The four-dimensional Sklyanin algebras revisited

Abstract

We give a theta-function and projective-geometric account of the four-dimensional Sklyanin algebras. We define a family of quadratic algebras by declaring that their relations are precisely the two-tensors vanishing on the graph of a translation automorphism on a quartic complex elliptic curve and the vertex pairs where the vertices are the singular points of the four singular quadrics that contain it. We then use Jacobi's addition formulas for theta functions to produce a set of six explicit relations that coincide with Sklyanin's RLL-relations associated to Baxter's elliptic solutions to the 2 x 2 Yang-Baxter equation. We use the geometric definition to show that these algebras coincide with Feigin and Odesskii's four-dimensional elliptic algebras. Our approach agrees with that of Rains-Ruijsenaars theta function perspective on the four-dimensional Sklyanin agebras. For example, the rank-two secant-line relations of Levasseur-Smith coincide with their two-variable theta function relations. We give explicit theta-function formulas for the degree-two central elements, previously obtained by Levassuer-Smith, that annihilate various families of line modules.