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Solvable lattice model and representation theory of quantum affine algebras. (English) Zbl 0973.82014
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.
##### MSC:
 82B23 Exactly solvable models; Bethe ansatz 81T10 Model quantum field theories 82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics 17B37 Quantum groups (quantized enveloping algebras) and related deformations 35R10 Partial functional-differential equations 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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