GENERALIZED QUANTUM YANG-BAXTER EQUATIONS

Abstract

The Quantum Yang-Baxter Equation (QYBE) appears as a master equation in statistical physics and integrable quantum field theory. It first manifests itself as a consistency equation for the Bethe Ansatz solutions of the many-body problem of a free fermion gas with delta function interactions. Many solutions for QYBE have been derived and in the case of two-state spectral-free QYBE, an exhaustive list of solutions has been found through computer algebraic approach. In this thesis, we study QYBE in higher dimensional forms. These equations are sometimes called d-simplex equations. Indeed, the QYBE turns out to be a 2- simplex equation. We suggest a possible labelling scheme for the generalization and contemplate the possible link with the permutation group. In particular, for the 3- simplex equation in which there are two popular versions, namely the Zamolodchikov tetrahedron equation (ZTE) and the Frenkel-Moore equation (FME), we attempt some systematic solutions to these difficult equations.