  We begin a study of higher-loop corrections to the dilatation generator of
N=4 SYM in non-compact sectors. In these sectors, the dilatation generator
contains infinitely many interactions, and therefore one expects very
complicated higher-loop corrections. Remarkably, we find a short and simple
expression for the two-loop dilatation generator. Our solution for the
non-compact su(1,1|2) sector consists of nested commutators of four O(g)
generators and one simple auxiliary generator. Moreover, the solution does not
require the planar limit; we conjecture that it is valid for any gauge group.
To obtain the two-loop dilatation generator, we find the complete O(g^3)
symmetry algebra for this sector, which is also given by concise expressions.
We check our solution using published results of direct field theory
calculations. By applying the expression for the two-loop dilatation generator
to compute selected anomalous dimensions and the bosonic sl(2) sector internal
S-matrix, we confirm recent conjectures of the higher-loop Bethe ansatz of
hep-th/0412188.
