A simple construction of elliptic R-matrices

We show that Belavin’s solutions of the quantum Yang–Baxter equation can be obtained by restricting an infinite R-matrix to suitable finite dimensional subspaces. This infinite R-matrix is a modified version of the Shibukawa–Ueno R-matrix acting on functions of two variables. (hep-th/9402011) Shibukawa and Ueno [1] have defined an elliptic R-operator acting on the space of functions of two variables on the circle, and obeying the quantum Yang–Baxter equation, extending the work of Gaudin [2]. It has a very simple form. To describe it let us introduce the basic function σw(z, τ) uniquely characterized by having the following behaviour as a function of z for fixed (generic) w ∈ C and τ ∈ C, Im(τ) > 0: σw(z + 1, τ) = σw(z, τ), σw(z + τ , τ) = e σw(z, τ), (1) and being meromorphic with only simple poles on the lattice Z + τZ, and unit residue at the origin. In terms of Jacobi’s theta function θ1(z, τ) = − ∑ n∈Z+ 1 2 e τ+2πin(z+ 1 2 ,