2003/03/03 by Alexander Odesskii
Mathematics · Physics and Astronomy · #math.QA #hep-th #math.AG #nlin.SI
published as Russ.Math.Surveys 57 (2002) 6 · Latex, 50 pages, to appear in Russian Math. Surveys 57:6 (2002)
arxiv created 2003/03/03 · arxiv updated 2009/11/30
The survey is devoted to associative \Z≥0-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which 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.