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Elliptic algebras

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  • 14 Algebraic Geometry
  • 16 Associative Rings And Algebras
  • 17 Nonassociative Rings And Algebras
  • 53 Differential Geometry
  • Mathematics


This survey is devoted to associative -graded algebras presented by n generators and 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.

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