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TMF, 2002, Volume 133, Number 1, Pages 3–23 (Mi tmf377)  

This article is cited in 20 scientific papers (total in 20 papers)

Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves

A. V. Odesskiiab, V. N. Rubtsovcb

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b Université d'Angers
c Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We study polynomial Poisson algebras with some regularity conditions. Linear (Lie–Berezin–Kirillov) structures on dual spaces of semisimple Lie algebras, quadratic Sklyanin elliptic algebras, and the polynomial algebras recently described by Bondal, Dubrovin, and Ugaglia belong to this class. We establish some simple determinant relations between the brackets and Casimir functions of these algebras. In particular, these relations imply that the sum of degrees of the Casimir functions coincides with the dimension of the algebra in the Sklyanin elliptic algebras. We present some interesting examples of these algebras and show that some of them arise naturally in the Hamiltonian integrable systems. A new class of two-body integrable systems admitting an elliptic dependence on both coordinates and momenta is among these examples.

Keywords: polynomial Poisson structures, elliptic algebras, integrable systems

DOI: https://doi.org/10.4213/tmf377

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English version:
Theoretical and Mathematical Physics, 2002, 133:1, 1321–1337

Bibliographic databases:

Received: 14.12.2001

Citation: A. V. Odesskii, V. N. Rubtsov, “Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves”, TMF, 133:1 (2002), 3–23; Theoret. and Math. Phys., 133:1 (2002), 1321–1337

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Odesskii, “Elliptic algebras”, Russian Math. Surveys, 57:6 (2002), 1127–1162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Spiridonov VP, Zhedanov AS, “Poisson algebras for some generalized eigenvalue problems”, Journal of Physics A-Mathematical and General, 37:43 (2004), 10429–10443  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. Panyushev, D, “On symmetric invariants of centralisers in reductive Lie algebras”, Journal of Algebra, 313:1 (2007), 343  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Panyushev, DI, “On the coadjoint representation of Z(2)-contractions of reductive Lie algebras”, Advances in Mathematics, 213:1 (2007), 380  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Luc Vinet, Alexei Zhedanov, “Quasi-Linear Algebras and Integrability (the Heisenberg Picture)”, SIGMA, 4 (2008), 015, 22 pp.  mathnet  crossref  mathscinet  zmath
    6. Joseph, A, “Slices for biparabolic coadjoint actions in type A”, Journal of Algebra, 319:12 (2008), 5060  crossref  mathscinet  zmath  isi  scopus  scopus
    7. Molev, AI, “Symmetries and invariants of twisted quantum algebras and associated Poisson algebras”, Reviews in Mathematical Physics, 20:2 (2008), 173  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Jordan, DA, “REVERSIBLE SKEW Laurent POLYNOMIAL RINGS AND DEFORMATIONS OF Poisson AUTOMORPHISMS”, Journal of Algebra and Its Applications, 8:5 (2009), 733  crossref  mathscinet  zmath  isi  scopus  scopus
    9. Pelap, SRT, “Poisson (co)homology of polynomial Poisson algebras in dimension four: Sklyanin's case”, Journal of Algebra, 322:4 (2009), 1151  crossref  mathscinet  zmath  isi  scopus  scopus
    10. Panyushev, DI, “PERIODIC AUTOMORPHISMS OF TAKIFF ALGEBRAS, CONTRACTIONS, AND theta-GROUPS”, Transformation Groups, 14:2 (2009), 463  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    11. Pelap, SRT, “On the Hochschild Homology of Elliptic Sklyanin Algebras”, Letters in Mathematical Physics, 87:3 (2009), 267  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. Dolgushev, V, “The Van den Bergh duality and the modular symmetry of a Poisson variety”, Selecta Mathematica-New Series, 14:2 (2009), 199  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. Pelap S.R.T., “Homological properties of certain Generalized Jacobian Poisson Structures in dimension 3”, J Geom Phys, 61:12 (2011), 2352–2368  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    14. 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
    15. Ortenzi G., Rubtsov V., Pelap S.R.T., “Integer Solutions of Integral Inequalities and H-Invariant Jacobian Poisson Structures”, Adv. Math. Phys., 2011, 252186  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Damianou P.A., Petalidou F., “Poisson Brackets with Prescribed Casimirs”, Can. J. Math.-J. Can. Math., 64:5 (2012), 991–1018  crossref  mathscinet  zmath  isi  scopus  scopus
    17. Jordan D.A., Oh S.-Q., “Poisson Brackets and Poisson Spectra in Polynomial Algebras”, New Trends in Noncommutative Algebra, Contemporary Mathematics, 562, eds. Ara P., Brown K., Lenagan T., Letzter E., Stafford J., Zhang J., Amer Mathematical Soc, 2012, 169–187  crossref  mathscinet  zmath  isi
    18. Gualtieri M., Pym B., “Poisson Modules and Degeneracy Loci”, Proc. London Math. Soc., 107:3 (2013), 627–654  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Yakimova O., “One-Parameter Contractions of Lie-Poisson Brackets”, J. Eur. Math. Soc., 16:2 (2014), 387–407  crossref  mathscinet  zmath  isi  scopus  scopus
    20. Pantelis A. Damianou, “Poisson Brackets after Jacobi and Plücker”, Regul. Chaotic Dyn., 23:6 (2018), 720–734  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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