Free field realizations of the elliptic affine Lie algebra sl(2,R)⊕(ΩR/dR)

In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2,R)⊕ΩR/dR where R=C[t,t−1,u∣u2=t3−2bt2+t]. The first realization provides an analogue of Wakimoto's construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new types of representations analogous to Imaginary Verma modules in the Affine setting.