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 TMF, 2000, Volume 122, Number 2, Pages 251–271 (Mi tmf567)

Graded Lie algebras, representation theory, integrable mappings, and integrable systems

A. N. Leznovab

a Institute for High Energy Physics
b National Autonomous University of Mexico, Institute of Applied Mathematics and Systems

Abstract: A new class of integrable mappings and chains is introduced. The corresponding $1+2$ integrable systems that are invariant under such integrable mappings are presented in an explicit form. Soliton-type solutions of these systems are constructed in terms of matrix elements of fundamental representations of semisimple $A_n$ algebras for a given group element. The possibility of generalizing this construction to the multidimensional case is discussed.

DOI: https://doi.org/10.4213/tmf567

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English version:
Theoretical and Mathematical Physics, 2000, 122:2, 211–228

Bibliographic databases:

Citation: A. N. Leznov, “Graded Lie algebras, representation theory, integrable mappings, and integrable systems”, TMF, 122:2 (2000), 251–271; Theoret. and Math. Phys., 122:2 (2000), 211–228

Citation in format AMSBIB
\Bibitem{Lez00} \by A.~N.~Leznov \paper Graded Lie algebras, representation theory, integrable mappings, and integrable systems \jour TMF \yr 2000 \vol 122 \issue 2 \pages 251--271 \mathnet{http://mi.mathnet.ru/tmf567} \crossref{https://doi.org/10.4213/tmf567} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1776521} \zmath{https://zbmath.org/?q=an:0962.37036} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 122 \issue 2 \pages 211--228 \crossref{https://doi.org/10.1007/BF02551198} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000086555000009} 

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This publication is cited in the following articles:
1. Hu, XB, “Application of Hirota's bilinear formalism to a two-dimensional lattice by Leznov”, Physics Letters A, 276:1–4 (2000), 65
2. A. N. Leznov, “Discrete Symmetries of the $n$-Wave Problem”, Theoret. and Math. Phys., 132:1 (2002), 955–969
3. Maruno, K, “Bilinear forms of integrable lattices related to Toda and Lotka-Volterra lattices”, Journal of Nonlinear Mathematical Physics, 9 (2002), 127
4. Tam, HW, “A generalized Leznov lattice: Bilinear form, Backlund transformation, and Lax pair”, Applied Mathematics Letters, 17:1 (2004), 35
5. J. Zhao, X. Hu, H. Tam, “Applying the Pfaffianization Procedure to the Two-Dimensional Leznov Lattice”, Theoret. and Math. Phys., 144:3 (2005), 1288–1295
6. Wang, HY, “On the two-dimensional Leznov lattice equation with self-consistent sources”, Journal of Physics A-Mathematical and Theoretical, 40:42 (2007), 12691
7. Yu, GF, “On the integrable discrete versions of the Leznov lattice: Determinant solutions and Pfaffianization”, Journal of Mathematical Analysis and Applications, 335:1 (2007), 377
8. Leznov, AN, “Resolving of discrete transformation chains and multisoliton solution of the 3-wave problem”, Journal of Nonlinear Mathematical Physics, 14:2 (2007), 238
9. Li, CX, “COMMUTATIVITY OF PFAFFIANIZATION AND BACKLUND TRANSFORMATIONS: THE LEZNOV LATTICE”, Journal of Nonlinear Mathematical Physics, 16:2 (2009), 169
10. Hu, XB, “Integrable semi-discretizations and full-discretization of the two-dimensional Leznov lattice”, Journal of Difference Equations and Applications, 15:3 (2009), 233
11. Hu J., Yu G.-F., Tam H.-W., “Commutativity of the Source Generation Procedure and Integrable Semi-Discretizations: the Two-Dimensional Leznov Lattice”, J. Phys. A-Math. Theor., 45:14 (2012), 145208
12. Qin B., Tian B., Wang Yu.-F., Shen Yu.-J., Wang M., “Bell-Polynomial Approach and Wronskian Determinant Solutions For Three Sets of Differential-Difference Nonlinear Evolution Equations With Symbolic Computation”, Z. Angew. Math. Phys., 68:5 (2017), 111
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