Geodesic
Classical double, $R$-operators, and negative flows of integrable hierarchies
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 1, pp. 40-63 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

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