International Journal of Sustainable and Green Energy

| Peer-Reviewed |

Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems

Received: 1 November 2014    Accepted: 5 November 2014    Published: 11 November 2014
Views:       Downloads:

Share This Article

Abstract

The 6=3×2 huge Lie algebra Ξ of all local and non-local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket scheme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter (GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consistent way a wide class of integrable systems. Other algebraic properties are also presented.

DOI 10.11648/j.ijrse.s.2015040302.13
Published in International Journal of Sustainable and Green Energy (Volume 4, Issue 3-2, May 2015)

This article belongs to the Special Issue Wind-Generated Waves, 2D Integrable KdV Hierarchies and Solitons

Page(s) 10-16
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Huge Lie Algebra, Graded Modified Classical Yang-Baxter Equations, Integrable Hamiltonian Systems

References
[1] E.H. Saidi and M.B. Sedra, "On The Gelfand-Dickey sln Algebra and W_n-Symmetry: The Bosonic case". J. Math. Phys.35, 3190 (1994);
[2] B. Kostant, London Math. Soc. Lect. Notes, Ser.34 (1979)287; M. Adler, "On a Trace Functional for Pseudo-Differential Operators and the Symplectic structure of the KdV Equations". Invent. Math. 50 (1979) 219; A.G. Reyman, M.A. Semenov-Tian-Shansky and I.B. Frenkel, J. Soviet. Math.247 (1979)802; A.G. Reyman and M.A. Semenov-Tian-Shansky, Invent. Math.54 (1979)81; 63 (1981)423;W. Symes, Invent. Math.59 (1980)13.H. Aratyn, E. Nissimov, S. Pacheva and I. Vaysburd, "R-matrix Formulation of KP Hierarchies and Their Gauge Equivalence". Phys. Lett. 294B (1992) 167(also in hep-th/9209006)
[3] H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, "On W_∞ Algebras, Gauge Equivalence of KP Hierarchies, Two-Boson Realizations and Their KdV Reduction". hep-th/9304152
[4] M.A. Semenov-Tian-Shansky, "What-is a classical r-matrix ?" Funct. Anal. and Its Appl. 17(1983)259-272A.B. Zamolodchikov, Ther. Math. Phys. 65(1985) 1205;A.B. Zamolodchikov and V.A. Fateev, Nucl. Phys. B280 [FS 18] (1987)644.
[5] I. Bakas, Commun. Math. Phys. 123, 627 (1989).A. Leznov and M. Saviliev, Lett. Math. Phys. 3(1979) 489; Commun. Math. Phys. 74, 111(1980);P. Mansfield, Nucl. Phys. B208 (1982)277.
[6] M.B. Sedra, "On The Huge Lie Superalgebra of Pseudo-Superdifferential Operators and Super KP-Hierarchies". J. Math. Phys.37, 3483(1996) K. Ikeda, "The Higher Order Hamiltonian Structures for the Modified Classical Yang-Baxter Equation" Commun. Math. Phys. 180, 757-777 (1996).
Cite This Article
  • APA Style

    Mahmoud Akdi, Amina Boulahoual, Moulay Brahim Sedra. (2014). Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems. International Journal of Sustainable and Green Energy, 4(3-2), 10-16. https://doi.org/10.11648/j.ijrse.s.2015040302.13

    Copy | Download

    ACS Style

    Mahmoud Akdi; Amina Boulahoual; Moulay Brahim Sedra. Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems. Int. J. Sustain. Green Energy 2014, 4(3-2), 10-16. doi: 10.11648/j.ijrse.s.2015040302.13

    Copy | Download

    AMA Style

    Mahmoud Akdi, Amina Boulahoual, Moulay Brahim Sedra. Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems. Int J Sustain Green Energy. 2014;4(3-2):10-16. doi: 10.11648/j.ijrse.s.2015040302.13

    Copy | Download

  • @article{10.11648/j.ijrse.s.2015040302.13,
      author = {Mahmoud Akdi and Amina Boulahoual and Moulay Brahim Sedra},
      title = {Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems},
      journal = {International Journal of Sustainable and Green Energy},
      volume = {4},
      number = {3-2},
      pages = {10-16},
      doi = {10.11648/j.ijrse.s.2015040302.13},
      url = {https://doi.org/10.11648/j.ijrse.s.2015040302.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijrse.s.2015040302.13},
      abstract = {The 6=3×2 huge Lie algebra Ξ of all local and non-local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket scheme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter (GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consistent way a wide class of integrable systems. Other algebraic properties are also presented.},
     year = {2014}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems
    AU  - Mahmoud Akdi
    AU  - Amina Boulahoual
    AU  - Moulay Brahim Sedra
    Y1  - 2014/11/11
    PY  - 2014
    N1  - https://doi.org/10.11648/j.ijrse.s.2015040302.13
    DO  - 10.11648/j.ijrse.s.2015040302.13
    T2  - International Journal of Sustainable and Green Energy
    JF  - International Journal of Sustainable and Green Energy
    JO  - International Journal of Sustainable and Green Energy
    SP  - 10
    EP  - 16
    PB  - Science Publishing Group
    SN  - 2575-1549
    UR  - https://doi.org/10.11648/j.ijrse.s.2015040302.13
    AB  - The 6=3×2 huge Lie algebra Ξ of all local and non-local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket scheme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter (GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consistent way a wide class of integrable systems. Other algebraic properties are also presented.
    VL  - 4
    IS  - 3-2
    ER  - 

    Copy | Download

Author Information
  • LHESIR, Faculty of Science of Kenitra, Ibn Toufail University, Kenitra, Morocco

  • LHESIR, Faculty of Science of Kenitra, Ibn Toufail University, Kenitra, Morocco

  • LHESIR, Faculty of Science of Kenitra, Ibn Toufail University, Kenitra, Morocco; ENSAH, Mohammed First University, Al Hoceima, Morocco

  • Sections