ar X iv : 1 20 1 . 04 19 v 3 [ he p - th ] 2 1 O ct 2 01 3 The tensor structure on the representation category of the W p triplet algebra

We study the braided monoidal structure that the fusion product induces on the abelian category Wp-mod, the category of representations of the triplet W -algebra Wp. The Wp-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of Wp-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of Wp-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective Wp-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel. Email: [email protected] Email: [email protected]