Iterative Structure of the N=4 SYM Spin Chain

We develop algebraic methods for finding loop corrections to the N=4 SYM dilatation generator, within the noncompact psu(1,1|2) sector. This sector gives a 't Hooft coupling lambda-dependent representation of psu(1,1|2) times psu(1|1)^2. At first working independently of the representation, we present an all-order algebraic ansatz for the lambda-dependence of this Lie algebra's generators. The ansatz solves the symmetry constraints if an auxiliary generator, h, satisfies certain simple commutation relations with the Lie algebra generators. Applying this to the psu(1,1|2) sector leads to an iterative solution for the planar three-loop dilatation generator in terms of leading order symmetry generators and h, which passes a thorough set of spectral tests. We argue also that this algebraic ansatz may be applicable to the nonplanar theory as well.