Systems of Correlation Functions, Coinvariants, and the Verlinde Algebra

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