Classical double, $R$-operators, and negative flows of integrable hierarchies
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 1, pp. 40-63
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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.
@article{TMF_2012_172_1_a3,
author = {B. A. Dubrovin and T. V. Skrypnik},
title = {Classical double, $R$-operators, and negative flows of integrable hierarchies},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {40--63},
publisher = {mathdoc},
volume = {172},
number = {1},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a3/}
}
TY - JOUR AU - B. A. Dubrovin AU - T. V. Skrypnik TI - Classical double, $R$-operators, and negative flows of integrable hierarchies JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2012 SP - 40 EP - 63 VL - 172 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a3/ LA - ru ID - TMF_2012_172_1_a3 ER -
B. A. Dubrovin; T. V. Skrypnik. Classical double, $R$-operators, and negative flows of integrable hierarchies. Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 1, pp. 40-63. http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a3/