# Algebraic Aspects of the Bethe Ansatz

@article{Faddeev1994AlgebraicAO,
title={Algebraic Aspects of the Bethe Ansatz},
journal={International Journal of Modern Physics A},
year={1994},
volume={10},
pages={1845-1878}
}
• Published 4 April 1994
• Mathematics
• International Journal of Modern Physics A
In this article an introduction to the algebraic aspects of the Bethe ansatz is given. The applications to the seminal spin 1/2 XXX model are discussed in detail and the generalization to higher spin as well as XXZ and the lattice sine-Gordon model are indicated. The origin of quantum groups and their appearance in CFT models are explained. The text can be considered as a guide to the research papers in this field.
117 Citations

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## References

SHOWING 1-10 OF 11 REFERENCES
Quantum Groups
Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups
Hamiltonian methods in the theory of solitons
• Mathematics
• 1987
The Nonlinear Schrodinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples
Quantum Groups
• Mathematics
• 1993
This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions
Int. J. Mod. Phys. B7
• Int. J. Mod. Phys. B7
• 1993
Lectures in Salamanka Summer School, NATO ASI series
• Lectures in Salamanka Summer School, NATO ASI series
• 1993
Integrable Models, Quantum Groups and Conformal Field Theory
• Integrable Models, Quantum Groups and Conformal Field Theory
• 1992
J. 1
• J. 1
• 1990
Sov. J. Math
• Sov. J. Math
• 1984
Theor. Math. Phys
• Theor. Math. Phys
• 1983
Math. Phys
• Math. Phys
• 1982