Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\mathrm{\ell }$. The aim of the present work is to investigate the lowest weight representations of CGA with $d=1$ for any integer value of $\mathrm{\ell }$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\mathrm{\ell }=1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\mathrm{\ell }\mathrm{\ne }1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.