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Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to $C_k$, all irreducible $C_k$- and $B_k$-modules are obtained from those of $L_k(\\mathfrak{sl}_2)$, as are the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible $B_k$-modules. 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