Strings on plane waves, super-Yang-Mills in four dimensions, quantum groups at roots of one

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 similar to e(2ipi/J). The standard quantum coproduct 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 cor-relators of BMN operators should have a geometrical meaning in terms of spaces with quantum group symmetry. (C) 2003 Elsevier B.V. All rights reserved.