  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.
