Representations of a_∞ and d_∞ with central charge 1 on the single neutral fermion Fock space F⊗½

We construct a new representation of the infinite rank Lie algebra a_∞ with central charge c = 1 on the Fock space ⊗½ of a single neutral fermion. We show that ⊗½ is a direct sum of irreducible integrable highest weight modules for a_∞ with central charge c = 1. We prove that as a_∞ modules ⊗½ is isomorphic to the Fock space F¹ of the charged free fermions. As a corollary we obtain the decompositions of certain irreducible highest weight modules for d_∞ with central charge c = ½ into irreducible highest weight modules for d_∞ with central charge c = 1.