Communications in Mathematical Physics

, Volume 146, Issue 1, pp 1–60 | Cite as

Quantum affine algebras and holonomic difference equations

  • I. B. Frenkel
  • N. Yu. Reshetikhin


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.


Neural Network Complex System Nonlinear Dynamics Statistical Mechanic Quantum Computing 
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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • I. B. Frenkel
    • 1
  • N. Yu. Reshetikhin
    • 2
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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