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Feigin and Odesskii's elliptic algebras

We study the elliptic algebras $Q_{n,k}(E,τ)$ introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers $n>k\geq 1$, an elliptic curve $E$, and a point $τ\in E$. 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 $Q_{n,n-1}(E,τ)$ are polynomial rings on $n$ variables. We also show that $Q_{n,k}(E,τ+ζ)$ is a twist of $Q_{n,k}(E,τ)$ when $ζ$ is an $n$-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,τ)$.