a deformable familyof modules of the semi-restricted quantum group that specialized at a special point to aprojective indecomposable but reducible module and that only at this special point alsobecame a module of U q ( sl ), while in [Mu1], Murakami varied the q -parameter of U q ( sl )such that in the limit q = e πi r the module specialized to projective indecomposable but educible module. In both cases it was then used that the employed R -matrix of the involvedquantum groups is basically the same. We however recall again, that U q ( sl ) is not braided.Presently, the only vertex operator algebras with infinitely many simple objects and nonsemi-simple representations whose categories of modules are known to be braided is theHeisenberg vertex algebra (see Remark 18) and the Kazhdan-Lusztig category [KL]. The lastsection present an attempt to push the tensor product theory beyond the Heisenberg vertexalgebra. We think that the singlet algebra is an excellent candidate in this direction. Inorder to apply the Huang-Lepowsky-Zhang theory, several highly non-trivial conditions haveto be verified including the C -cofiniteness of a suitable subcategory of modules. Althoughat this stage we do not have a full proof that M ( r )-mod is braided, we reduced this problemto a purely representation theoretic condition (assumptions (a) and (b) below): Theorem 3.
All finite-length M ( r ) -modules are C -cofinite. Assume that (a) every C -cofinite, N -graded module is of finite length, and (b) every finitely generated, generalized N -graded M ( r ) -module is C -cofinite. Then, the category of N -graded, C -cofinite M ( r ) -modules can be equipped with a braided tensor category structure. Future work.
This work has several ramifications and extensions. In [CM2], we intro-duced and studied regularized characters of modules of certain non C -cofinite W -algebrasdenoted by W ( Q ) r , where r ≥ Q is the root lattice of a simply-laced simple Liealgebra g . These vertex algebras are ”higher rank” generalization of the singlet vertex al-gebra M ( r ). Their categories of modules are expected to be closely related to the categoryof modules for higher rank unrolled quantum groups U h q ( g ) at 2 r -th root of unity. We planto study asymptotic dimensions of W ( Q ) r -modules in connection to quantum invariants ofknots and links colored with representations of U h q ( g ). Acknowledgements.
T.C. is very grateful to Terry Gannon for collaboration on C -cofinite VOAs and modular tensor categories resulting in the works [CG1, CG2] where theimportance of logarithmic Hopf links in C -cofinite VOAs has been realized. He is alsothankful to Azat Gainutdinov and Ingo Runkel for valuable discussions on the relation of U Hq ( sl ) and U q ( sl ) [CGR]. A.M. would like to thank Yi-Zhi Huang for discussions onvarious aspects of the vertex tensor product theory [HLZ] and to Draˇzen Adamovi´c. Finally,we are grateful to Jun Murakami for a correspondence.2. The singlet vertex operator algebra M ( r )Let r ∈ N ≥ . Here, we review necessary information of the vertex algebra M ( r ) following[CM1]; see also [AD, AM1, AM2]. The vertex algebra M ( r ) is realized as a subalgebra ofthe rank one Heisenberg vertex algebra F . It is strongly generated by the Virasoro vector ω (suitably chosen such that the central charge is 1 − p − p ), together with one primary fieldof conformal weight 2 r −
1, usually denoted by H . Introduce α + = √ r, α − = − p /r, and α = α − + α + . Denote by F λ the Fock space with the highest weight λ ∈ C , which is also an F -module. Allirreducible M -modules are realized as subquotients of F λ , which we now describe briefly.Let L = √ r Z , viewed as a rank one lattice, and L ′ denotes its dual lattice. Then typical simple modules F λ are parameterized by λ in ( C \ L ′ ) ∪ L , while atypical simple modules M t,s ⊂ F α t,s are parameterized by integers t, s with 1 ≤ s ≤ r −
1. Characters of irreducible ( r )-modules can be easily computed. Irreducible characters admit an ǫ -regularization, asexplained in [CM1], where they are denoted by ch[ X ǫ ], where X is an M ( r )-module. Here ǫ ∈ C . The main result of [CM1] is a formula for the modular transformation of regularizedpartial and false theta functions, which then gives modular properties of regularized char-acters. These in turn give a Verlinde-type algebra for the regularized characters, where theproduct is defined (for Re( ǫ ) > ) asch[ X ǫa ] × ch[ X ǫb ] = Z R Z R S ǫaρ S ǫbρ S − ǫρµ S ǫ (1 , ρ ch[ F ǫµ ] dµdρ, and where S · , · defines the S -kernel. For irreducible characters this formula readsch[ F ǫλ ] × ch[ F ǫµ ] = p − X ℓ =0 ch[ F ǫλ + µ + ℓα − ] , ch[ M ǫt,s ] × ch[ F ǫµ ] = s X ℓ = − s +2 ℓ + s =0 mod 2 ch[ F ǫµ + α r,ℓ ]ch[ M ǫt,s ] × ch[ M ǫt ′ ,s ′ ] = min { s + s ′ − ,p } X ℓ = | s − s ′ | +1 ℓ + s + s ′ =1 mod 2 ch[ M ǫt + t ′ − ,ℓ ]+ s + s ′ − X ℓ = p +1 ℓ + s + s ′ =1 mod 2 (cid:16) ch[ M ǫt + t ′ − ,ℓ − p ] + ch[ M ǫt + t ′ − , p − ℓ ] + ch[ M ǫt + t ′ ,ℓ − p ] (cid:17) . (1)Then regularized asymptotic dimensions are introduced asqdim[ X ǫ ] := lim τ → ch[ X ǫ ( τ )]ch[ M ǫ , ]( τ ) . (2)Following [CMW] introduce B rǫ := − min (cid:26)(cid:12)(cid:12)(cid:12) m √ r − Im ( ǫ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) m ∈ Z \ r Z (cid:27) . (3)Then for Re( ǫ ) > B rǫ the regularized asymptotic dimensions areqdim[ F ǫλ ] = q λ − α ǫ sin( − πα + ǫi )sin( πα − ǫi ) = q λ − α ǫ p − X ℓ = − p +1 ℓ + p =1 mod 2 q α − ℓǫ qdim[ M ǫt,s ] = q − ( t − α + ǫ sin( πsα − ǫi )sin( πα − ǫi ) = q − ( t − α + ǫ s − X ℓ = − s +1 ℓ + s =1 mod 2 q α − ℓǫ (4)and for Re( ǫ ) < B rǫ the answer is qdim[ F ǫλ ] = 0 and for ǫ ∈ S ( k, m ) , k ∈ Z , m = 0 . . . , r − M ǫt,s ] = ( − m ( t − sin( πms/r )sin( πm/r ) if m = 0 , r, ( − ( m +1)( t − mr ( s − sin( πs/r )sin( π/r ) if m = 0 , r. (5) In [CM1], we used − ǫ instead of ǫ so the formula was given in the Re( ǫ ) < ith S ( k, m ) = (cid:26) ǫ ∈ C (cid:12)(cid:12)(cid:12)(cid:12) k + 2 m − r < Im( ǫ ) √ r < k + 2 m + 14 r (cid:27) ,k ∈ Z where m = 0 , . . . , r −
1. It turns out that the algebra of quantum dimensions andabove Verlinde algebra of characters coincide. Further the conjecture of [CM1] is that theserelations also hold in the Grothendieck ring of the module category of M ( r ). Note, that thelatter is also only conjectured to be braided.3. The unrolled restricted quantum group U Hq ( sl )Throughout q = e πi/r , r ≥
2. Also, { α } = q α − q − α , [ α ] = { α }{ } , and [ n ]! = [ n ] · · · [1].In this section we review basic facts about the unrolled quantum group U Hq ( sl ) followingprimarily [CGP1]. The quantum group U Hq ( sl ) as a unital associative algebra is generated E , F , K , K − and H , with the following relations: KE = q EK, KF = q − F K, HK = KH [ H, E ] = 2 E, [ H, F ] = − F, [ E, F ] = K − K − q − q − .E r = 0 , F r = 0 . The Hopf algebra structure is defined by using the standard comultiplication formulas for E , F and K , while H is primitive, that is ∆( H ) = 1 ⊗ H + H ⊗
1. Thus, the antipode map S is induced by letting S ( H ) = − H and S ( E ), S ( K ) and S ( F ) are defined as usual. Notall U Hq ( sl ) representations are of interest. We say that an U Hq ( sl )-module M is a weight module if M is finite-dimensional and H -diagonalizable such that q K = H (as an operatoron M ). As in [CGP1], we denote by C the category of weight U Hq ( sl )-modules. Irreducibleobjects in C are easy to classify. They are clearly of highest weight and belong to three types: S n , n = 0 , .., r −
1, of dimension n + 1, V α , where α ∈ ¨ C := ( C \ Z ) ∪ r Z are of dimension r , and one-dimensional modules C Hkr . Then a complete list of irreps is given by: (atypicals) S i ⊗ C Hkr , k ∈ Z , n = 0 , .., r − V α , α ∈ ( C \ Z ) ∪ r Z . All the irreduciblemodules can be constructed explicitly in terms of their bases [CGP1].3.1. Beyond the category C . Here we discuss an enlargement of the category C . Aswe shall explain below, it is actually not true that the (full) category of finitely generated M ( r )-modules is equivalent to C . This is why it is interesting and important to consider U Hq ( sl )-modules outside the category C .Next we introduce the cateogry C log (here log is meant to indicate inclusion of logarithmic modules - this terminology is motivated by a related notion in logarithmic conformal fieldtheory [HLZ, Mi]) . Objects in C log are finite-dimensional M ( p )-modules such that q H = K (as operators) but H does not act necessarily semisimple (of course, here q H = P n ≥ πiH ) n r n n ! ).Now we show that this category admits self-extensions of generic modules V α but no self-extension of simple modules of dimension < r (this is in agreement with the singlet algebracase [AM1, Mi]). Consider a 2 r -dimensional module ˜ V λ with a basis v i , v i , i = 0 , ..., r − .v i = ( λ − i ) v i + v i , H.v i = ( λ − i ) v i ; 0 ≤ i ≤ r − K.v i = q λ − i v i + πir q λ − i v i , Kv i = q λ − i v i E.v i = { i }{ λ + 1 − i }{ i } v i − + β i v i − , E.v i = { i }{ λ + 1 − i }{ i } v i − , E.v = E.v = 0 ,F.v i = v i +1 , F.v i = v i +1 ; 0 ≤ i ≤ r − , F.v r − = F.v r − = 0 , where β = 0 , β i = πir q − q − i X j ≥ (cid:0) q λ − j − + q j − − λ (cid:1) . Claim: ˜ V λ is a U Hq ( sl )-module, a self-extension of V λ . In order to show that ˜ V λ is an U Hq ( sl )-module the only non-trivial relation relation to verify is( EF − F E ) v i = K − K − q − q − v i . For this it is essential that β r = r X j =1 q j = 0 . It is clear that we now get a non-split short exact sequece:0 −→ V λ −→ ˜ V λ −→ V λ −→ . Any irreducible modules of dimension i < r , is isomorphic to S i ⊗ C Hkr . In order to rule outits self extension it is enough to follows steps in the proof of the claim and observe that P ij =0 q j = 0 . We conclude with the conjecture that the category of finite-length M ( r )-modules is equiva-lent to the category C log . Moreover, the full subcategory C ⊂ C log is expected to be equivalentto the subcategory of M ( r )-mod generated by irreducible objects (as a tensor category). Wealso plan to investigate possible braided category structure on C log .3.2. Projective modules.
Projective modules in C are classified in [CGP1]. Although thispaper did not discuss projective covers, it can be easily shown that the projective modulesdenoted by P i are projective covers of S i , P i ⊗ C Hkr are projective covers of S i ⊗ C Hkr , and V α , α ∈ ¨ C , are their own projective covers. Their Jordan-H¨older filtration is described by Figure1. We have End U Hq ( sl ) ( V α ) = C Id V α and it can be shown that End U Hq ( sl ) ( P i ) = C Id P i ⊗ C Hkr ⊕ C x i,k where x i,k is the nilpotent endomorphism of P i ⊗ C Hkr uniquely determined by w Hi w Si (see Figure 1).3.3. Modified quantum dimension.
Let P be the full subtensor category of modules gen-erated by projective U Hq ( sl )-modules. There exists a unique trace on P , up to multiplicationby an element of C . In particular, there is a unique trace t = { t V } , on P , such that forany f ∈ End C ( V ) we have t V ( f ) = ( − r − h f i . For such choice of t , following [CGP1], wedefine the modified quantum dimension function as d : Ob( P ) → C , d ( V ) := t V (Id V ). i ⊗ C Hkr : S i ⊗ C Hkr S r − i − ⊗ C H ( k +1) r S r − i − ⊗ C H ( k − r S i ⊗ C Hkr V i +( k − r V r − i − kr Figure 1.
Loewy diagram of P i ⊗ C Hkr in terms ofsimple (left) and typical (right) composition factors4.
Logarithmic invariants via deformable modules
Open Hopf link invariants have been computed in [CGP1]. However for those involvingprojective modules knowledge of the tensor ring is required. Here we find a new way ofcomputation that does not require this knowledge. For this, we introduce a deformablefamily of modules that then will be used to compute logarithmic tangle invariants.
Theorem 4.
Let ǫ ∈ ( − , ) , i ∈ { , ..., r − } and let ℓ ∈ Z . Denote by X ǫ the modulewith vector space basis { w Li +2 − r , w Li +4 − r , ..., w L − i − , w H − i , ..., w Hi , w S − i , ..., w Si , w Ri +2 , ..., w R r − − i } and action given by w Ri +2 = ( − ℓ E w Hi , w L − i − = F w H − i , F w Ri +2 = w Si + [1 + i ][ ǫ ] w Hi w Hi − k = F k w Hi and w Si − k = F k w Si for k ∈ { , ..., i } w L − i − − k = F k w L − i − and w Ri +2+2 k = ( − kℓ E k w Ri +2 for k ∈ { , ..., r − − i } H w Xk = ( k + ℓr + ǫ ) w Xk , K w Xk = ( − ℓ q k + ǫ w Xk , for X ∈ { L, H, S, R } E w Rk = ( − ℓ w Rk +2 , F w Xk = w Xk − , for X ∈ { L, S, H } F w S − i = [1 + i ][ ǫ ] w L − i − , E w Si = 2( − ℓ +1 [1 + i ][ ǫ ] w Ri +2 , E w R r − − i = F w Li +2 − r = 0 E w L − i − = 2( − ℓ [ i + 1][ ǫ ] w H − i + ( − ℓ w S − i ,E w L − i − − k = ( − ℓ +1 [1 + i + k ][ k − ǫ ] w L − i − k − ,F w Ri +2+2 k = − [1 + i + k ][ k + ǫ ] w Ri +2+2( k − ,E w Hi − k = (2[1 + i − k + ǫ ][ k ] − [1 + i − k ][ k − ǫ ])( − ℓ w Hi − k − + ( − ℓ w Si − k − ,E w Si − k = (2[1 + i − k ][ k − ǫ ] − [1 + i − k + ǫ ][ k ])( − ℓ w Si − k − + 2( − ℓ +1 [1 + i ] [ ǫ ] w Hi − k − . Then X ǫ = (cid:26) V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ if ǫ = 0 P i ⊗ C Hℓr if ǫ = 0 roof: Let ǫ ∈ ( − , ) \ { } . Let { x , ..., x r − } denote the standard basis for V i − r + ℓr + ǫ and { y , ..., y r − } the standard basis for V − − i + r + ℓr + ǫ . Define a new basis { w Li +2 − r , w Li +4 − r , ..., w L − i − , w H − i , ..., w Hi , w S − i , ..., w Si , w Ri +2 , ..., w R r − − i } for V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ by w Hi − k = 2 x k − i ][ ǫ ] y r − − i + k w Si − k = − i ][ ǫ ] x k + y r − − i + k w L − i − − k = 2 x i + k w Ri +2+2 k = − i ][ ǫ ] k Y s =0 [1 + i + s ][ − s − ǫ ] ! y r − − i − k We will first prove the statement for ǫ = 0. We will show that the action of U Hq ( sl ) onthe new basis is precisely as stated in the theorem. Recall that the action on the standardbasis element v k ∈ V α is given by Hv k = ( α + r − − k ) v k , Ev k = [ k ][ k − α ] v k − , F v k = v k +1 . By direct computation, we obtain the following: E w Hi = 2 Ex − i ][ ǫ ] Ey r − − i = − [ r − − i ][ − ℓr − ǫ ]2[1 + i ][ ǫ ] y r − − i = 12 ( − ℓ y r − − i = ( − ℓ w Ri +2 ,F w H − i = 2 F x i − i ][ ǫ ] F y r − = 2 x i +1 = w L − i − , w Si + [1 + i ][ ǫ ] w Hi = − i ][ ǫ ] x + y r − − i + 2[1 + i ][ ǫ ] x − y r − − i = 12 y r − − i = 12 F y r − − = F w Ri +2 . Hence, we have shown w Ri +2 = ( − ℓ E w Hi , w L − i − = F w H − i , and F w Ri +2 = w Si + [1 + i ][ ǫ ] w Hi . Itis easily seen that F w Xk = w Xk − for all X ∈ { L, S, H } and that E w Ri +2+2 k = − i ][ ǫ ] k Y s =0 [1 + i + s ][ − s − ǫ ] ! Ey r − − i − k = − [ r − − i − k ][ − − k − ℓr − ǫ ]2[1 + i ][ ǫ ] k Y s =0 [1 + i + s ][ − s − ǫ ] ! y r − − i − ( k +1) = − ( − ℓ i ][ ǫ ] k +1 Y s =0 [1 + i + s ][ − s − ǫ ] ! y r − − i − ( k +1) = ( − ℓ w Ri +2+2( k +1) so E w Rk = ( − ℓ w Rk +2 , which gives w Ri +2+2 k = ( − kℓ E k w Ri +2 , w Hi − k = F k w Hi , w Si − k = F k w Si ,and w L − i − − k = F k w L − i − . H acts on a standard basis vector v k ∈ V α by Hv k = ( α + r − − k ) v k , so we have Hx k = (1 + i − r + ℓr + ǫ + r − − k ) x k = ( i − k + ℓr + ǫ ) x k Hy r − − i + k = ( − − i + r + ℓr + ǫ + r − − r − − i + k )) y r − − i + k = ( i − k + ℓr + ǫ ) y r − − i + k Hx i + k = (1 + i − r + ℓr + ǫ + r − − i + k )) x i + k ( − i − − k + ℓr + ǫ ) x i + k Hy r − − i − ( k +1) = ( − − i + r + ℓr + ǫ + r − − r − − i − ( k + 1)) y r − − i − ( k +1) = ( i + 2 + 2 k + ℓr + ǫ ) y r − − i − ( k +1) From this, it immediately follows that H w Hi − k = ( i − k + ℓr + ǫ ) w Hi − k , H w Si − k = ( i − k + ℓr + ǫ ) w Si − k ,H w L − i − − k = ( − i − − k + ℓr + ǫ ) w L − i − − k , H w Ri +2+2 k = ( i + 2 + 2 k + ℓr + ǫ ) w Ri +2+2 k and K acts as q H , so we have shown that H w Xk = ( k + ℓr + ǫ ) w Xk and K w Xk = q k + ℓr + ǫ w Xk =( − ℓ q k + ǫ w Xk for all X ∈ { L, H, S, R } . It is easy to see that E w R r − − i = F w Li +2 − r = 0 as F x r − = Ey = 0. We also have F w S − i = − i ][ ǫ ] F x i + F y r − = − i ][ ǫ ] x i +1 = − [1 + i ][ ǫ ] w L − i − ,E w Si = − i ][ ǫ ] Ex + Ey r − − i = [1 + i ][ − ℓr − ǫ ] y r − − i = 2( − ℓ +1 [1 + i ][ ǫ ] w Ri +2 ,E w L − i − − k = 2 Ex i + k = 2[1 + i + k ][ r + k − ℓr − ǫ ] x i + k = ( − ℓ +1 [1 + i + k ][ k − ǫ ] w L − i − − k − ,F w Ri +2+2 k = − i ][ ǫ ] k Y s =0 [1 + i + s ][ − s − ǫ ] ! y r − − i − k = − [1 + i + k ][ − k − ǫ ]2[1 + i ][ ǫ ] k − Y s =0 [1 + i + s ][ − s − ǫ ] ! y r − − i − ( k − = − [1 + i + k ][ k + ǫ ] w Ri +2+2( k − and ( − ℓ (cid:0) w S − i + 2[1 + i ][ ǫ ] w H − i (cid:1) = ( − ℓ ( − i ][ ǫ ] x i + y r − + 4[1 + i ][ ǫ ] x i − y r − )= 2( − ℓ [1 + i ][ ǫ ] x i = 2[1 + i ][ ℓr + ǫ ] x i = E w L − i − . From the definition of w Hi − k and w Si − k it is easy to show that x k = w Hi − k + 12[1 + i ][ ǫ ] w Si − k , y r − − i + k = 2 w Si − k + 2[1 + i ][ ǫ ] w Hi − k . From this, we see that E w Hi − k = 2 Ex k − i ][ ǫ ] Ey r − − i + k = 2[ k ][1 + i − k + ℓr + ǫ ] x k − − [1 + i − k ][ k − ℓr − ǫ ]2[1 + i ][ ǫ ] y r − − i + k = 2[ k ][1 + i − k + ǫ ]( − ℓ (cid:18) w Hi − k − + 12[1 + i ][ ǫ ] w Si − k − (cid:19) − [1 + i − k ][ k − ǫ ]2[1 + i ][ ǫ ] ( − ℓ (cid:0) w Si − k − + 2[1 + i ][ ǫ ] w Hi − k − (cid:1) = (2[ k ][1 + i − k + ǫ ] − [1 + i − k ][ k − ǫ ]) ( − ℓ w Hi − k − (cid:18) [ k ][1 + i − k + ǫ ] − [1 + i − k ][ k − ǫ ][1 + i ][ ǫ ] (cid:19) ( − ℓ w Si − k − and E w Si − k = − i ][ ǫ ] Ex k + Ey r − − i + k = − i ][ ǫ ][ k ][1 + i − k + ℓr + ǫ ] x k − + [1 + i − k ][ k − ℓr − ǫ ] y r − − i + k = − i ][ ǫ ][ k ][1 + i − k + ǫ ]( − ℓ (cid:18) w Hi − k − + 12[1 + i ][ ǫ ] w Si − k − (cid:19) + [1 + i − k ][ k − ǫ ]( − ℓ (cid:0) w Si − k − + 2[1 + i ][ ǫ ] w Hi − k − (cid:1) = (2[1 + i − k ][ k − ǫ ] − [1 + i − k + ǫ ][ k ]) ( − ℓ w Si − k − + 2[1 + i ][ ǫ ] ([1 + i − k ][ k − ǫ ] − [1 + i − k + ǫ ][ k ]) ( − ℓ w Hi − k − . However, by expanding the brackets one has [ k ][1 + i − k + ǫ ] − [1 + i − k ][ k − ǫ ] = [1 + i ][ ǫ ].Hence, the above equations give w Hi − k = (2[ k ][1 + i − k + ǫ ] − [1 + i − k ][ k − ǫ ]) ( − ℓ w Hi − k − + ( − ℓ w Si − k − , w Si − k = (2[1 + i − k ][ k − ǫ ] − [1 + i − k + ǫ ][ k ]) ( − ℓ w Si − k − + 2( − ℓ +1 [1 + i ] [ ǫ ] w Hi − k − as desired. This proves that X ǫ = V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ when ǫ = 0. As ǫ →
0, it is easyto see that the action on X is exactly the action on P i ⊗ C Hℓr (see [CGP1]) by identifying w Xk ∈ X with w Xk ⊗ v ∈ P i ⊗ C Hkr (here C Hkr = C v ). Hence, we have shown X ǫ = (cid:26) V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ if ǫ = 0 P i ⊗ C Hℓr if ǫ = 0 . (cid:3) Corollary 5. lim ǫ → a AX ǫ = A lim ǫ → a X ǫ ∀ A ∈ U Hq ( sl ) Proof:
The Theorem holds for
E, F, H and K by construction of X ǫ and hence holds for allpolynomials in E, F, H and K . (cid:3) Corollary 6.
Morphisms that consist of compositions of braidings, twists, evaluations andco-evaluations commute with limits.
Proof:
This follows from the previous Theorem as braiding, twist, evaluation and co-evaluation are expressed in terms of the action of elements of U Hq ( sl ) or in the H -completionof U Hq ( sl ) (for the braiding). (cid:3) Proposition 7.
The modified quantum dimension satisfies lim ǫ → d ( X ǫ ) = d ( X ) . Proof:
Set λ = 1 + i − r , then we have d ( X ǫ ) = d ( V λ + ℓr + ǫ ⊕ V − λ + ℓr + ǫ ) = d ( V λ + ℓr + ǫ ) + d ( V − λ + ℓr + ǫ ))= ( − r − r (cid:18) { λ + ℓr + ǫ }{ r ( λ + ℓr + ǫ ) } + {− λ + ℓr + ǫ }{ r ( − λ + ℓr + ǫ ) } (cid:19) = ( − ℓ ( r − r − r (cid:18) { λ + ǫ }{ r ( λ + ǫ ) } + {− λ + ǫ }{ r ( − λ + ǫ ) } (cid:19) ( − ( ℓ +1)( r − r ( q λ + ǫ − q − ( λ + ǫ ) )( q r ( − λ + ǫ ) − q − r ( − λ + ǫ ) ) + ( q − λ + ǫ − q − ( − λ + ǫ ) )( q r ( λ + ǫ ) − q − r ( λ + ǫ ) )( q r ( λ + ǫ ) − q − r ( λ + ǫ ) )( q r ( − λ + ǫ ) − q − r ( − λ + ǫ ) ) . We have to evaluate this expression for ǫ →
0. Both the denominator and nominator vanishin this limit. It turns out that the same happens for the derivatives of both denominatorand nominator and so we have to apply the rule of L’Hˆopital twice and we get as nominator8 r ( − i − r ( q i − r + q − (1+ i − r ) ), and a denominator of 8 r . Hence,lim ǫ → d ( X ǫ ) = ( − ℓ ( r − r − r (cid:18) r ( − i − r ( q i − r + q − (1+ i − r ) )8 r (cid:19) = ( − ℓ ( r − i +1 ( q i +1 + q − i − ) = d ( P i ⊗ C Hℓr ) . (cid:3) Colored Alexander invariants.
We now apply the above construction. For this weneed to recall Section 3 of [GPT1]. Let L be a C -colored ribbon graph, such that at leastone of the colors is a simple V λ . Let T λ be the colored (1 , V λ . Then the re-normalized Reshetikhin-Turaev link invariant is F ′ ( L ) = t V λ ( T λ ) . These where shown in [GPT1] to coincide with Murakami’s Alexander invariants [Mu1]provided all colors are simple projective modules. We can now extend these results to anyprojective module. For this let L be as above but with at least one of the colors the module P i ⊗ C Hℓr . Let T P i ⊗ C Hℓr be the colored (1 , P i ⊗ C Hℓr . Let T λ be the colored (1 , P i ⊗ C Hℓr ) with V λ . Theorem 8.
The colored (1 , -ribbon graph T P i ⊗ C Hℓr satisfies t P i ⊗ C Hℓr (cid:16) T P i ⊗ C Hℓr (cid:17) = lim ǫ → (cid:0) t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) + t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) (cid:1) and T P i ⊗ C Hℓr = aId P i ⊗ C Hℓr + bx i,ℓr with coefficients a = lim ǫ → t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) = lim ǫ → t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) and b = lim ǫ → − i ][ ǫ ] (cid:18) t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) − t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) (cid:19) = r πi { i } (cid:18) ddλ t V λ ( T λ ) d ( V λ ) (cid:12)(cid:12)(cid:12) λ = ℓr + r − i − − ddλ t V λ ( T λ ) d ( V λ ) (cid:12)(cid:12)(cid:12) λ =1+ i − r + ℓr (cid:19) . Before proving this theorem, we remark that this result nicely relates to the work ofMurakami and Nagatomo on logarithmic link invariants obtained using different quantumgroups but the same R -matrix [Mu1, Mu2, MN].We also remark that T λ = t V λ ( T λ ) d ( V λ ) Id V λ . roof: For the first statement, we use the identity lim ǫ → V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ = P i ⊗ C Hℓr which gives t P i ⊗ C Hℓr (cid:16) T P i ⊗ C Hℓr (cid:17) = lim ǫ → T V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ (cid:0) T V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ (cid:1) where we pulled the limit out of the function using the fact that limits commute with thepartial trace and the action of U Hq ( sl ). The relation follows since coloring with the direct sumof two objects X ⊕ Y amounts to computing the sum of the individually colored components t X ⊕ Y ( T X ⊕ Y ) = t X ( T X ) + t Y ( T Y ) . For the second statement, since w Hi generates P i ⊗ C Hℓr , it is enough to find the action of T P i ⊗ C Hℓr on w Hi . For this we compute the action of T V i − r + ℓr + ǫ ⊕ V − − i + r + ℓr + ǫ on w Hi and thentake the limit ǫ to zero. Recall that we have w Hi = a ǫ x + b ǫ y r − − i and w Si = c ǫ x + d ǫ y r − − i where x , y r − − i , a ǫ = 2, b ǫ = − i ][ ǫ ] , c ǫ = − i ][ ǫ ], and d ǫ = 1 are as in the constructionof X ǫ . Notice that x = d ǫ w Hi − b ǫ w Si and y r − − i = a ǫ w Si − c ǫ w Hi . We can now compute theaction of T P i ⊗ C Hℓr on w Hi : T P i ⊗ C Hℓr ( w Hi ) = T P i ⊗ C Hℓr (cid:16) lim ǫ → a ǫ x + b ǫ y r − − i (cid:17) = lim ǫ → ( a ǫ T i − r + ℓr + ǫ ( x ) + b ǫ T − − i + r + ℓr + ǫ ( y r − − i ))= lim ǫ → (cid:18) a ǫ t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) ( d ǫ w Hi − b ǫ w Si )+ b ǫ t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) ( a ǫ w Si − c ǫ w Hi ) (cid:19) . It follows that a = lim ǫ → (cid:18) a ǫ d ǫ t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) − b ǫ c ǫ t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) (cid:19) = lim ǫ → (cid:18) t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) − t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) (cid:19) and b = − lim ǫ → (cid:18) a ǫ b ǫ t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) − a ǫ b ǫ t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) (cid:19) = lim ǫ → i ][ ǫ ] (cid:18) t V i − r + ℓr + ǫ ( T i − r + ℓr + ǫ ) d ( V i − r + ℓr + ǫ ) − t V − − i − r + ℓr + ǫ ( T − − i − r + ℓr + ǫ ) d ( V − − i + r + ℓr + ǫ ) (cid:19) . Evaluating the limits (using L’Hˆopital’s rule for b ) give the result. (cid:3) Recall the definition of general Hopf link invariants. Given two modules
V, W in C , defineΦ V,W = (Id W ⊗ ev ′ V ) ◦ ( c V,W ⊗ Id V ∗ ) ◦ ( c W,V ⊗ Id V ∗ ) ◦ (Id W ⊗ coev V ) ∈ End( W ) . where ev ′ V and coev V are the right evaluation and left coevaluation, respectively. Theseinvariants have been computed in [CGP1]. If W is simple then this computation is relativelystraight forward as one only needs to know the action on a highest-weight state. One gets or example Φ V α ,V β = { rβ }{ β } q αβ Id V β , Φ S i ⊗ C Hkr ,V β = { ( i + 1) β }{ β } q krβ Id V β , Φ P i ⊗ C Hkr ,V β = { rβ }{ β } q krβ (cid:0) q ( r − − i ) β + q − ( r − − i ) β (cid:1) Id V β . (6)Also note that d ( V β ) = { β }{ rβ } ( − r − r . If W is not simple the computation is more in-volved and [CGP1] needs tensor product decomposition. Using above theorem and (6) weimmediately get Corollary 9.
The colored Hopf links Φ Z,P j with Z ∈ { S i ⊗ C Hkr , V α , P i ⊗ C Hkr } are given by Φ Z,P j ⊗ C Hℓr = a Z Id P j ⊗ C Hℓr + b Z x j,ℓ , where a P i ⊗ C Hkr = a V α = 0 and a S i ⊗ C Hkr = ( − i + ℓi + kℓr { ( i + 1)( j + 1) }{ j + 1 } ,b S i ⊗ C Hkr = ( − i + ℓi + kℓr i { ( i + 2)( j + 1) } − ( i + 2) { i ( j + 1) } [ j + 1] { j + 1 } ,b V α = q αℓr ( − r − j r [ j + 1] ( q ( r − − j ) α + q − ( r − − j ) α ) ,b P i ⊗ C Hkr = 2 r ( − i + ℓi + kℓr [ j + 1] ( q ( i +1)( j +1) + q − ( i +1)( j +1) ) . Open Hopf links and asymptotic dimensions
We will see that continuous quantum/asymptotic dimensions of the singlet algebra cor-respond to semi-simple parts of logarithmic Hopf links of U Hq ( sl ) and the discrete onescorrespond to the nilpotent ones; thus giving a categorical interpretation of asymptotic di-mensions as suggested by [CM1, CMW] (see also [BFM] for another derivation of asymptoticdimensions). Proposition 10.
The quantum dimensions of the typical and atypical modules for the sin-glet vertex algebra are in agreement with the modified traces of their corresponding U Hq ( sl ) modules in the following sense: Let β ∈ ¨ C and i, j ∈ { , . . . , r − } and k, k ′ ∈ Z then for Re ( ǫ ) > B rǫ and α = √ riǫ qdim[ F − iα √ rβ + r − √ r ] = t V α (Φ V β ,V α ) t V α (Φ S ,V α ) , qdim[ M − iα √ r − k,j +1 ] = t V α (Φ S j ⊗ C Hkr ,V α ) t V α (Φ S ⊗ V α ) , and for Re ( ǫ ) < B rǫ and ǫ ∈ S ( k, j + 1 + r ( k + 1)) then qdim[ F ǫ β + r − √ r ] = t P j ⊗ C Hkr (Φ V β ,P j ⊗ C Hkr ◦ x j,k ) t P j ⊗ C Hkr (Φ S ,P j ⊗ C Hkr ◦ x j,k ) , qdim[ M ǫ − k ′ ,i +1 ] = t P j ⊗ C Hkr (Φ S i ⊗ C Hk ′ r ,P j ⊗ C Hkr ◦ x j,k ) t P j ⊗ C Hkr (Φ S ,P j ⊗ C Hkr ◦ x j,k ) . roof: The proof follows directly by comparing Corollary 9 and the quantum dimensionslisted in section 2, namelyqdim[ F − iα √ rβ + r − √ r ] = e π − iα √ r (2 β + r − √ r − α ) ( e − π √ r − iα √ r i − e π √ r − iα √ r i )( e − π √ √ r − iα √ r i − e π √ √ r − iα √ r i ) = e − πiαβr ( e − πiαrr − e πiαrr )( e − πiαr − e πiαr )= q αβ { rα }{ α } = t V α (Φ V β ,V α ) t V α (Φ S ,V α ) , qdim[ M − iα √ r − k,j +1 ] = e πk √ r − iα √ r ( e − π ( j +1) √ √ r − iα √ r i − e π ( j +1) √ √ r − iα √ r i )( e − π √ √ r − iα √ r i − e π √ √ r − iα √ r i )= e − πkiα ( e − π ( j +1) iαr − e π ( j +1) iαr )( e − πiαr − e πiαr ) = q rkα { ( j + 1) α }{ α } = t V α (Φ S j ⊗ C Hkr ,V α ) t V α (Φ S ⊗ V α )and for Re ( ǫ ) < B rǫ together with ǫ ∈ S ( k, j + 1 + r ( k + 1))qdim[ M ǫ − k ′ ,i +1 ] = ( − ( j + kr +1 − r ) k ′ { ( i + 1)( j + 1 + r ( k + 1)) }{ j + 1 + r ( k + 1) } = ( − i ( k +1)+( j + kr +1 − r ) k ′ { ( i + 1)( j + 1) }{ j + 1 } = t P j ⊗ C Hkr (Φ S i ⊗ C Hk ′ r ,P j ⊗ C Hkr ◦ x j,k ) t P j ⊗ C Hkr (Φ S ,P j ⊗ C Hkr ◦ x j,k ) . Finally in this region both qdim[ F ǫ β + r − √ r ] and t Pj ⊗ C Hkr (Φ Vβ,Pj ⊗ C Hkr ◦ x j,k ) t Pj ⊗ C Hkr (Φ S ,Pj ⊗ C Hkr ◦ x j,k ) vanish. (cid:3) Corollary 11.
Let α ∈ ¨ C . Then the map ϕ : V α F α + r − √ r , S i ⊗ C Hkr M − k,i +1 extendedlinearly over direct sums for α ∈ ¨ C is a morphism up to equality of characters. Proof:
This can be directly verified via computation. It however also follows since the Hopflinks as well as the asymptotic dimensions uniquely specify the (conjectured) tensor ring upto equality of characters; see Theorem 28 of [CM1] and [CGP1]. (cid:3) Towards braided tensor category structure on M ( r ) -Mod C -cofiniteness. In this section we obtain sufficient conditions for the existence of abraided tensor category (and more) structure on a suitable category of M ( r )-modules. Inother words, we discuss applicability of the Huang-Lepowsky-Zhang tensor product theory[HLZ] to M ( r )-Mod. We also comment on the r = 2 case due to recent rigorous derivationof the fusion ring for M (2) [AM3]. For r ≥
3, this ring is known only conjecturally [CM1].We believe that the powerful technique from [AM3] can be extended to all r .Recall that a V -module M is C -cofinite if the subspace C ( M ) := h v − m : v ∈ V, wt( v ) > , m ∈ M i is of finite-codimension in M . We shall focus on the category of finite-length C -cofinitemodules, denoted by M ( r )-Mod. Observe that this category is closed under finite prod-ucts/coproducts and taking quotients. So it has an abelian category structure and thus weare in the position to apply parts of [HLZ] theory. The next result is needed in the proof ofthe main result. emma 12. Let L ( c, h ) be a non-generic (i.e. not isomorphic to the Verma module) Virasoromodule for the Virasoro vertex operator algebra L ( c, . Then L ( c, h ) is C -cofinite, viewedas an L ( c, -module. Proof:
By definition, every non-generic Verma module admits a singular vector 0 = w ∈ M ( c, h ) of weight n (depending on c and h ). From the structure of singular vectors for theVirasoro algebra (see Chapter 5, [IK]) we get a decomposition w = L ( − n v c,h + w ′ v c,h where v c,h is the highest weight vector in L ( c, h ) and w ′ ∈ U (Vir < ), which is lower in thefiltration than L ( − n . Clearly, we have w ′ v c,h ∈ C ( L ( c, h )). We claim that L ( c, h ) = C ( L ( c, h )) + n − X i =0 L ( − i v c,h . Denote the right hand-side by U . Clearly we only have to prove that L ( − i v c,h ∈ U for i ≥
0. This follows immediately because of L ( − C ( L ( c, h ))) ⊂ C ( L ( c, h )). (cid:3) Theorem 13.
All irreducible M ( r ) -modules are C -cofinite. Proof:
We first consider atypical modules. It is known that M t,s = ∞ M n =0 L Vir ( c r, , h t,sn ) , where L V ir ( c r, , h t,sn ) are certain irreducible Virasoro modules of central charge c r, ; for explicitformulas see for instance [AM1, AM2, CM1]. These singlet modules can be realized insidethe lattice vertex algebra modules by using the short screening Q = e α acting on specialhighest weight vectors e γ inside the generalized lattice vertex algebra V L ′ . More precisely,we have M t,s = ∞ M n =0 U (Vir < ) .v ( n ) = ∞ M n =0 U (Vir < ) .Q n e β t,s − nα , β t,s ∈ L ′ . Claim:
For every n ≥
1, there is a nonzero constant C and j ≤ − n ) suchthat H j v ( n ) = Cv ( n +1) + w, where w ∈ V n := n M i =0 U (Vir < ) .v ( i ) . This claim follows along the lines of Lemma 5.3 in [AM1].It is now sufficient to show that M t,s = C ( M t,s ) + k t,s M i =0 L ( − i v (0) , (7)where k t,s ∈ N depend only on L ( c r, , h t,s ). Notice that this follows from Lemma 12 once weprove that H − i · · · H − i k L − j · · · L − j ℓ v (0) , (8) m ≥
1, 1 ≤ m ≤ k , j n ≥
1, 1 ≤ n ≤ ℓ , is a spanning set for M t,s . Let V n be defined as above.We shall prove by induction on n ≥
0, that V n is spanned by vectors in (8). For n = 0, thisfollows from Lemma 12. Now the above claim together with the inductive hypothesis implythat V n +1 is spanned by vectors of the form L − k · · · L − k p H − i · · · H − i k L − j · · · L − j ℓ v (0) . By using the bracket relations among L i and H j , we can move all Virasoro generators thatare on the left across the H -generators to the right. The resulting vector is clearly insidethe span of vectors in (8).Now we switch to typical modules F λ , λ / ∈ L ′ . At first we let λ to be arbitrary. Wealready know that all atypicals are C -cofinite. In particular, this family includes infinitelymany Fock spaces F λ , λ ∈ L . We will need the following two known facts. Fact 1. If A = C \ B , where B is a countable set, then any complex number can be written as a finitesum of elements in A . Fact 2 (Miyamoto [Miy1]) Suppose that M and N are C -cofinite and Y ∈ (cid:0)
WM N (cid:1) is surjective, then W is also C -cofinite. We shall apply the latter for M = F λ , N = F λ ′ and W = F λ ′ + λ , with Y the obvious Heisenberg VOA intertwining operator Y acting among three modules.Now we proceed with the proof. Clearly, it is sufficient to show that F λ is C -cofinite for λ ∈ B , where B is as above. Let F λ = ∞ M n =0 F λ ( n ) , graded space decomposition of F λ . Then F λ being C -cofinite means that F λ ( n ) ⊆ C ( F λ ) , n ≥ k, for some fixed k ∈ N . Consider C F λ ( n ) := C ( F λ ) ∩ F λ ( n ) . Because C ( F λ )( n ) depends polynomially on λ (due to the fact that it picks up λ when we actwith the Heisenberg generator ϕ (0) on v λ ) there will be only finitely many λ values for which dim ( C ( F λ )( n )) will drop ( non-generic values) and for all other values dim ( C ( F λ )( n )) willbe constant ( generic values). We know that there is λ for which F λ is C -cofinite and thus dim ( C ( F λ )( n )) = p ( n ), here p ( n ) is the number of partitions of n , for all n ≥ k λ . For every n ≥ k λ , denote by B i the set of all non-generic λ -values for dim ( C ( F λ )( i )) < p ( i ) (this setis always finite). We let A := C \ ∞ [ i = k λ B i and B = ∞ S i = k λ B i is a desired countable set. The proof follows. (cid:3) Corollary 14.
All finite-length M ( r ) -modules are C -cofinite. Remark 15.
It is not hard to see that F λ is not C -cofinite if viewed as a module for theVirasoro vertex operator algebra. .2. Fusion rules.
Formula (1), when specialized to p = 2, gives the following conjecturalrelations in the Grothendieck ring (here λ, µ / ∈ L ′ ):[ F λ ] × [ F µ ] = [ F λ + µ ] + [ F λ + µ + α − ] , (9)[ M t, ] × [ F µ ] = [ F µ + α r, ] , (10)[ M t, ] × [ M ǫt ′ , ] = [ M t + t ′ − , ] , (11)[ M ǫt, ] × [ M ǫt ′ , ] = [ M ǫt + t ′ − , ] , (12)[ M ǫt, ] × [ M ǫt ′ , ] = 2[ M ǫt + t ′ − , ] + [ M ǫt + t ′ − , ] + [ M ǫt + t ′ , ] (13)= [ F α t + t ′− , ] + [ F α t + t ′− , ] . Next we discuss rigorous results pertaining to fusion rules of irreducible M (2)-modules.As in [AM3], for a triple of equivalence classes of irreducible M (2)-modules, [ M ], [ N ] and[ K ], where M , N and K are representatives of classes, respectively, we define[ M ] × [ N ] := X [ K ] ∈ Irrep dim I (cid:18) KM N (cid:19) [ K ] , (14)where Irrep denotes the set of all equivalence classes of irreducible modules. In [AM3], D.Adamovi´c and the second author essentially proved the following result, thus verifying thecorrectness of fusion rules formulas obtained conje[CM1] for r = 2: Theorem 16.
We have:(i) All typical modules M t, are simple currents, in the sense that for a given irreduciblemodule N there is a unique irreducible module M such that I (cid:0) MM ,t N (cid:1) is nontrivial andone-dimensional. Moreover, under the product (14), formula (9) holds if λ + µ / ∈ L ′ , andformulas (10)-(12) also hold.(ii) If λ + µ ∈ L ′ , then there exists a logarithmic module P such that in the Grothendieckgroup [ P ] = [ F λ + µ ] + [ F λ + µ + α − ] and I (cid:0) PF λ F µ (cid:1) = 0 .(iii) There exists a logarithmic module Q , with [ Q ] = 2[ M ǫt + t ′ − , ] + [ M ǫt + t ′ − , ] + [ M ǫt + t ′ , ] suchthat I (cid:0) QM t, M t ′ , (cid:1) = 0 . Parts (i) and (ii) are already proven in [AM3] in the setup of lattice vertex algebras.Part (iii) can be also proven by using [AM3] (additional details will appear elsewhere).Alternatively, we can also argue as follows. As we know, the r = 2 case corresponds to asubalgebra of the rank one symplectic fermions. Logarithmic intertwining operator basedon symplectic fermions modules was constructed in [Ru], albeit in a slightly different setup.By virtue of restriction to the singlet algebra modules we obtain a family of (logarithmic)intertwining operators, leading to (iii).6.3. Tensor category structure.
According to [HLZ], in order for a suitably (sub)categoryof V -modules C to have the structure of a braided tensor category, it suffices that assumptions10.1,12.1 and 12.2 in [HLZ] hold. Let C denote the category of N -gradable C -cofinite M ( r )-modules. By [Miy1] any such module is logarithmic, that is its graded componentsdecompose into generalized spaces with respect to the Virasoro generator L (0). Denote itsfull subcategory of finite-length (1 , r )-modules by C fin . Now we find sufficient conditions forapplicability of the HLZ-tensor product theory. ssumptions 10.1 (i)-10.1 (v) are satisfied in both categories. Assumptions 12.1 and 12.2are also satisfied if every finitely generated N -graded, generalized module is C -cofinite (see[HLZ] for details), which we assume below. It remains to analyze:Assumption 10.1 (vi): For any object of C , the (generalized) weights are real numbers andin addition there exists K ∈ Z + such that ( L (0) − L (0) ss ) K = 0 on the generalized module.Notice that this condition holds in C fin . Indeed, a finite-length module M must satisfy thiscondition because of K ≤ ℓ ( M ), where ℓ ( M ) is the length of M . This condition might nothold in C ; see however below.Assumption 10.1 (vii): C is closed under images, under the contragredient functor, undertaking finite direct sums, and under P ( z )-tensor products for some z ∈ C × . Because the dualof a finite length modules is of finite length, C fin is closed under the contragredient functor.However, in the category C , it is a priori not clear that the dual of a module remains C -cofinite. Finally, we have to check whether our categories are closed under the P ( z )-tensorproduct. For the category C , this is implicitly verified in the paper of Miyamoto (see MainTheorem in [Miy1]), albeit in a slightly different formulation. But for C fin , it is not clearwhether their tensor product (which exists) is still in C fin . Based on what we already knowabout C we believe that Theorem 17.
Assume that C = C fin , that is, every C -cofinite N -graded module is of finitelength and that every finitely generated generalized, N -graded M ( r ) -module is C -cofinite.Then, by virtue of [HLZ] , the category C can be equipped with a braided tensor structure. Remark 18.
It is perhaps unclear why the Heisenberg vertex algebra has not been muchdiscussed in the context of tensor categories. For one, this vertex algebras is much easier tostudy. The reason is that the category of F -modules is much less interesting compared tothe singlet algebra and in addition does not produce any non-trivial quantum invariants. Thecategory of modules for the Heisenberg vertex algebra F that are diagonalizable under thezero mode subalgebra h is known to be semisimple. Intertwining operators, tensor productand associativity in the sense of [HLZ] can be easily verified and all irreducible modulesare clearly C -cofinite, see [CKLR] for details. There is an enlargement of this categoryby inclusion of generalized (or logarithmic) F -modules [Mi]. (Logarithmic) intertwiningoperators of logarithmic F -modules are completely classified and explicitly constructed in[Mi] (see also [Ru]). Now we consider the subcategory of finite-length F -modules. It is nothard to see that any finitely generated, generalized N -graded F -module is of finite-lenght.By combining results of [Mi, Ru] with [HLZ], we infer that this category is indeed braided.Again, this new category is not terribly interesting; for instance, as a tensor category, it isequivalent to the category of finite-dimensional h -modules equipped with the usual tensorproduct. Of course, it would be desirable to work out complete details even in the Heisenbergcase, but that is outside the scope of the present paper (see however [Ru]). References [AC] C. Alfes and T. Creutzig, The mock modular data of a family of superalgebras. Proc. Amer. Math.Soc. 142 (2014), no. 7, 2265-2280.[AD] D. Adamovi´c, Classification of irreducible modules of certain subalgebras of free boson vertexalgebra. Journal of Algebra, 270(1), 115–132.[AM1] D. Adamovi´c and A. Milas , Logarithmic intertwining operators and W (2 , p − AM2] D. Adamovi´c, and A. Milas, On the triplet vertex algebra W ( p ), Adv. Math.
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Rupert, Master Thesis at the University of Alberta.[T] V. G. Turaev, Quantum Invariants of Knots and 3-manifolds, Volume 18 of De Gruyter studies inmathematics. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton,Alberta T6G 2G1, Canada. emails: [email protected], [email protected]
Department of Mathematics and Statistics, SUNY-Albany, 1400 Washington Avenue, Al-bany, NY 12222, USA. email: [email protected]: [email protected]