The relation between duality and quantum groups is discussed. The fusion matrix of the SU(2) WZW model is explicitly computed in terms of a quantum group 6j-symbol and a normalization factor. The commutators of charge screening operators are computed, resulting in a deformation of SU (2)q algebra.
Two topics in two-dimensional quantum field theory are presented. The first is a classification of 2- and 3-field rational conformal field theories. Using the fact that the fusion algebra of a RCFT is specified in terms of integers that are related to modular transformation properties, we classify 2- and 3-field chiral RCFT's. We show that the only possibilities for the non-trivial fusion rule in the 2-field case are φ x φ = 1 or φ x φ = 1 + φ. Similar results are obtained for the 3-field case. A partial classification of possible conformal dimensions and central charges for these theories is also obtained. The second topic is in two-dimensional quantum gravity. Explicit computation of the non-perturbative correlation functions of the (1, q) models of KdV-gravity is presented. This computation includes contributions from high genus as well as correlation functions of descendant fields. A ghost number conservation law for these models is derived from purely algebraic considerations. A hint of further selection rules is found.
In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.