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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 S 1 , S 2 , and S 3 generating the permutation group of four parameters 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 S j are determined uniquely with the help of the elliptic modular double.

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Abstract. The problem of constructing the R-matrix is considered in the case of an integrable spin chain with symmetry group SL(n, C). A fairly complete study of general R-matrices acting in the tensor product of two continuous series representations of SL(n, C) is presented. On this basis, R-matrices are constructed that act in the tensor product of Verma modules (which are infinite-dimensional representations of the Lie algebra sl(n)), and also R-matrices acting in the tensor product of finite-dimensional representations of the Lie algebra sl(n).