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Journal ArticleDOI

A q-analogue of U(g[(N+1)), Hecke algebra, and the Yang-Baxter equation

Michio Jimbo1
01 Apr 1986-Letters in Mathematical Physics (Kluwer Academic Publishers)-Vol. 11, Iss: 3, pp 247-252
TL;DR: In this article, the structure and representations of the universal enveloping algebra U(g) were studied for g = g[(N+1) the structure of the algebra Ŭ(g), a q-analogue of the Universal Enveloping Algebra (U(g)).
Abstract: We study for g=g[(N+1) the structure and representations of the algebra Ŭ(g), a q-analogue of the universal enveloping algebra U(g). Applying the result, we construct trigonometric solutions of the Yang-Baxter equation associated with higher representations of g.
Citations
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Book ChapterDOI
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

Journal ArticleDOI
TL;DR: The quantum group SU(2)q is discussed in this paper 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.
Abstract: 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

1,555 citations

Journal ArticleDOI
TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Abstract: The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.

1,248 citations


Cites methods from "A q-analogue of U(g[(N+1)), Hecke a..."

  • ...On the other hand, the approach to quantum group theory presented in [3] and [ 4 ] uses the quantized enveloping algebra introduced by relations involving entire analytic functions....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the standard q-deformation of the Cartan-Weyl basis for o(3,2) ⋍ sp (4| R ) (real form of C2) is calculated.

882 citations

Journal ArticleDOI
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

References
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Book
01 Jan 1971
TL;DR: Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements as mentioned in this paper.
Abstract: Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971."

3,256 citations

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this paper, the ground-state problem of spin-textonehalf{} fermions is reduced to a generalized Fredholm equation, in a generalized form, by using Bethe's hypothesis.
Abstract: The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for $N$ particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations $R$ of the permutation group ${S}_{N}$. For some $R'\mathrm{s}$ this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.

2,135 citations

Journal ArticleDOI
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