The Tetrahedron Zamolodchikov Algebra and the AdS5 x S5 S-matrix

The S-matrix of the $Ad{S}_5\times S^5$ string theory is a tensor product of two centrally extended su(2|2) S-matrices, each of which is related to the R-matrix of the Hubbard model. The R-matrix of the Hubbard model was first found by Shastry, who ingeniously exploited the fact that, for zero coupling, the Hubbard model can be decomposed into two XX models. In this article, we review and clarify this construction from the AdS/CFT perspective and investigate the implications this has for the $Ad{S}_5\times S^5$ S-matrix.