Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V . We employ tensor categories to study untwisted (also called local) Amodules in C, using results of Huang-Kirillov-Lepowsky that show that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang.Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V -modules to A-modules is a vertex tensor functor. Two applications are given:First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z-graded components, respectively.Second, we analyze parafermionic cosets C = Com(VL, V ) where L is a positive definite even lattice and V is regular. If the vertex tensor category of either V -modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples. * T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460).

Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com (H,V) be the coset of H in V. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational and C 2 -cofinite and CFT-type, and Com (C,V) is a rational lattice vertex operator algebra, then so is C. These results are illustrated with many examples and the C 1 -cofiniteness of certain interesting classes of modules is established. is exact, for M ′′ ∼ = (J ⊠ M)/M ′ = 0. But, fusion is right-exact [HLZ, Prop. 4.26], sois exact. However, M ′′ = 0 implies that J −1 ⊠ M ′′ is a non-zero quotient of M, by (1), so we must have J −1 ⊠ M ′′ ∼ = M, as M is simple. Fusing with J now gives J ⊠ M ∼ = M ′′ , so we conclude that M ′ = 0 and that J ⊠ M is simple. The simplicity of J ∼ = J ⊠ V now follows from that of V, completing the proof of (3).To prove (4), note that applying right-exactness to the short exact sequence 0where f is the induced map from J ⊠ M ′ to J ⊠ M that might not be an inclusion. Fusing with J −1 and applying (2.4), we arrive at 5)

A. We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the principally specialized characters of certain level 2 modules for the affine Lie algebra A (2) 9 . Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic sum-sides which we believe to be new. All of the new conjectures presented in this article are supported by a strong mathematical evidence.

The Rogers-Ramanujan identities and various analogous identities (Gordon, Andrews-Bressoud, Capparelli, etc.) form a family of very deep identities concerned with integer partitions. These identities (written in generating function form) are typically of the form "product side" equals "sum side," with the product side enumerating partitions obeying certain congruence conditions and the sum side obeying certain initial conditions and difference conditions (along with possibly other restrictions). We use symbolic computation to generate various such sum sides and then use Euler's algorithm to see which of them actually do produce elegant conjectured product sides. We not only rediscover many of the known identities but also discover some apparently new ones, as conjectures supported by strong mathematical evidence.

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that V ⊕ J is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are C2-cofinite and nonrational are then given and induced modules listed.• Which generalized modules 1 of V lift to those of the extension V e ?We will use our answers to these questions to construct three new families of C 2 -cofinite VOAs together with all modules that lift at the end of this work. Our main intention is however to find genuinely new C 2 -cofinite VOAs, that is VOAs that are not directly related to the well-known triplet VOA. For example consider the following diagramHere A k is a family of VOAs with one-dimensional associated variety [A1] containing a Heisenberg sub VOA H and having a simple current J of infinite order. Our picture is that the Heisenberg coset Com (H, A k ) still has a one-dimensional associated variety while the extension E k only has finitely many irreducible objects. Especially Com (H, E k ) is our candidate for new C 2 -cofinite VOAs. A natural example is. . } as it has one-dimensional variety of modules [AdM3]. In [CR1, CR2] it is conjectured that these VOAs allow for infinite order simple current extensions E k , which then would only have finitely many simple modules. These VOAs would be somehow unusual as they would not be of CFT-type, and these VOAs are not our final goal, but rather Com (H, E k ). In the example of k = −1/2 this construction would just yield W(2) and in the case of k = −4/3 it would just give W(3) [Ad1, CRW, R2]. In all other cases we expect new C 2 -cofinite VOAs. Another potential candidate is the Bershadsky-Polyakov algebra whose Heisenberg coset is studied in [ACL]. In a subsequent work, we will thus develop general properties of Heisenberg cosets beyond semi-simplicity [CKLR]. This work will rely on our findings here and will then further be used for interesting examples.In the present work, we will translate results into the corresponding statements in a braided tensor category using the theory of [KO, HKL]. The advantage is that the categorical picture is much better suited for proving properties of the representation category. For us it provides a very nice way to understand the problem of the two questions: Does a module lift to a module of the extended VOA? Is the extension a VOA or a super VOA? The work [KO] assumes categories to be semi-simple and focuses on algebras with trivial twist....

We construct the unique two-parameter vertex algebra which is freely generated of type W(2, 4, 6, . . . ), and generated by the weights 2 and 4 fields. Subject to some mild constraints, all vertex algebras of type W(2, 4, . . . , 2N ) for some N , can be obtained as quotients of this universal algebra. This includes the B and C type principal W-algebras, the Z 2 -orbifolds of the D type principal W-algebras, and many others which arise as cosets of affine vertex algebras inside larger structures. As an application, we classify all coincidences among the simple quotients of the B and C type principal W-algebras, as well as the Z 2 -orbifolds of the D type principal W-algebras. Finally, we use our classification to give new examples of principal W-algebras of B, C, and D types, which are lisse and rational.Quotients of W ev (c, λ) and the classification of vertex algebras of type W(2, 4, . . . , 2N). W ev (c, λ) has a conformal weight gradingThere is a symmetric bilinear form on W ev (c, λ)[n] given byThe level n Shapovalov determinant det n ∈ C[c, λ] is just the determinant of this form. It turns out that det n is nonzero for all n; equivalently, W ev (c, λ) is a simple vertex algebra over C [c, λ]. Let p be an irreducible factor of det 2N +2 and let I = (p) ⊆ C[c, λ] ∼ = W ev (c, λ)[0] be the ideal generated by p. Consider the quotient W ev,I (c, λ) = W ev (c, λ)/I · W ev (c, λ),where I · W ev (c, λ) is the vertex algebra ideal generated by I. This is a vertex algebra over the ring C[c, λ]/I, which is no longer simple. It contains a singular vector ω in weight 2N + 2, that is, a nonzero element of the maximal proper ideal I ⊆ W ev,I (c, λ) graded by conformal weight. If p does not divide det m for any m < 2N + 2, ω will have minimal 3

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A. We study the minimal models associated to osp(1 |2), otherwise known as the fractional-level Wess-Zumino-Witten models of osp(1 |2). Since these minimal models are extensions of the tensor product of certain Virasoro and sl 2 minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of osp(1 |2). In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles. IThis project is part of a programme to understand the admissible-level Wess-Zumino-Witten (WZW) models for a Lie algebra or superalgebra g. While the theories with non-negative integer levels and simple Lie algebras lead to rational conformal field theories and, as such, are very well understood, the situation is much more complicated and rich for other levels or when superalgebras are involved. Indeed, the non-rational admissible-level WZW models are expected to be prime examples of logarithmic conformal field theories, these being models that admit representations on which the hamiltonian acts non-diagonalisably, leading to correlation functions with logarithmic singularities.Another interesting feature of these models is that they have a continuous spectrum of modules.We view our programme as complementary to older approaches. In particular, Quella, Saleur, Schomerus et al.[1-9] approached supergroup WZW theories via free field realisations and semiclassical limits (the minisuperspace analysis), the interest being rather in features of the WZW theory of the supergroup at integer levels. Another approach employed was to learn more about the conformal field theory using the mock modular behaviour of certain irreducible characters [10][11][12]. The relatively accessible case of g = gl(1|1) has also been studied from a more algebraic perspective by two of us [13,14].Presently, we have a very good picture in the case of g = sl 2 [15][16][17][18][19][20][21][22][23]. In order to extend our understanding to more sophisticated theories, one has to develop some basic strategies. First, one has to study the general theory of relaxed highest-weight modules. These natural generalisations of the usual highest-weight modules were introduced in the conformal field theory literature in [24] for g = sl 2 , though they had already appeared in mathematics classifications such as [15], but have only recently been formalised in a general setting [25]. Since then, the role played by irreducible relaxed highest-weight modules in facilitating the study of general admissible-level WZW models has been widely appreciated and the field has been rapidly developing, see [26][27][28][29] for example.Second, one should develop techniques to reconstruct, at least in favourable circumstances, the representation theory of the algebra of interest in terms of those of subalgebras. We call this techniq...