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TMF, 2012, Volume 172, Number 1, Pages 40–63 (Mi tmf6903)  

This article is cited in 2 scientific papers (total in 2 papers)

Classical double, $R$-operators, and negative flows of integrable hierarchies

B. A. Dubrovinab, T. V. Skrypnikacd

a Lomonosov Moscow State University, Moscow, Russia
b International School for Advanced Studies, Trieste, Italy
c Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
d Universita di Milano Bicocca, Milan, Italy

Abstract: Using the classical double $\mathcal G$ of a Lie algebra $\mathfrak g$ equipped with the classical $R$-operator, we define two sets of functions commuting with respect to the initial Lie–Poisson bracket on $\mathfrak g^*$ and its extensions. We consider examples of Lie algebras $\mathfrak g$ with the “Adler–Kostant–Symes” $R$-operators and the two corresponding sets of mutually commuting functions in detail. Using the constructed commutative Hamiltonian flows on different extensions of $\mathfrak g$, we obtain zero-curvature equations with $\mathfrak g$-valued $U$$V$ pairs. The so-called negative flows of soliton hierarchies are among such equations. We illustrate the proposed approach with examples of two-dimensional Abelian and non-Abelian Toda field equations.

Keywords: classical $R$-operator, integrable hierarchy

DOI: https://doi.org/10.4213/tmf6903

Full text: PDF file (624 kB)
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English version:
Theoretical and Mathematical Physics, 2012, 172:1, 911–931

Bibliographic databases:

Received: 28.04.2011
Revised: 13.11.2011

Citation: B. A. Dubrovin, T. V. Skrypnik, “Classical double, $R$-operators, and negative flows of integrable hierarchies”, TMF, 172:1 (2012), 40–63; Theoret. and Math. Phys., 172:1 (2012), 911–931

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Dobrogowska A., “R-Matrix, Lax pair, and Multiparameter Decompositions of Lie Algebras”, J. Math. Phys., 56:11 (2015), 113508  crossref  mathscinet  zmath  adsnasa  isi
    2. Skrypnyk T., “Reduction in Soliton Hierarchies and Special Points of Classical R-Matrices”, J. Geom. Phys., 130 (2018), 260–287  crossref  mathscinet  zmath  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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