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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 $\gh$. We prove that these spaces are isomorphic to the spaces of coinvariants with respect to certain subalgebras of $\gh$. This allows to describe the Gaberdiel-Goddard spaces as direct sums of tensor products of irreducible $\g$-modules with multiplicities given by fusion coefficients. We thus reprove and generalize Frenkel-Zhu's theorem.