Modules over the algebra [formula omitted].

For any two complex numbers a and b , V i r ( 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 { L n , W n | n ∈ Z } and relations [ L m , L n ] = ( n − m ) L m + n , [ L m , W n ] = ( a + n + b m ) W m + n , [ W m , W n ] = 0 . In this paper, we construct and classify a class of non-weight modules over the algebra V i r ( a , b ) which are free U ( C L 0 ⊕ C W 0 ) -modules of rank 1. It is proved that such modules can only exist for a ∈ Z . [ABSTRACT FROM AUTHOR]