Modules over the algebra Vir(a,b)

For any two complex numbers a and b, Vir(a,b) is a central extension of W(a,b) which is universal in the case (a,b)≠(0,1), where W(a,b) is the Lie algebra with basis {Ln,Wn|n∈Z} and relations [Lm,Ln]=(n−m)Lm+n, [Lm,Wn]=(a+n+bm)Wm+n, [Wm,Wn]=0. In this paper, we construct and classify a class of non-weight modules over the algebra Vir(a,b) which are free U(CL0⊕CW0)-modules of rank 1. It is proved that such modules can only exist for a∈Z.