In this paper, under the assumption that the diagonal coset vertex operator algebra $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is rational and $C_2$-cofinite, the global dimension of $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is obtained, the quantum dimensions of multiplicity spaces viewed as $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$-modules are also obtained. As an application, a method to classify irreducible modules of $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is provided... (read more)

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