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 $n>k\ge 1$, an elliptic curve E, and a point $\tau \in E$. 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 $Q_{n,n-1}(E,\tau )$ 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 ζ is an n-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,\tau )$.

我们研究椭圆代数 ${问}_{ñ，ķ}（E，\tau ）$由Feigin和Odesskii引入，是Sklyanin代数的泛化。它们形成由二次质数参数化的二次代数族$ñ>ķ\ge \mathrm{1个}$，椭圆曲线E和一个点$\tau \in E$。我们考虑并比较了代数的几种不同定义，并提供了Feigin和Odesskii关于它们的各种陈述的证明。例如，我们表明${问}_{ñ，ķ}（E，0）$， 和 ${问}_{ñ，ñ-\mathrm{1个}}（E，\tau ）$是n个变量上的多项式环。我们还表明${问}_{ñ，ķ}（E，\tau +\zeta ）$ 是一个转折 ${问}_{ñ，ķ}（E，\tau ）$当ζ为n扭转点时。本文是我们正在撰写的有关代数的第一篇${问}_{ñ，ķ}（E，\tau ）$。