Systems of correlation functions, coinvariants and the Verlinde algebra

We study the Gaberdiel-Goddard spaces of systems of correlation functions attached to an affine Kac-Moody Lie algebra $\text{gh}$. We prove that these spaces are isomorphic to the spaces of coinvariants with respect to certain subalgebras of $\text{gh}$. This allows to describe the Gaberdiel-Goddard spaces as direct sums of tensor products of irreducible $\text{g}$-modules with multiplicities given by fusion coefficients. We thus reprove and generalize Frenkel-Zhu’s theorem.