  We consider the dimensional reductions of N=4 Supersymmetric Yang-Mills
theory on R x S^3 to the three-dimensional theory on R x S^2, the orbifolded
theory on R x S^3/Z_k, and the plane-wave matrix model. With explicit emphasis
on the three-dimensional theory, we demonstrate the realization of the SU(2|3)
algebra in a radial Hamiltonian framework. Using this structure we constrain
the form of the spin chains, their S-matrices, and the corresponding one- and
two-loop Hamiltonian of the three dimensional theory and find putative signs of
integrability up to the two-loop order. The string duals of these theories
admit the IIA plane-wave geometry as their Penrose limit. Using known results
for strings quantized on this background, we explicitly construct the
strong-coupling dual extended SU(2|2) algebra and discuss its implications for
the gauge theories.
