On the isomorphism between four-dimensional Sklyanin and elliptic algebras

Abstract

Let [Formula: see text] be an elliptic curve, and let [Formula: see text] be a point on [Formula: see text]. In his groundbreaking paper [E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional. Anal. i Prilozhen. 17(4) (1983) 34–48], Sklyanin introduced a family of algebras [Formula: see text], consisting of four generators and six relations. These algebras hold profound connections with his work on the Quantum Scattering Method. Subsequently, Feigin and Odesskii defined algebras [Formula: see text] for any number of generators [Formula: see text] [Sklyanin’s algebras associated to elliptic curves, Inst. Theor. Phys., Akad. Nauk Ukrain. SSR, Kiev (1988)] and asserted that, for [Formula: see text], they coincide with Sklyanin’s algebras. In this paper, we provide an explicit isomorphism between the algebras [Formula: see text] and [Formula: see text]. As an application, we present a new proof of the Calabi–Yau property for [Formula: see text].