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

S. E. Derkachov, V. P. Spiridonov

Published 2012-05-15, updated 2013-05-31Version 2

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.