Decomposition of ${\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,k}\oplus {\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,1}$ highest weight representations for generic level $k$ and equivalence between two dimensional CFT models

We construct highest weight vectors of ${\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,k+1}\oplus \mathsf{V}\mathsf{i}\mathsf{r}$ in tensor products of highest weight modules of ${\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,k}$ and ${\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,1}$, and thus for generic weights we find the decomposition of the tensor product into irreducibles of ${\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,k+1}\oplus \mathsf{V}\mathsf{i}\mathsf{r}$. The construction uses Wakimoto representations of ${\widehat{{\mathfrak{s}\mathfrak{l}}_2}}_{,k}$, but the obtained vectors can be mapped back to Verma modules. Singularities of this mapping are cancelled by a renormalization. A detailed study of ``degenerations’’ of Wakimoto modules allowed to find the renormalization factor explicitly. The obtained result is a significant step forward in a proof of equivalence of certain two-dimesnional CFT models.