Author
N. Yu. Reshetikhin
Other affiliations: University of California, Berkeley
Bio: N. Yu. Reshetikhin is an academic researcher from Steklov Mathematical Institute. The author has contributed to research in topics: Bethe ansatz & Integrable system. The author has an hindex of 28, co-authored 38 publications receiving 8629 citations. Previous affiliations of N. Yu. Reshetikhin include University of California, Berkeley.
Papers
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24 Sep 1987
TL;DR: The quantum 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 as discussed by the authors emerged as a natural development of the various directions in mathematical physics.
Abstract: 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.
1,584 citations
TL;DR: In this paper, the generalization of Jones polynomial of links to the case of graphs in R 3 is presented as the functor from the category of graphs to the categories of representations of the quantum groups.
Abstract: The generalization of Jones polynomial of links to the case of graphs inR
3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.
1,281 citations
TL;DR: In this article, the problem of constructing the GL(N,ℂ) solutions to the Yang-Baxter equation (factorizedS-matrices) is considered.
Abstract: 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.
979 citations
TL;DR: In this paper, the authors derived new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk.
Abstract: We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra
representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra
$$\mathfrak{g}$$
and arbitrary finite-dimensional representations of
. We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into
solutions with
$$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$
. We also study the simples examples of solutions of our holonomic difference equations associated to
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$
and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.
683 citations
TL;DR: In this paper, 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 and geometry.
Abstract: 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).
537 citations
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TL;DR: Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced in this article, and its structure and representations are studied in the simplest case g=sl(2).
Abstract: 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.
2,767 citations
TL;DR: In this paper, the authors construct topological invariants of compact oriented 3-manifolds and of framed links in such manifolds, where the terms of the sequence are equale to the values of the Jones polynomial of the link in the corresponding roots of 1.
Abstract: The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory
1,709 citations
TL;DR: Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944 as mentioned in this paper, and 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.
Abstract: 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.
1,658 citations
TL;DR: In this paper, the authors derived the one-loop mixing matrix for anomalous dimensions in = 4 super Yang-Mills and showed that this matrix can be identified with the hamiltonian of an integrable SO(6) spin chain with vector sites.
Abstract: We derive the one loop mixing matrix for anomalous dimensions in = 4 Super Yang-Mills. We show that this matrix can be identified with the hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.
1,585 citations
24 Sep 1987
TL;DR: The quantum 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 as discussed by the authors emerged as a natural development of the various directions in mathematical physics.
Abstract: 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.
1,584 citations