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

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