Non-weight modules over algebras related to the Virasoro algebra

In this paper, we study some non-weight modules over the loop-Virasoro algebra L and Block type Lie algebras B(q), where q is a nonzero complex number. Those modules when restricted to the Cartan subalgebra (modulo center) are free of rank one. We provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules.