Geodesic
Integrable $N$-dimensional systems on the Hopf algebra and $q$-deformations
Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 3, pp. 373-390
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct the class of integrable classical and quantum systems on the Hopf algebras describing $n$ interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra $A(g)$ of a simple Lie algebra $g$ and prove that the integrals of motion depend only on linear combinations of $k$ coordinates of the phase space, $2\cdot\mathrm{ind}g\leq k\leq\mathbf g\cdot\mathrm{ind}g$, where $\mathrm{ind} g$ and $\mathbf g$ are the respective index and Coxeter number of the Lie algebra $g$. The standard procedure of $q$-deformation results in the quantum integrable system. We apply this general scheme to the algebras $sl(2)$, $sl(3)$, and $o(3,1)$. An exact solution for the quantum analogue of the $N$-dimensional Hamiltonian system on the Hopf algebra $A\bigl(sl(2)\bigr)$ is constructed using the method of noncommutative integration of linear differential equations. 
@article{TMF_2000_124_3_a1,
     author = {Ya. V. Lisitsyn and A. V. Shapovalov},
     title = {Integrable $N$-dimensional systems on the {Hopf} algebra and $q$-deformations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {373--390},
     year = {2000},
     volume = {124},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a1/}
}
TY  - JOUR
AU  - Ya. V. Lisitsyn
AU  - A. V. Shapovalov
TI  - Integrable $N$-dimensional systems on the Hopf algebra and $q$-deformations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2000
SP  - 373
EP  - 390
VL  - 124
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a1/
LA  - ru
ID  - TMF_2000_124_3_a1
ER  - 
%0 Journal Article
%A Ya. V. Lisitsyn
%A A. V. Shapovalov
%T Integrable $N$-dimensional systems on the Hopf algebra and $q$-deformations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2000
%P 373-390
%V 124
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a1/
%G ru
%F TMF_2000_124_3_a1
Ya. V. Lisitsyn; A. V. Shapovalov. Integrable $N$-dimensional systems on the Hopf algebra and $q$-deformations. Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 3, pp. 373-390. http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a1/

[1] A. S. Mischenko, A. T. Fomenko, Funkts. analiz i ego prilozh., 12:2 (1978), 46–56 | MR | Zbl

[2] A. V. Brailov, DAN SSSR, 271:2 (1983), 273–276 | MR

[3] A. T. Fomenko, Simplekticheskaya geometriya. Metody i prilozhenie, Izd-vo MGU, M., 1988 | MR | Zbl

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

[5] M. V. Karasev, V. P. Maslov, Nelineinye skobki Puassona. Geometriya i kvantovanie, Nauka, M., 1991 | MR | Zbl

[6] O. Babelon, C. M. Viallet, Phys. Lett. B, 237 (1990), 411–415 | DOI | MR

[7] E. K. Sklyanin, Funkts. analiz i ego prilozh., 16:4 (1982), 27–34 | MR | Zbl

[8] E. K. Sklyanin, Funkts. analiz i ego prilozh., 17:4 (1983), 34–48 | MR | Zbl

[9] V. B. Kuznetsov, A. V. Tsyganov, Zap. nauchn. semin. LOMI, 172, 1989, 88–98 | MR

[10] A. Ballesteros, O. Ragnisco, A systematic construction of completely integrable Hamiltonians from coalgebras, E-print solv-int/9802008 | MR

[11] A. Ballesteros, F. J. Herranz, Long range integrable oscillator chains from quantum algebras, E-print solv-int/9805004

[12] V. N. Shapovalov, Diff. uravneniya, 16 (1980), 1864–1874 | MR | Zbl

[13] A. V. Shapovalov, I. V. Shirokov, TMF, 104:2 (1995), 195–213 | MR | Zbl

[14] A. V. Shapovalov, I. V. Shirokov, TMF, 106:1 (1996), 3–15 | DOI | MR | Zbl

[15] E. K. Sklyanin, Progr. Theor. Phys. Suppl., 118 (1995), 35–60 | DOI | MR | Zbl

[16] E. K. Sklyanin, Separation of variables in the quantum integrable models related to the Yangian ${\mathcal Y}\bigl[sl(3)\bigr]$, E-print hep-th/9212076 | MR

[17] J. C. Eilbeck, V. Z. Enol'skii, V. B. Kuznetsov, A. V. Tsiganov, J. Phys. A, 27 (1994), 567–578 | DOI | MR | Zbl

[18] J. Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge University Press, Cambridge, 1995 | MR | Zbl