Quasimodular forms and Jacobi-like forms

We study various properties of quasimodular forms by using their connections with Jacobi-like forms. Such connections are made by identifying quasimodular forms for a discrete subgroup of with certain polynomials over the ring of holomorphic functions of the Poincar, upper half plane that are Gamma-invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms.