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

This article is cited in 19 scientific papers (total in 19 papers)

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

Full text: PDF file (515 kB)
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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
Received: 15.05.2002

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

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    This publication is cited in the following articles:
    1. A. V. Odesskii, “Bihamiltonian elliptic structures”, Mosc. Math. J., 4:4 (2004), 941–946  mathnet  mathscinet  zmath
    2. Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., “Traces on the Sklyanin algebra and correlation functions of the eight-vertex model”, J. Phys. A, 38:35 (2005), 7629–7659  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    3. A. V. Odesskii, V. V. Sokolov, “Compatible Lie brackets related to elliptic curve”, J Math Phys (N Y ), 47:1 (2006), 013506  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. S. A. Zelenova, “Large Commutative Subalgebras of Quantum Algebras”, Journal of Mathematical Sciences (New York), 134:1 (2006), 1879  crossref  mathscinet  zmath  elib  scopus  scopus
    5. Etingof P., Oblomkov A., “Quantization, orbifold cohomology, and Cherednik algebras”, Jack, Hall-Littlewood and Macdonald Polynomials, Contemporary Mathematics Series, 417, 2006, 171–182  crossref  mathscinet  zmath  isi
    6. Nevins T.A., Stafford J.T., “Sklyanin algebras and Hilbert schemes of points”, Adv. Math., 210:2 (2007), 405–478  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Wijnholt, M, “Parameter space of quiver gauge theories”, Advances in Theoretical and Mathematical Physics, 12:4 (2008), 711  crossref  mathscinet  zmath  isi  scopus  scopus
    8. B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida, “A commutative algebra on degenerate CP[sup 1] and Macdonald polynomials”, J Math Phys (N Y ), 50:9 (2009), 095215  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    9. Makar-Limanov L., Turusbekova U., Umirbaev U., “Automorphisms of elliptic Poisson algebras”, Algebras, Representations and Applications, Contemporary Mathematics Series, 483, 2009, 169–177  crossref  mathscinet  zmath  isi
    10. A. Zabrodin, “Intertwining operators for Sklyanin algebra and elliptic hypergeometric series”, Journal of Geometry and Physics, 61:9 (2011), 1733  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    11. Ortenzi G., Rubtsov V., Pelap Serge Romeo Tagne, “On the Heisenberg Invariance and the Elliptic Poisson Tensors”, Lett Math Phys, 96:1–3 (2011), 263–284  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. Alexander Odesskii, Thomas Wolf, “Compatible quadratic Poisson brackets related to a family of elliptic curves”, Journal of Geometry and Physics, 2012  crossref  mathscinet  isi  scopus  scopus
    13. Eric Rains, Simon Ruijsenaars, “Difference Operators of Sklyanin and van Diejen Type”, Commun. Math. Phys, 2013  crossref  mathscinet  isi  scopus  scopus
    14. Davies A., “Cocycle twists of 4-dimensional Sklyanin algebras”, J. Algebra, 457 (2016), 323–360  crossref  mathscinet  zmath  isi  scopus
    15. Etingof P., Walton Ch., “Finite dimensional Hopf actions on algebraic quantizations”, Algebr. Number Theory, 10:10 (2016), 2287–2310  crossref  mathscinet  zmath  isi  scopus
    16. Iyudu N., Shkarin S., “Three dimensional Sklyanin algebras and Gröbner bases”, J. Algebra, 470 (2017), 379–419  crossref  mathscinet  zmath  isi  scopus
    17. Davies A., “Cocycle twists of algebras”, Commun. Algebr., 45:3 (2017), 1347–1363  crossref  mathscinet  zmath  isi  scopus
    18. Hua Zh., Polishchuk A., “Shifted Poisson Structures and Moduli Spaces of Complexes”, Adv. Math., 338 (2018), 991–1037  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus
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