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We begin a study of higher-loop corrections to the dilatation generator ofN=4 SYM in non-compact sectors. In these sectors, the dilatation generatorcontains infinitely many interactions, and therefore one expects verycomplicated higher-loop corrections. Remarkably, we find a short and simpleexpression for the two-loop dilatation generator. Our solution for thenon-compact su(1,1|2) sector consists of nested commutators of four O(g)generators and one simple auxiliary generator. Moreover, the solution does notrequire 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 theorycalculations. By applying the expression for the two-loop dilatation generatorto compute selected anomalous dimensions and the bosonic sl(2) sector internalS-matrix, we confirm recent conjectures of the higher-loop Bethe ansatz ofhep-th/0412188.Comment: 28 pages, v2: additional checks against direct field theory  calculations, references added, minor corrections, v3: additional minor  correction

Script N = 4 SYM to two loops: compact expressions for the non-compact symmetry algebra of the fraktur sfraktur u(1,1|2) sector