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 $$\tau $$ , $$Q_{n,k}(E,\tau )$$ 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 $$R_{\tau }(z):{\mathbb {C}}^n\otimes {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n\otimes {\mathbb {C}}^n$$ , $$z\in {\mathbb {C}}$$ , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.