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

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 11-01-0057011-01-1203712-02-9105211-01-00980 Deutsche Forschungsgemeinschaft KI 623/8-1 Ministry of Education and Science of the Russian Federation 12-09-0064

DOI: https://doi.org/10.4213/rm9552

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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

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
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