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 Uspekhi Mat. Nauk, 2002, Volume 57, Issue 6(348), Pages 87–122 (Mi umn573)

Elliptic algebras

A. V. Odesskii

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/rm573

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English version:
Russian Mathematical Surveys, 2002, 57:6, 1127–1162

Bibliographic databases:

UDC: 512.552.8
MSC: Primary 16W50, 14H52; Secondary 16S80, 17B63, 17B37, 53D30, 53D55, 16S37, 53D17, 1

Citation: A. V. Odesskii, “Elliptic algebras”, Uspekhi Mat. Nauk, 57:6(348) (2002), 87–122; Russian Math. Surveys, 57:6 (2002), 1127–1162

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Odesskii, “Bihamiltonian elliptic structures”, Mosc. Math. J., 4:4 (2004), 941–946
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3. A. V. Odesskii, V. V. Sokolov, “Compatible Lie brackets related to elliptic curve”, J Math Phys (N Y ), 47:1 (2006), 013506
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5. Etingof P., Oblomkov A., “Quantization, orbifold cohomology, and Cherednik algebras”, Jack, Hall-Littlewood and Macdonald Polynomials, Contemporary Mathematics Series, 417, 2006, 171–182
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9. Makar-Limanov L., Turusbekova U., Umirbaev U., “Automorphisms of elliptic Poisson algebras”, Algebras, Representations and Applications, Contemporary Mathematics Series, 483, 2009, 169–177
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14. Davies A., “Cocycle twists of 4-dimensional Sklyanin algebras”, J. Algebra, 457 (2016), 323–360
15. Etingof P., Walton Ch., “Finite dimensional Hopf actions on algebraic quantizations”, Algebr. Number Theory, 10:10 (2016), 2287–2310
16. Iyudu N., Shkarin S., “Three dimensional Sklyanin algebras and Gröbner bases”, J. Algebra, 470 (2017), 379–419
17. Davies A., “Cocycle twists of algebras”, Commun. Algebr., 45:3 (2017), 1347–1363
18. Hua Zh., Polishchuk A., “Shifted Poisson Structures and Moduli Spaces of Complexes”, Adv. Math., 338 (2018), 991–1037
19. Odesskii A., “Poisson Structures on Loop Spaces of Cpn and An R-Matrix Associated With the Universal Elliptic Curve”, J. Geom. Phys., 140 (2019), 152–160
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