Leon A. Takhtajan

Quantization of Lie Groups and Lie Algebras

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.

Liouville model on the lattice

Liouville equation is put on the lattice in a completely integrable way. The classical version is investigated in details and a lattice deformation of the Virasoro algebra is obtained. The quantum version still lacks a satisfactory definition of the Hamiltonian.

The quantum inverse problem method and the XYZ Heisenberg model

Abstract The XYZ Heisenberg model is considered from the stand-point of the quantum inverse problem method. It is shown that this model is a completely integrable quantum system. Algebraic generalization of the Bethe Ansatz for finding eigenvectors and eigenvalues of the Hamiltonian of the XYZ model is given.

Scattering theory for the Korteweg-De Vries (KdV) equation and its Hamiltonian interpretation

Abstract It is shown that the scattering transform for the KdV equation is not canonical in the naive sense. An explanation of this phenomenon is found, and a correct Hamiltonian formulation of the scattering theory is proposed.

Fundamental Models on the Lattice

Here we shall give a complete list of results pertaining to the Toda model, a fundamental model on the lattice. We will show that the r-matrix approach applies to this case and may be used to prove the complete integrability of the model in the quasi-periodic case. For the rapidly decreasing boundary conditions we will analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and outline a method for solving the inverse problem, i.e. for F−1 constructing. On the basis of the r-matrix approach it will be shown that F is a canonical trans formation to action-angle type variables establishing the complete integrability of the Toda model in the rapidly decreasing case. We shall also define a lattice version of the LL model, the most general integrable lattice system with two-dimensional auxiliary space.

Zero Curvature Representation

The dynamical system to be considered is generated by the nonlinear equation (1.1) with the initial condition (1.2) .

Basic Examples and Their General Properties

In this chapter we shall give a list of typical examples and establish their general properties: the zero curvature representation and the Hamiltonian formulation. Then, motivated by these examples, we shall outline a general scheme for constructing integrable equations and their solutions based on the matrix Riemann problem. A detailed study of the most important models and the Hamiltonian interpretation of the general scheme will be presented in the following chapters. The examples to be considered fall into two classes: dynamical systems generated by partial differential evolution equations (continuous models), and evolution systems of difference type (lattice models).

The Riemann Problem

In Chapter I we analyzed the mapping from the functions Ψ(x), to the transition coefficients and discrete spectrum of the auxiliary linear problem. We saw that for both rapidly decreasing and finite density boundary conditions this “change of variables” makes the dynamics quite simple because the time evolution of the transition coefficients for the continuous and discrete spectra becomes linear.

Fundamental Continuous Models

We shall give a complete list of results pertaining to two fundamental continuous models, the HM and SG models. For the rapidly decreasing boundary conditions we shall analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and the discrete spectrum, and show how to solve the inverse problem, i. e. how to construct the mapping F−1. We shall see that these models allow an r-matrix approach, which will enable us to show that F is a canonical transformation to variables of action-angle type. It will thus be proved that the HM and SG models are completely integrable Hamiltonian systems. We shall also present a Hamiltonian interpretation of the change to light-cone coordinates in the SG model. To conclude this chapter, we shall explain that in some sense the LL model is the most universal integrable system with two-dimensional auxiliary space.

Lie-Algebraic Approach to the Classification and Analysis of Integrable Models

In this chapter we shall summarize and generalize our experience in describing integrable models gained from the study of particular examples. The principal entities of the inverse scattering method and its Hamiltonian interpretation were the auxiliary linear problem operator L = d/dx − U(x, λ) and the fundamental Poisson brackets for U(x, λ) involving the r-matrix. Similar objects were introduced for lattice models. We will show that these notions have a simple geometric interpretation.