  We show that the BMN operators in D=4 N=4 super Yang Mills theory proposed as
duals of stringy oscillators in a plane wave background have a natural quantum
group construction in terms of the quantum deformation of the SO(6) $R$
symmetry. We describe in detail how a q-deformed U(2) subalgebra generates BMN
operators, with $ q \sim e^{2 i \pi \over J}$. The standard quantum co-product
as well as generalized traces which use $q$-cyclic operators acting on tensor
products of Higgs fields are the ingredients in this construction. They
generate the oscillators with the correct (undeformed) permutation symmetries
of Fock space oscillators. The quantum group can be viewed as a spectrum
generating algebra, and suggests that correlators of BMN operators should have
a geometrical meaning in terms of spaces with quantum group symmetry.
