Non-weight modules over the affine-Virasoro algebra of type A 1

In this paper, we study a class of nonweight modules over the affine-Virasoro algebra of type A1, which are free modules of rank one when restricted to the universal enveloping algebra of the Cartan subalgebra (modulo center). We give the classification of such modules. Moreover, the simplicity and the isomorphism classes of these modules are determined.