Geodesic
Dual $R$-matrix integrability
Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 147-160

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Using the $R$-operator on a Lie algebra $\mathfrak{g}$ satisfying the modified classical Yang–Baxter equation, we define two sets of functions that mutually commute with respect to the initial Lie–Poisson bracket on $\mathfrak{g}^*$. We consider examples of the Lie algebras $\mathfrak{g}$ with the Kostant–Adler–Symes and triangular decompositions, their $R$-operators, and the corresponding two sets of mutually commuting functions in detail. We answer the question for which $R$-operators the constructed sets of functions also commute with respect to the $R$-bracket. We briefly discuss the Euler–Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals.
Keywords: Lie algebra, classical $R$-matrix, classical integrable system. 
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     author = {T. V. Skrypnik},
     title = {Dual $R$-matrix integrability},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {147--160},
     publisher = {mathdoc},
     volume = {155},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a12/}
}
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T. V. Skrypnik. Dual $R$-matrix integrability. Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 147-160. http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a12/