Antilinear deformations of Coxeter groups with application to Hamiltonian systems

In this thesis we provide several different systematic methods for constructing complex root spaces that remain invariant under an antilinear transformation. The first method is based on any element of the Weyl group, which is extended to factorizations of the Coxeter element and a reduced Coxeter element thereafter. An antilinear deformation method for the longest element of the Weyl group is given as well. Our last construction method leads to an alternative construction for q-deformed roots. For each of these construction methods we provide examples. In addition, we show a method of construction that for some special cases leads to rotations in the dual space and vice versa, starting from a rotation we find the root space involved. We then continue to apply these deformations to a generalized Calogero model and Affine Toda field theory. We provide a general solution for the ground state wave function of the Calogero model that is independent of a root representation and we extend this to the deformed case. An important property of this deformed Calogero model is that the amount of singularities in its potential is significantly reduced. We find that the exchange of particles in this model then leads to anyonic exchange factors. Following this we solve the model and find the ground state eigenvalues and eigenfunctions for the deformed Calogero model. We apply the q-deformed roots to an Affine Toda field theory and find that one may formulate a classical theory respecting the mass renormalisation of the quantum case.EThOS - Electronic Theses Online ServiceGBUnited Kingdo