The tensor structure on the representation category of the W_p triplet algebra

Physics – High Energy Physics – High Energy Physics - Theory

Scientific paper

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56 pages; edit: fixed minor typos and some formatting

Scientific paper

We study the braided monoidal structure that the fusion product induces on the abelian category W_p-mod, the category of representations of the triplet W-algebra W_p. The W_p-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 that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We combine these methods with the general theory of braided monoidal categories to prove that W_p-mod is a rigid braided monoidal category and that therefore the fusion product is bi-exact. The rigidity of W_p-mod allows us to provide explicit formulae for the fusion product on the set of all simple and all projective W_p-modules.

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