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

Abstract

We construct an infinite-dimensional solution of the Yang-Baxter equation (YBE) 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 R-operator intertwines the product of two standard L-operators associated with the Sklyanin algebra, an elliptic deformation of sl(2)-algebra. It is built from three basic operators ${\mathrm{S}}_1,{\mathrm{S}}_2$, and ${\mathrm{S}}_3$ generating the permutation group of four parameters ${\mathfrak{S}}_4$. Validity of the key Coxeter relations (including the star-triangle relation) is based on the elliptic beta integral evaluation formula 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.

43 pp., to appear in Russian Math. Surveys

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