Feigin and Odesskii's elliptic algebras

Abstract

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 $\zeta$ is an $n$-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,\tau )$.

36 pages + references; a number of changes + updated cross-references to other papers in the same series