A new class of simple smooth modules over the affine algebra A1(1)

In this paper, we construct a new class of simple smooth modules over the affine algebra A1(1). Specifically, we present a family of sl(2)ˆ-modules Wφ for functions φ which are not Whittaker functions, and provide a simplicity criterion as well as an isomorphism theorem for such modules. For each simple sl(2)ˆ-module Wφ, a class of simple sl(2)˜-module Wφ,γ is also constructed for any constant γ. The sl(2)ˆ-modules Wφ are weight modules, while the sl(2)˜-modules Wφ,γ are not.