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.
| Published in |
International Journal of Sustainable and Green Energy (Volume 4, Issue 3-2)
This article belongs to the Special Issue Wind-Generated Waves, 2D Integrable KdV Hierarchies and Solitons |
| DOI | 10.11648/j.ijrse.s.2015040302.13 |
| 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), 2014. Published by Science Publishing Group |
Huge Lie Algebra, Graded Modified Classical Yang-Baxter Equations, Integrable Hamiltonian Systems
| [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). |
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
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
Share This Article
Twitter
Linked In
Facebook
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}
}
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 -
Email: