Elliptic R-matrices and Feigin and Odesskii's elliptic algebras

TL;DR

This paper proves that Feigin and Odesskii's elliptic algebras $Q_{n,k}(E, au)$ have the same Hilbert series as polynomial rings when $ au$ is not a torsion point, and establishes their Koszul and Artin-Schelter regular properties.

Contribution

It demonstrates that these elliptic algebras are Koszul, have the same Hilbert series as polynomial rings, and are Artin-Schelter regular for generic parameters, using quantum Yang-Baxter operators.

Findings

$Q_{n,k}(E, au)$ has the same Hilbert series as polynomial rings when $ au$ is not torsion.

$Q_{n,k}(E, au)$ is a Koszul algebra with global dimension $n$.

For all but countably many $ au$, $Q_{n,k}(E, au)$ is Artin-Schelter regular.

Abstract

The algebras $Q_{n,k}(E,\tau)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>k\ge 1$, a complex elliptic curve $E$, and a point $\tau\in E$. The main result in this paper is that $Q_{n,k}(E,\tau)$ has the same Hilbert series as the polynomial ring on $n$ variables when $\tau$ is not a torsion point. We also show that $Q_{n,k}(E,\tau)$ is a Koszul algebra, hence of global dimension $n$ when $\tau$ is not a torsion point, and, for all but countably many $\tau$, it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining $Q_{n,k}(E,\tau)$ is the image of an operator $R_{\tau}(\tau)$ that belongs to a family of operators $R_{\tau}(z):\mathbb{C}^n\otimes\mathbb{C}^n\to\mathbb{C}^n\otimes\mathbb{C}^n$, $z\in\mathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.