Geodesic
Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics
Teoretičeskaâ i matematičeskaâ fizika, Tome 75 (1988) no. 1, pp. 3-17 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A new and extremely important property of the algebraic structure of symmetries of nonlinear infinite-dimensional integrable Hamiltonian dynamical systems is described. It is that their invariance groups are isomorphic to a unique universal Banach Lie group of currents $G=\mathcal I\odot\mathrm{diff}(T^n)$ on an $n$-dimensional torus $T^n$. Applications of this phenomenon to the problem of constructing general criteria of integrability of nonlinear dynamical systems of theoretical and mathematical physics are considered. 
@article{TMF_1988_75_1_a0,
     author = {N. N. Bogolyubov (Jr.) and A. K. Prikarpatskii},
     title = {Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--17},
     year = {1988},
     volume = {75},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1988_75_1_a0/}
}
TY  - JOUR
AU  - N. N. Bogolyubov (Jr.)
AU  - A. K. Prikarpatskii
TI  - Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1988
SP  - 3
EP  - 17
VL  - 75
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1988_75_1_a0/
LA  - ru
ID  - TMF_1988_75_1_a0
ER  - 
%0 Journal Article
%A N. N. Bogolyubov (Jr.)
%A A. K. Prikarpatskii
%T Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1988
%P 3-17
%V 75
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1988_75_1_a0/
%G ru
%F TMF_1988_75_1_a0
N. N. Bogolyubov (Jr.); A. K. Prikarpatskii. Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics. Teoretičeskaâ i matematičeskaâ fizika, Tome 75 (1988) no. 1, pp. 3-17. http://geodesic.mathdoc.fr/item/TMF_1988_75_1_a0/

[1] Bogolyubov N. N., Bogolyubov N. N. (ml.), Vvedenie v kvantovuyu statisticheskuyu mekhaniku, Nauka, M., 1984 | MR

[2] Bogolyubov N. N. (ml.), Prikarpatskii A. K., EChAYa, 17:4 (1986), 789–827 | MR

[3] Goldin G. A., Sharp D. H., Commun. Math. Phys., 92:2 (1983), 217–228 | DOI | MR | Zbl

[4] Bogolyubov N. N. (ml.), Prikarpatskii A. K., UMZh, 38:3 (1986), 284–289 | MR

[5] Prikarpatskii A. K., Samoilenko V. G., Integriruemost dinamicheskikh sistem Kaupa i tipa Neimana i ikh tochnye resheniya, Preprint No 86-56, IM AN USSR, K., 1986

[6] Bogolyubov N. N. (ml.) i dr., TMF, 65:2 (1985), 271–284 | MR | Zbl

[7] Bogolyubov N. N. (ml.), Prikarpatskii A. K., TMF, 67:3 (1986), 410–425 | MR | Zbl

[8] Prikarpatskii A. K., DAN SSSR, 287:4 (1986), 827–832 | MR

[9] Mitropolskii Yu. A., Prikarpatskii A. K., Samoilenko V. G., DAN SSSR, 287:6 (1986), 1312–1317 | MR

[10] Olive D. I., Kac–Moody algebras for Physicists, Preprint 184-85/14, Imperial College, Cambridge, 1985 | MR

[11] Godbiion K., Differentsialnaya geometriya i analiticheskaya mekhanika, Mir, M., 1973

[12] Gelfand I. M., Dorfman I. Ya., Funkts. analiz i ego prilozh., 13:4 (1979), 13–30 | MR | Zbl

[13] Fucssteiner B., Fokas A. S., Stud. in Appl. Mathem., 68:1 (1983), 1–10 ; Phys. Lett., 86A:6/7 (1981), 341–345 | DOI | MR

[14] Magri F., J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[15] Chen H. H., Lee Y. C., Lin J. E., Physica D, 9:3 (1983), 439–445 ; Прикарпатский А. А., Самойленко В. Г., Функциональные уравнения в статистической механике и интегрируемые системы, Препринт No 86-53, ИМ АН УССР, К., 1986 | DOI | MR | Zbl

[16] Ablowitz M. J., Fokas A. S., Physica, 4D:1 (1981), 47–66 | MR

[17] Sidorenko Yu. N., DAN USSR. Ser. A, 1985, no. 11, 21–24; 1986, No 3, 21–23 | MR

[18] Sidorenko Yu. N., Sb. nauchn. trudov LOMI, 150, LOMI, L., 1986, 105–111; Орлов А. Ю., Шульман Е. И., ТМФ, 64:2 (1985), 323–328 | MR | Zbl

[19] Melnikov V. K., Matem. sb., 121:4 (1983), 469–498 | MR | Zbl

[20] Bogolyubov N. N. (ml.) i dr., Kvantovaya i klassicheskaya integriruemost nelineinoi modeli tipa Shredingera, Preprint No 84-53, IM AN USSR, K., 1984 | MR

[21] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[22] Korepin V. N., Izergin A. G., EChAYa, 14:2 (1983), 3–58

[23] Faddeev L. D., Sov. Sci. Rev. Sec. C. Math. Phys., 1 (1980), 107–155 | Zbl