Elliptic algebras. (English. Russian original) Zbl 1062.16035
Russ. Math. Surv. 57, No. 6, 1127-1162 (2002); translation from Usp. Mat. Nauk 57, No. 6, 87-122 (2002).
From the author’s abstract: This survey is devoted to associative \(\mathbb{Z}_{\geq 0}\)-graded algebras presented by \(n\) generators and \(\tfrac{n(n-1)}2\) 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.
Reviewer: Zhang Hechun (Beijing)
MSC:
16S37 | Quadratic and Koszul algebras |
16W50 | Graded rings and modules (associative rings and algebras) |
14H52 | Elliptic curves |
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
53D55 | Deformation quantization, star products |
81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
17B63 | Poisson algebras |