  We describe an embedding of the Poincare Lie algebra into an extension of the
Lie field of SO(2,3) (the anti-de Sitter group). We also describe higher
dimensional analogs of this embedding, which connect SO(p,q) groups to their
associated motion groups (higher dimensional Poincare groups). Further some
results on q generalizations of these results are given. We apply our results
to certain unitary, continuous series representations of SO(2,3) associated
with functions on SO(2,3)/SO(1,3). From the embedding theorem we obtain
representations of the Poincare Lie algebra which are associated with unitary,
tachyonic representations of the Poincare group, provided these representations
are integrable to group representations.
