Representations of affine Nappi-Witten Lie algebras over polynomial algebras

In this paper, we study the representation theory of affine Nappi-Witten Lie algebra H4̂ corresponding to the Nappi-Witten Lie algebra H4. We completely classify Cartan-free modules of rank one for the Nappi-Witten Lie algebra H4. With the help of Cartan free H4 modules we classify Cartan-free modules of rank one over affine Nappi Witten Lie algebras. We also give a necessary and sufficient condition for these modules to be irreducible. Finally as an application we classify Cartan free modules of rank one for affine-Virasoro Nappi-Witten Lie algebras.