Feigin and Odesskii's elliptic algebras

Abstract

We study the elliptic algebras Qn,k(E,τ) 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 τ∈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 Qn,k(E,0), and Qn,n−1(E,τ) are polynomial rings on n variables. We also show that Qn,k(E,τ+ζ) is a twist of Qn,k(E,τ) when ζ is an n-torsion point. This paper is the first of several we are writing about the algebras Qn,k(E,τ).