  We give a review of some group-theoretical results related to
non-relativistic holography. Our main playgrounds are the Schr\"odinger
equation and the Schr\"odinger algebra. We first recall the interpretation of
non-relativistic holography as equivalence between representations of the
Schr\"odinger algebra describing bulk fields and boundary fields. One important
result is the explicit construction of the boundary-to-bulk operators in the
framework of representation theory, and that these operators and the
bulk-to-boundary operators are intertwining operators. Further, we recall the
fact that there is a hierarchy of equations on the boundary, invariant w.r.t.
Schr\"odinger algebra. We also review the explicit construction of an analogous
hierarchy of invariant equations in the bulk, and that the two hierarchies are
equivalent via the bulk-to-boundary intertwining operators. The derivation of
these hierarchies uses a mechanism introduced first for semi-simple Lie groups
and adapted to the non-semisimple Schr\"odinger algebra. These requires
development of the representation theory of the Schr\"odinger algebra which is
reviewed in some detail. We also recall the $q$-deformation of the
Schr\"odinger algebra. Finally, the realization of the Schr\"odinger algebra
via difference operators is reviewed.
