Akihiro Tsuchiya, Simon Wood
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.
View original:
http://arxiv.org/abs/1201.0419
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