Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced. Its structure and representations are studied in the simplest case g=sl(2). It is then applied to determine the eigenvalues of the trigonometric solution of the Yang-Baxter equation related to sl(2) in an arbitrary finite-dimensional irreducible representation.

A q -difference analogue of U(g) and the Yang-Baxter equation

A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

https://math.berkeley.edu/~reshetik/Sklyanin-boundary.pdf

Boundary conditions for integrable quantum systems

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Publisher Summary
This chapter focuses on the quantization of lie groups and lie algebras. The Algebraic Bethe Ansatz—the quantum inverse scattering method—emerges as a natural development of the various directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution, the quantum theory of magnets, the method of commuting transfer-matrices in classical statistical mechanics, and factorizable scattering theory. The chapter discusses quantum formal groups, a finite-dimensional example, an infinite-dimensional example, and reviews the deformation theory and quantum groups.

https://inspirehep.net/files/33b6ead3f196ec7bd10f771d606096d9

Quantization of Lie Groups and Lie Algebras

The quantum group SU(2)q is discussed by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum Such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose, is developed

https://iopscience.iop.org/article/10.1088/0305-4470/22/21/020/pdf

On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q

Quantum inverse problem method. I

The problem of constructing the GL(N,ℂ) solutions to the Yang-Baxter equation (factorizedS-matrices) is considered. In caseN=2 all the solutions for arbitrarily finite-dimensional irreducible representations of GL(2,ℂ) are obtained and their eigenvalues are calculated. Some results for the caseN>2 are also presented.

Yang-baxter equation and representation theory: i

Some algebraic structures connected with the Yang—Baxter equation

We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle S-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y-B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y-B equation to the case ofZ
2-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.

Evgeny Sklyanin

Papers

Boundary conditions for integrable quantum systems

Quantum inverse problem method. I

Yang-baxter equation and representation theory: i

Some algebraic structures connected with the Yang—Baxter equation

Solutions of the Yang-Baxter equation