In the following dissertation, 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 dissertation 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.
This version of the thesis includes minor corrections following the viva on
17 September 2010.