Geodesic
Yang--Baxter equation, parameter permutations, and the elliptic beta integral
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 6, pp. 1027-1072

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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, elliptic beta integral.
Mots-clés : permutation group 
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S. È. Derkachev; V. P. Spiridonov. Yang--Baxter equation, parameter permutations, and the elliptic beta integral. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 6, pp. 1027-1072. http://geodesic.mathdoc.fr/item/RM_2013_68_6_a1/