Geodesic
Integrable $N$-dimensional systems on the Hopf algebra and $q$-deformations
Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 3, pp. 373-390 
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/
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     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},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a1/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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