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Difference equations with elliptic coefficients and quantum affine algebras
High Energy Physics - Theory · Submitted 8 Dec 1993
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main result of the paper states that these functions satisfy two holonomic systems of difference equations -- the first one has shift parameter p and elliptic modulus s, and the second one has shift parameter s and elliptic modulus p.
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The abstract, sentence by sentence
The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic deformation of the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. In hep-th 9303018, it was shown that solutions of the elliptic r-matrix equations admit a representation as traces of products of intertwining operators between certain modules over affine sl(N). In this paper, this construction is generalized to quantum affine sl(N). The main object of study in the paper is a family of meromorphic matrix functions of n complex variables z_1,...,z_n and three additional parameters p,q,s -- (modified) traces of products of intertwiners between modules over quantum affine sl(N). They are a new class of transcendental functions which can be degenerated into many interesting special functions -- hypergeometric and q-hypergeometric functions, elliptic and modular functions, transcendental functions of an elliptic curve, vector-valued modular forms, solutions of the Bethe ansatz equations etc. The main result of the paper states that these functions satisfy two holonomic systems of difference equations -- the first one has shift parameter p and elliptic modulus s, and the second one has shift parameter s and elliptic modulus p. The paper also contains a short proof of the quantum KZ equations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic deformation of the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. | contribution 0.99 | 0.64 | 0.06 | asserted 0.93 |
| s1 | In hep-th 9303018, it was shown that solutions of the elliptic r-matrix equations admit a representation as traces of products of intertwining operators between certain modules over affine sl(N). | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| s2 | In this paper, this construction is generalized to quantum affine sl(N). | contribution 0.93 | 0.85 | 0.03 | asserted 1.00 |
| s3 | The main object of study in the paper is a family of meromorphic matrix functions of n complex variables z_1,...,z_n and three additional parameters p,q,s -- (modified) traces of products of intertwiners between modules over quantum affine sl(N). | contribution 0.99 | 0.80 | 0.04 | asserted 1.00 |
| s4 | They are a new class of transcendental functions which can be degenerated into many interesting special functions -- hypergeometric and q-hypergeometric functions, elliptic and modular functions, transcendental functions of an elliptic curve, vector-valued modular forms, solutions of the Bethe ansatz equations etc. | interpretation 0.76 | 0.60 | 0.03 | asserted 0.92 |
| s5 | The main result of the paper states that these functions satisfy two holonomic systems of difference equations -- the first one has shift parameter p and elliptic modulus s, and the second one has shift parameter s and elliptic modulus p. | result 1.00 | 0.94 | 0.98 | asserted 1.00 |
| s6 | The paper also contains a short proof of the quantum KZ equations. | contribution 0.80 | 0.90 | 0.03 | asserted 1.00 |
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Original abstract
The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic deformation of the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. In hep-th 9303018, it was shown that solutions of the elliptic r-matrix equations admit a representation as traces of products of intertwining operators between certain modules over affine sl(N). In this paper, this construction is generalized to quantum affine sl(N). The main object of study in the paper is a family of meromorphic matrix functions of n complex variables z_1,...,z_n and three additional parameters p,q,s -- (modified) traces of products of intertwiners between modules over quantum affine sl(N). They are a new class of transcendental functions which can be degenerated into many interesting special functions -- hypergeometric and q-hypergeometric functions, elliptic and modular functions, transcendental functions of an elliptic curve, vector-valued modular forms, solutions of the Bethe ansatz equations etc. The main result of the paper states that these functions satisfy two holonomic systems of difference equations -- the first one has shift parameter p and elliptic modulus s, and the second one has shift parameter s and elliptic modulus p. The paper also contains a short proof of the quantum KZ equations.
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3 arXiv papers cite this in 6 sentences
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Cites it as support 1 paper
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By [ref] , the quantum KZ equations [ref] are consistent for all values $ζ∈T$ of the central character.
Double affine Hecke algebras and bispectral quantum Knizhnik-Zamolodchikov equations · Specialized central character and Harish-Chandra series · 0.96
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