  In the following paper, which is based on the authors PhD thesis submitted to
Imperial College London, we explore the applicability of Yangian symmetry to
various integrable models, in particular, in relation with S-matrices. One of
the main themes in this work is that, after a careful study of the mathematics
of the symmetry algebras one finds that in an integrable model, one can
directly reconstruct S-matrices just from the algebra. It has been known for a
long time that S-matrices in integrable models are fixed by symmetry. However,
Lie algebra symmetry, the Yang-Baxter equation, crossing and unitarity, which
are what constrains the S-matrix in integrable models, are often taken to be
separate, independent properties of the S-matrix. Here, we construct scattering
matrices purely from the Yangian, showing that the Yangian is the right
algebraic object to unify all required symmetries of many integrable models. In
particular, we reconstruct the S-matrix of the principal chiral field, and, up
to a CDD factor, of other integrable field theories with su(n) symmetry.
Furthermore, we study the AdS/CFT correspondence, which is also believed to be
integrable in the planar limit. We reconstruct the S-matrices at weak and at
strong coupling from the Yangian or its classical limit. We give a pedagogical
introduction into the subject, presenting a unified perspective of Yangians and
their applications in physics. This paper should hence be accessible to
mathematicians who would like to explore the application of algebraic objects
to physics as well as to physicists interested in a deeper understanding of the
mathematical origin of physical quantities.
