Fusion Rules of the ${\cal W}_{p,q}$ Triplet Models

In this paper we determine the fusion rules of the logarithmic ${\calW}_{p,q}$ triplet theory and construct the Grothendieck group with subgroups for which consistent product structures can be defined. The fusion rules are then used to determine projective covers. This allows us also to write down a candidate for a modular invariant partition function. Our results demonstrate that recent work on the ${\cal W}_{2,3}$ model generalises naturally to arbitrary (p,q).