We show that the Kazhdan-Lusztig category KL_k of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1)ˆ has vertex algebraic braided tensor supercategory structure, and that its full subcategory O_k^fin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)ˆ-module in KL_k has a projective cover in O_k^fin, and we determine all fusion rules involving simple and projective objects in O_k^fin. Then using Knizhnik-Zamolodchikov equations, we prove that KL_k and O_k^fin are rigid. As an application of the tensor supercategory structure on O_k^fin, we study certain module categories for the affine Lie superalgebra sl(2|1)ˆ at levels 1 and −1/2. In particular, we obtain a tensor category of sl(2|1)ˆ-modules at level −1/2 that includes relaxed highest-weight modules and their images under spectral flow.

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This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.

For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

Nous montrons que le morphisme de Frobenius quantique construit par Lusztig dans le cadre des algèbres enveloppantes quantiques $U_{\mathcal{B}}$ spécialisées en une racine de l’unité admet un scindage multiplicatif (non unitaire). Nous utilisons pour ce faire une base de la partie torique de la petite algèbre quantique constituée d’idempotents orthogonaux deux à deux et de somme 1 et faisons de même dans le cas «modulaire» pour l’algèbre des distributions d’un groupe algébrique semi-simple.