  We study irreducible representations of two classes of conformal Galilei
algebras in 1-spatial dimension. We construct a functor which transforms simple
modules with nonzero central charge over the Heisenberg subalgebra into simple
modules over the conformal Galilei algebras. This can be viewed as an analogue
of oscillator representations. We use oscillator representations to describe
the structure of simple highest weight modules over conformal Galilei algebras.
We classify simple weight modules with finite dimensional weight spaces over
finite dimensional Heisenberg algebras and use this classification and
properties of oscillator representations to classify simple weight modules with
finite dimensional weight spaces over conformal Galilei algebras.

Representation Theory