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Yang-Baxter algebras, integrable theories and Bethe Ansatz. (English) Zbl 0714.17021

Summary: The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropriate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter.
To conclude, some recent developments in the domain of integrable theories are summarized.

MSC:

17B81 Applications of Lie (super)algebras to physics, etc.
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
53D50 Geometric quantization
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