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

The algebras Q(n,k)(E, t) 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= 1, a complex elliptic curve E, and a point t ? E. The main result in this paper is that Q(n,k)(E, t) has the same Hilbert series as the polynomial ring on n variables when t is not a torsion point. We also show that Q(n,k)(E, t) is a Koszul algebra, hence of global dimension n when t is not a torsion point, and, for all but countably many t, Q(n,k)(E, t) is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining Q(n,k)(E, t) is the image of an operator R-t (t) that belongs to a family of operators R-t (z) : C-n ? C-n? C-n ? C-n, z ? C, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.