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

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.

51 pages + index + references