Abstract
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 · ind g ≤ k ≤ g · ind g. where ind g and 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(sl(2)) is constructed using the method of noncommutative integration of linear differential equations.
| Original language | English |
|---|---|
| Pages (from-to) | 1172-1186 |
| Number of pages | 15 |
| Journal | Theoretical and Mathematical Physics |
| Volume | 124 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - Sep 2000 |
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ASJC Scopus subject areas
- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
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Integrable N-dimensional systems on the hopf algebra and q-deformations. / Lisitsyn, Ya V.; Shapovalov, A. V.
In: Theoretical and Mathematical Physics, Vol. 124, No. 3, 09.2000, p. 1172-1186.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Integrable N-dimensional systems on the hopf algebra and q-deformations
AU - Lisitsyn, Ya V.
AU - Shapovalov, A. V.
PY - 2000/9
Y1 - 2000/9
N2 - 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 · ind g ≤ k ≤ g · ind g. where ind g and 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(sl(2)) is constructed using the method of noncommutative integration of linear differential equations.
AB - 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 · ind g ≤ k ≤ g · ind g. where ind g and 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(sl(2)) is constructed using the method of noncommutative integration of linear differential equations.
UR - http://www.scopus.com/inward/record.url?scp=0034348048&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0034348048&partnerID=8YFLogxK
U2 - 10.1007/BF02550996
DO - 10.1007/BF02550996
M3 - Article
AN - SCOPUS:0034348048
VL - 124
SP - 1172
EP - 1186
JO - Theoretical and Mathematical Physics(Russian Federation)
JF - Theoretical and Mathematical Physics(Russian Federation)
SN - 0040-5779
IS - 3
ER -