# Feigin and Odesskii's elliptic algebras

## TL;DR

This paper investigates Feigin and Odesskii's elliptic algebras, exploring their various definitions, properties, and special cases, establishing foundational results about their structure and symmetries.

## Contribution

It clarifies multiple definitions of the elliptic algebras and proves key properties, including polynomiality and twisting behavior under torsion points.

## Key findings

- $Q_{n,0}(E,0)$ and $Q_{n,n-1}(E, au)$ are polynomial rings
- $Q_{n,k}(E, au+ ext{torsion})$ is a twist of $Q_{n,k}(E, au)$
- Various definitions of the algebras are equivalent or comparable

## 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 $n>k\geq 1$, 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 $Q_{n,k}(E,\tau)$.