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

The ℓ-conformal Galilei algebra, denoted by gl(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 gl(d) with a central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of gl(d) and vector field representations for d = 1, 2. (paper)