Solvable lattice model and representation theory of quantum affine algebras (Q1129854)

The paper is an excellent review of some recent developments of solvable lattice models in connection with representation theory of the quantum affine algebras. The following topics are considered. In Sec. 1 the \(XYZ\) model as an example of a solvable system of infinite degrees of freedom is considered. The vacuum and excited states, correlation functions as vacuum-to-vacuum matrix element of local operators are described. In Sec. 2 the problems of integrability of the model are considered, and then the structure of the transfer matrix and the six-vertex model are recalled. The treatment of the \(R\)-matrices as intertwiners and \(U_q(\widehat{{\mathfrak {sl}}_2})\) symmetry of the \(XYZ\) model are given. Sec. 3 is devoted to integrable models of quantum field theories. The connection between QFT and statistical mechanics is considered. Then the primary fields and vertex operators, the Knizhnik-Zamolodchikov equations, the form-factors and the sine-Gordon model are shortly discussed. In conclusion (Sec. 4) the problems of the algebraic structure of the \(XXZ\) and the six vertex models in the language of the representation theory are considered. The identification of the principal operators of the above models to the operators of the representation theory permits to solve the main problems of the diagonalization of the \(XXZ\) Hamiltonian and the computation of the form-factors and the correlation functions.