REALIZATION OF ${\widehat{\mathfrak{sl}}_2({\mathbb C})}$ AT THE CRITICAL LEVEL

An explicit realization of the affine Lie algebra at the critical level is constructed using a mixture of bosons and parafermions. Subsequently, a representation of the associated Lepowsky–Wilson Z-algebra is given on a space of the tensor product of bosonic fields and certain semi-infinite wedge products.