Feigin and Odesskii's elliptic algebras

We study the elliptic algebras Q(n,k) (E, tau) introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers n > k >= 1, an elliptic curve E, and a point tau is an element of E. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that Q(n,k) (E, 0), and Q(n, n-1) (E, tau) are polynomial rings on n variables. We also show that Q(n,k) (E, tau + zeta) is a twist of Q(n,k) (E, tau) when zeta is an n-torsion point. This paper is the first of several we are writing about the algebras Q(n,k) (E, tau). (C) 2021 Elsevier Inc. All rights reserved.