Integrable theories and generalized graded Maillet algebras

We present a general formalism to investigate the integrable properties of a large class of non-ultralocal models which in principle allows the construction of the corresponding lattice versions. Our main motivation comes from the su(1|1) subsector of the string theory on AdS_5 x S^5 in the uniform gauge, where such type of non-ultralocality appears in the resulting Alday-Arutyunov-Frolov (AAF) model. We first show how to account for the second derivative of the delta function in the Lax algebra of the AAF model by modifying Maillet's r- and s-matrices formalism, and derive a well-defined algebra of transition matrices, which allows for the lattice formulation of the theory. We illustrate our formalism on the examples of the bosonic Wadati-Konno-Ichikawa-Shimizu (WKIS) model and the two-dimensional free massive Dirac fermion model, which can be obtained by a consistent reduction of the full AAF model, and give the explicit forms of their corresponding r-matrices.