Elliptic algebras

This survey is devoted to associative $\mathbb Z_{\geqslant 0}$-graded algebras presented by $n$ generators and $\frac{n(n-1)}2$ quadratic relations and satisfying the so-called Poincaré–Birkhoff–Witt condition (PBW-algebras). Examples are considered of such algebras, depending on two continuous parameters (namely, on an elliptic curve and a point on it), that are flat deformations of the polynomial ring in $n$ variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces, and other directions of modern investigations.