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International Journal of Modern Physics B

SYMMETRIES RELATED TO YANG–BAXTER EQUATION AND REFLECTION EQUATIONS

P. P. Kulish

Published Oct 10, 1999• 3 citations
Publisher

Abstract

Some of main achievements obtained recently in investigations of the symmetry properties of systems solved by the quantum inverse scattering methods and connected with the Yang–Baxter equation and reflection equations are reviewed. Special attention is devoted to the twist procedure. Twist relates three constant R-matrices corresponding to Lie super-algebra gl(1|1).

DOI
10.1142/s0217979299002770

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Some of main achievements obtained recently in investigations of the symmetry properties of systems solved by the quantum inverse scattering methods and connected with the Yang–Baxter equation and reflection equations are reviewed. Special attention is devoted to the twist procedure. Twist relates three constant R-matrices corresponding to Lie super-algebra gl(1|1).
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Some special solutions to the reflection equation are considered. These boundary matrices are defined on the common quantum space with the other operators in the chain. The relations with the Drinfeld twist are discussed. PACS: 02.90.+p, 03.20.+i,03.65.Fd, 11.30.-j 0 1 On a covariance property of the reflection equation Let the operator-valued function R(u) : C → Aut(V ⊗ V) be a solution to the quantum Yang-Baxter equation [1] in a finite-dimensional linear space V. Let us connect with a given matrix R(u) two associative algebras TR and UR generated by noncommutative entries of the square matrices T(u) and K(u) satisfying the fundamental commutator relation [1, 2] or the reflection equation [3, 4] R12(u − v) 1
The Yang-Baxter equation appear in various situations in physics and mathematics. For example it arises as a consistency condition in integrable models. The reflection equation (boundary Yang-Baxter equation) is a generalization of the Yang-Baxter equation to systems with a boundary. A further generalization to systems with defects which admits both reflection and transmission can be made, which results in reflection-transmission Yang-Baxter equations.In this thesis the Yang-Baxter equation and the reflection equation are presented. Representations of the Temperley-Lieb algebra and the blob algebra are used to construct matrices which solve the respective equations. For the reflection-transmission Yang-Baxter equations, steps toward a solution are taken by using a similar approach as for the first two cases, namely by finding an algebra whose representations can be used to construct matrices which solve the equations.
In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary Yang-Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.
In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.