N. Yu. Reshetikhin

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.

YANG-BAXTER EQUATION AND REPRESENTATION THEORY: 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.

Quantum linear problem for the sine-Gordon equation and higher representations

A quantum linear problem is constructed which permits the investigation of the sine-Gordon equation within the framework of the inverse scattering method in an arbitrary representation of algebra. The corresponding R-matrix is found, satisfying the Yang-Baxter equation (the condition for the factorization of the multiparticle matrices of the scattering of particles on a straight line).

Central extensions of quantum current groups

We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.

Quantum R-matrices and factorization problems

Abstract A relation between quantum R-matrices and certain factorization problem in Hopf algebras is established. A definition of dressinf transformation in the quantum case is also given.

Integrability of the principal chiral field model in 1 + 1 dimension

The quantum inverse scattering method is used to solve model of the principal chiral field in 1 + 1 dimension. A different approach based on fermionization was proposed before by A. M. Polyakov and P. B. Wiegmann (Phys. Lett. B131 (1981), 121). The main idea of our work is to get the relativistic model from the lattice magnetic model in the scaling limit with spin S going to infinity. Our method became realistic after the new Hamiltonian interpretation of the Pohlmeyer-Zakharov-Mikhailov L-M pair, (K. Pohlmeyer, Comm. Math. Phys.46 (1976), 207; V. E. Zakharov and A. V. Mikhailov, Sov. Phys. JETP (Engl. Transl.)47 (1978), 1017) was devised.

The Bethe Ansatz and the combinatorics of Young tableaux

The investigation of combinatorial aspects of the method of the inverse problem is continued in this paper.

Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras

New combinatorial formulas are obtained for the multiplicities in the decomposition of the tensor product of the representations of simple Lie algebras into irreducible components.

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).

Combinatorics, Bethe Ansatz, and representations of the symmetric group

Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations).