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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 6(414), Pages 59–106 (Mi umn9552)  

This article is cited in 23 scientific papers (total in 23 papers)

Yang–Baxter equation, parameter permutations, and the elliptic beta integral

S. È. Derkacheva, V. P. Spiridonovbc

a St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences
b Max Planck Institute for Mathematics, Bonn, Germany
c Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

Abstract: This paper presents a construction of an infinite-dimensional solution of the Yang–Baxter equation of rank 1 which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. This $\mathrm{R}$-operator intertwines the product of two standard $\mathrm{L}$-operators associated with the Sklyanin algebra, an elliptic deformation of the algebra $\operatorname{sl}(2)$. The solution is constructed from three basic operators $\mathrm{S}_1$$\mathrm{S}_2$, and $\mathrm{S}_3$ generating the permutation group $\mathfrak{S}_4$ on four parameters. Validity of the key Coxeter relations (including a star-triangle relation) is based on the formula for computing an elliptic beta integral and the Bailey lemma associated with an elliptic Fourier transformation. The operators $\mathrm{S}_j$ are determined uniquely with the help of the elliptic modular double.
Bibliography: 37 titles.

Keywords: Yang–Baxter equation, Sklyanin algebra, permutation group, elliptic beta integral.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00570
Deutsche Forschungsgemeinschaft KI 623/8-1
Ministry of Education and Science of the Russian Federation 12-09-0064


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English version:
Russian Mathematical Surveys, 2013, 68:6, 1027–1072

Bibliographic databases:

UDC: 517.3+517.9
MSC: Primary 16T25; Secondary 33E20
Received: 29.11.2012

Citation: S. È. Derkachev, V. P. Spiridonov, “Yang–Baxter equation, parameter permutations, and the elliptic beta integral”, Uspekhi Mat. Nauk, 68:6(414) (2013), 59–106; Russian Math. Surveys, 68:6 (2013), 1027–1072

Citation in format AMSBIB
\by S.~\`E.~Derkachev, V.~P.~Spiridonov
\paper Yang--Baxter equation, parameter permutations, and the elliptic beta integral
\jour Uspekhi Mat. Nauk
\yr 2013
\vol 68
\issue 6(414)
\pages 59--106
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 6
\pages 1027--1072

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