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

TL;DR

This paper constructs an infinite-dimensional solution to the Yang-Baxter equation using elliptic hypergeometric kernels, revealing deep connections between algebraic structures and integral identities.

Contribution

It introduces a novel integral operator solution to the YBE linked to the Sklyanin algebra and elliptic beta integrals, expanding the understanding of elliptic quantum groups.

Findings

Constructed an integral operator R-operator solving the YBE.

Linked the R-operator to elliptic hypergeometric functions and the Sklyanin algebra.

Established the validity of Coxeter relations via elliptic beta integral and Bailey lemma.

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.