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

Alex Chirvasitu, Ryo Kanda, S. Paul Smith

• The algebras Qn,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 Qn,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 Qn,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, Qn,k(E,τ)is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining Qn,k(E,τ) is the image of an operator Rτ(τ) that belongs to a family of operators Rτ(z):Cn⊗Cn→Cn⊗Cn, z∈C, that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.