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.