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TMF, 2008, Volume 155, Number 1, Pages 147–160 (Mi tmf6200)  

This article is cited in 7 scientific papers (total in 7 papers)

Dual $R$-matrix integrability

T. V. Skrypnik

N. N. Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine

Abstract: Using the $R$-operator on a Lie algebra $\mathfrak{g}$ satisfying the modified classical Yang–Baxter equation, we define two sets of functions that mutually commute with respect to the initial Lie–Poisson bracket on $\mathfrak{g}^*$. We consider examples of the Lie algebras $\mathfrak{g}$ with the Kostant–Adler–Symes and triangular decompositions, their $R$-operators, and the corresponding two sets of mutually commuting functions in detail. We answer the question for which $R$-operators the constructed sets of functions also commute with respect to the $R$-bracket. We briefly discuss the Euler–Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals.

Keywords: Lie algebra, classical $R$-matrix, classical integrable system

DOI: https://doi.org/10.4213/tmf6200

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English version:
Theoretical and Mathematical Physics, 2008, 155:1, 633–645

Bibliographic databases:


Citation: T. V. Skrypnik, “Dual $R$-matrix integrability”, TMF, 155:1 (2008), 147–160; Theoret. and Math. Phys., 155:1 (2008), 633–645

Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf6200
  • http://mi.mathnet.ru/eng/tmf/v155/i1/p147

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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. Taras V. Skrypnik, “Classical $R$-Operators and Integrable Generalizations of Thirring Equations”, SIGMA, 4 (2008), 011, 19 pp.  mathnet  crossref  mathscinet  zmath
    2. Skrypnyk T., “Lie algebras with triangular decompositions, non-skew-symmetric classical $r$-matrices and Gaudin-type integrable systems”, J. Geom. Phys., 60:3 (2010), 491–500  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. B. A. Dubrovin, T. V. Skrypnik, “Classical double, $R$-operators, and negative flows of integrable hierarchies”, Theoret. and Math. Phys., 172:1 (2012), 911–931  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    4. Skrypnyk T., “Infinite-Dimensional Lie Algebras, Classical R-Matrices, and Lax Operators: Two Approaches”, J. Math. Phys., 54:10 (2013), 103507  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. Skrypnyk T., ““Many-Poled” R-Matrix Lie Algebras, Lax Operators, and Integrable Systems”, J. Math. Phys., 55:8 (2014), 083507  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. Chervonyi Yu. Lunin O., “Generalized ?-deformations of AdSp?Sp”, Nucl. Phys. B, 913 (2016), 912–941  crossref  mathscinet  zmath  isi  elib  scopus
    7. Skrypnyk T., “Reduction in Soliton Hierarchies and Special Points of Classical R-Matrices”, J. Geom. Phys., 130 (2018), 260–287  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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