On fusion rules in logarithmic conformal field theories

We find the fusion rules for the cp,i series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c_3p,3. This leads to the conjecture that (almost) all minimal models with c = c_p,q, gcd(p,q) > 1, belong to the class of rational logarithmic conformal field theories.