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Introduction to the Yang-Baxter equation. (English) Zbl 0716.17011

Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys. 9, 111-134 (1989).
[For the entire collection see Zbl 0716.00010.]
From the author’s introduction: For over two decades the Yang-Baxter equation (YBE) has been studied as the master equation in integrable models in statistical mechanics and quantum field theory. Recent progress in other fields - \(C^*\)-algebras, link invariants, quantum groups, conformal field theory, etc. - shed new light to the significance of YBE, and has aroused interest among many people.
The present article is aimed to be an introduction to YBE for nonspecialists. About the state of the matter up to 1982 good review papers are available [P. P. Kulish, E. K. Sklyanin, J. Sov. Math. 19, 1596-1620 (1982); transl. from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 95, 129-160 (1980; Zbl 0553.58039) and Lect. Notes Phys. 151, 61-119 (1982)]. I have tried here to include some of the more recent developments (to within my limited knowledge), with the emphasis on the role of quantum groups.
The text is organized as follows. Section 1 is devoted to basic definitions, properties and elementary examples of solutions of YBE. In Section 2 the classical YBE is defined, and the structure of its solutions is depicted following Belavin-Drinfeld. In Section 3 the quantized universal enveloping algebra \(U_ q{\mathfrak g}\) is introduced. Known facts about its representations and Drinfeld’s universal R-matrix are briefly summarized. As an application, in Section 4 the trigonometric solutions of YBE related to the vector representation of classical Lie algebras are described. Solutions corresponding to ‘higher’ representations can be obtained by the fusion procedure. Section 5 outlines this method. In the last Section 6 the solutions of YBE of the ‘face-model’ type are discussed together with the braid representations induced by them.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
53D50 Geometric quantization
81T25 Quantum field theory on lattices
82B23 Exactly solvable models; Bethe ansatz