Intertwining operators for l-conformal Galilei algebras and hierarchy of invariant equations

The l-conformal Galilei algebra, denoted by g(l) (d), is a non-semisimple Lie algebra specified by a pair of parameters (d, l). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by g(l) (d) with a central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g(l) (d) and vector field representations for d = 1, 2.