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Funktsional. Anal. i Prilozhen., 2002, Volume 36, Issue 3, Pages 9–19 (Mi faa200)  

This article is cited in 30 scientific papers (total in 30 papers)

Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type

I. Z. Golubchika, V. V. Sokolovb

a Bashkir State Pedagogical University
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We consider two classes of integrable nonlinear hyperbolic systems on Lie algebras. These systems generalize the principal chiral model. Each system is related to a pair of compatible Lie brackets and has a Lax representation, which is determined by the direct sum decomposition of the Lie algebra of Laurent series into the subalgebra of Taylor series and the complementary subalgebra corresponding to the pair. New examples of compatible Lie brackets are given.

Keywords: compatible Lie brackets; principal chiral model; homogeneous subalgebras


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English version:
Functional Analysis and Its Applications, 2002, 36:3, 172–181

Bibliographic databases:

UDC: 517.958+512.81
Received: 23.04.2001

Citation: I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type”, Funktsional. Anal. i Prilozhen., 36:3 (2002), 9–19; Funct. Anal. Appl., 36:3 (2002), 172–181

Citation in format AMSBIB
\by I.~Z.~Golubchik, V.~V.~Sokolov
\paper Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type
\jour Funktsional. Anal. i Prilozhen.
\yr 2002
\vol 36
\issue 3
\pages 9--19
\jour Funct. Anal. Appl.
\yr 2002
\vol 36
\issue 3
\pages 172--181

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    This publication is cited in the following articles:
    1. O. V. Efimovskaya, V. V. Sokolov, “Decompositions of the loop algebra over $\mathrm{so}(4)$ and integrable models of the chiral equation type”, J. Math. Sci., 136:6 (2006), 4385–4391  mathnet  crossref  mathscinet  zmath
    2. I. Z. Golubchik, V. V. Sokolov, “Factorization of the Loop Algebra and Integrable Toplike Systems”, Theoret. and Math. Phys., 141:1 (2004), 1329–1347  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. A. V. Tsiganov, “Integrable Deformations of Tops Related to the Algebra $so(p,q)$”, Theoret. and Math. Phys., 141:1 (2004), 1348–1360  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Skrypnyk T., “Deformations of loop algebras and classical integrable systems: Finite-dimensional Hamiltonian systems”, Rev. Math. Phys., 16:7 (2004), 823–849  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Sokolov V.V., “On decompositions of the loop algebra over so(3) into a sum of two subalgebras”, Doklady Mathematics, 70:1 (2004), 568–570  mathnet  mathscinet  zmath  isi
    6. Lombardo S, Mikhailov A.V., “Reductions of integrable equations: dihedral group”, J. Phys. A, 37:31 (2004), 7727–7742  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. O. V. Efimovskaya, “Factorization of loop algebras over $\mathrm{so}(4)$ and integrable nonlinear differential equations”, J. Math. Sci., 144:2 (2007), 3926–3937  mathnet  crossref  mathscinet  zmath  elib
    8. T. V. Skrypnik, “Quasigraded lie algebras, Kostant–Adler scheme, and integrable hierarchies”, Theoret. and Math. Phys., 142:2 (2005), 275–288  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Golubchik, IZ, “Factorization of the loop algebras and compatible Lie brackets”, Journal of Nonlinear Mathematical Physics, 12 (2005), 343  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. Skrypnyk, T, “New integrable Gaudin-type systems, classical r-matrices and quasigraded Lie algebras”, Physics Letters A, 334:5–6 (2005), 390  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and the Yang–Baxter Equation”, Theoret. and Math. Phys., 146:2 (2006), 159–169  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Taras V. Skrypnyk, “Quasigraded Lie Algebras and Modified Toda Field Equations”, SIGMA, 2 (2006), 043, 14 pp.  mathnet  crossref  mathscinet  zmath
    13. Dimakis, A, “From AKNS to derivative NLS hierarchies via deformations of associative products”, Journal of Physics A-Mathematical and General, 39:45 (2006), 14015  crossref  mathscinet  zmath  adsnasa  isi  scopus
    14. Odesskii, A, “Algebraic structures connected with pairs of compatible associative algebras”, International Mathematics Research Notices, 2006, 43734  mathscinet  zmath  isi  elib
    15. Skrypnyk, T, “Integrable quantum spin chains, non-skew symmetric r-matrices and quasigraded Lie algebras”, Journal of Geometry and Physics, 57:1 (2006), 53  crossref  mathscinet  zmath  adsnasa  isi  scopus
    16. Odesskii, AV, “Integrable matrix equations related to pairs of compatible associative algebras”, Journal of Physics A-Mathematical and General, 39:40 (2006), 12447  crossref  mathscinet  zmath  isi  scopus
    17. Skrypnyk, T, “Modified non-Abelian Toda field equations and twisted quasigraded Lie algebras”, Journal of Mathematical Physics, 47:6 (2006), 063509  crossref  mathscinet  zmath  adsnasa  isi  scopus
    18. Skrypnyk, T, “Integrable deformations of the mKdV and SG hierarchies and quasigraded Lie algebras”, Physica D-Nonlinear Phenomena, 216:2 (2006), 247  crossref  mathscinet  zmath  adsnasa  isi  scopus
    19. Odesskii, AV, “Compatible Lie brackets related to elliptic curve”, Journal of Mathematical Physics, 47:1 (2006), 013506  crossref  mathscinet  zmath  adsnasa  isi  scopus
    20. Skrypnyk, T, “Special quasigraded Lie algebras and integrable Hamiltonian systems”, Acta Applicandae Mathematicae, 99:3 (2007), 261  crossref  mathscinet  zmath  isi  elib  scopus
    21. Odesskii, A, “Pairs of compatible associative algebras, classical Yang–Baxter equation and quiver representations”, Communications in Mathematical Physics, 278:1 (2008), 83  crossref  mathscinet  zmath  adsnasa  isi  scopus
    22. Roubtsov V., Skrypnyk T., “Compatible Poisson Brackets, Quadratic Poisson Algebras and Classical r-Matrices”, Differential Equations: Geometry, Symmetries and Integrability - the Abel Symposium 2008, Abel Symposia, 5, 2009, 311–333  mathscinet  zmath  isi
    23. R. A. Atnagulova, I. Z. Golubchik, “Novye resheniya uravneniya Yanga–Bakstera s kvadratom”, Ufimsk. matem. zhurn., 4:3 (2012), 6–16  mathnet  mathscinet
    24. Zhang Y., Bai Ch., Guo L., “The Category and Operad of Matching Dialgebras”, Appl. Categ. Struct., 21:6 (2013), 851–865  crossref  mathscinet  zmath  isi  scopus
    25. Skrypnyk T., “Decompositions of Quasigraded Lie Algebras, Non-Skew-Symmetric Classical R-Matrices and Generalized Gaudin Models”, J. Geom. Phys., 75 (2014), 98–112  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    26. Zhang Yong Bai ChengMing G.L., “Totally Compatible Associative and Lie Dialgebras, Tridendriform Algebras and Postlie Algebras”, Sci. China-Math., 57:2 (2014), 259–273  crossref  mathscinet  zmath  isi  scopus
    27. Panasyuk A., “Compatible Lie Brackets: Towards a Classification”, J. Lie Theory, 24:2 (2014), 561–623  mathscinet  zmath  isi
    28. Dobrogowska A., “R-Matrix, Lax pair, and Multiparameter Decompositions of Lie Algebras”, J. Math. Phys., 56:11 (2015), 113508  crossref  mathscinet  zmath  isi  scopus
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    30. Daniel J. F. Fox, “Symmetries of the Space of Linear Symplectic Connections”, SIGMA, 13 (2017), 002, 30 pp.  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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