Geodesic
Dual $R$-matrix integrability
Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 147-160 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{TMF_2008_155_1_a12,
     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},
     year = {2008},
     volume = {155},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a12/}
}
TY  - JOUR
AU  - T. V. Skrypnik
TI  - Dual $R$-matrix integrability
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2008
SP  - 147
EP  - 160
VL  - 155
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a12/
LA  - ru
ID  - TMF_2008_155_1_a12
ER  - 
%0 Journal Article
%A T. V. Skrypnik
%T Dual $R$-matrix integrability
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2008
%P 147-160
%V 155
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a12/
%G ru
%F TMF_2008_155_1_a12
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/

[1] M. A. Semënov-Tyan-Shanskii, Funkts. analiz i ego pril., 17:4 (1983), 17–33 | MR

[2] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, “Integriruemye sistemy. II. Glava 2. Teoretiko-gruppovye metody v teorii konechnomernykh integriruemykh sistem”, Dinamicheskie sistemy – 7, Itogi nauki i tekhniki. Seriya sovremennye problemy matematiki. Fundamentalnye napravleniya, 16, eds. V. I. Arnold, S. P. Novikov, VINITI, M., 1987, 119–193 | MR | Zbl

[3] A. Newell, Solitons in Mathematics and Physics, CBMS-NSF Regional Conference Series in Applied Mathematics, 48, SIAM, Philadelphia, PA, 1985 | MR | Zbl

[4] T. V. Skrypnik, TMF, 142:2 (2005), 329–345 | DOI | MR

[5] F. Guil, M. Manas, Lett. Math. Phys., 19 (1990), 89–95 | DOI | MR | Zbl

[6] T. Skrypnyk, SIGMA, 04 (2008), 011 | DOI

[7] A. Belavin, V. Drienfield, preprint ITF 1982-18, ITF, Chernogolovka | MR

[8] A. A. Belavin, V. G. Drinfeld, Funkts. analiz i ego pril., 16:3 (1982), 159–180 | MR