Integrable Systems and Finite-Dimensional Lie Algebras

  • M. A. Olshanetsky
  • A. M. Perelomov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 16)


In this survey we consider integrable systems whose construction makes use of root systems of simple (usually finite-dimensional) Lie algebras.


Toda Lattice Cartan Matrix Integrable Hamiltonian System Coadjoint Orbit Periodic Toda Lattice 
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  1. Arnol’d, V.I. [1974]: Mathematical Methods of Classical Mechanics. Nauka: Moscow, 431 pp. English transl.: Graduate Texts in Mathematics 60, Springer-Verlag: New York — Berlin-Heidelberg 1978,462 pp., 1978, Zbl.386.70001MATHGoogle Scholar
  2. Berezin, F.A. [1967]: Some remarks on the universal enveloping algebra of a Lie algebra. Funkts. Anal. Prilozh. 1, No. 2, 1–14. English transl.: Funct. Anal. Appl. 1, 91–102, Zbl.227.22020MathSciNetCrossRefGoogle Scholar
  3. Bogoyavlensky, O.I. (= Bogoyavlenskij, O.I.) [1976]: On perturbations of the periodic Toda lattice. Commun. Math. Phys. 51, 201–209MathSciNetCrossRefGoogle Scholar
  4. Borel, A. [1954]: Kählerian coset spaces of semisimple Lie groups. Proc. Natl. Acad. Sci. USA 40, 1147–1151, Zbl.58, 160MathSciNetCrossRefMATHGoogle Scholar
  5. Bourbaki, N. [1968]: Groupes et Algèbres de Lie. Ch. IV. Hermann: Paris, 288 pp, Zbl.186, 330Google Scholar
  6. Calogero, F. [1971]: Solution of the one-dimensional n-body problem with quadratic and/or inversely quadratic pair potential. J. Math. Phys. 12, 419–436MathSciNetCrossRefGoogle Scholar
  7. Calogero, F. [1975]: Exactly solvable one-dimensional many body-problems. Lett. Nuovo Cim. 13, 411–427MathSciNetCrossRefGoogle Scholar
  8. Calogero, F. [1976]: On a functional equation connected with many-body problem. Lett. Nuovo Cim. 16, 77–80MathSciNetCrossRefGoogle Scholar
  9. Calogero, F. [1978]: Integrable many-body problems. In: Nonlinear equations in physics and mathematics. Proc. NATO adv. Study Inst., Istanbul 1977 (Barut, A.O. (Ed.)), 3–53, Zbl.399.70022CrossRefGoogle Scholar
  10. Case, K.M. [1976]: Meromorphic solutions of the Benjamin-Ono equation. Physica A 96, 173–182MathSciNetCrossRefGoogle Scholar
  11. Dubrovin, B.A., Krichever, I.M., Novikov, S.P. [1985]: Dynamical Systems IV: Integrable systems. In: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 4, 179–285. English transl: Encyclopaedia of Mathematical Sciences 4, 173–280. Springer-Verlag: Berlin-Heidelberg-New York 1990, Zbl.591.58013Google Scholar
  12. Flaschka, H. [1974a]: The Toda lattice. I. Existence of integrals. Phys. Rev. B 9, 1924–1925MathSciNetCrossRefMATHGoogle Scholar
  13. Flaschka, H. [1974b]: On the Toda lattice. II. Inverse scattering solution. Progr. Theor. Phys. 51, 703–716MathSciNetCrossRefMATHGoogle Scholar
  14. Goodman, R., Wallach, N.R. [1984]: Classical and quantum mechanical systems of Toda lattice type II. Commun. Math. Phys. 94, 177–217, Zbl.592, 58028MathSciNetCrossRefMATHGoogle Scholar
  15. Helgason, S. [1978]: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press: New York, 643 pp., Zbl.451.53088MATHGoogle Scholar
  16. Kac, V.G. [1968]: Simple irreducible graded Lie algebras of finite growth. Izv. Akad. Nauk SSSR, Ser. Mat. 32, No. 6, 1323–1367. English transl.: Math. USSR, Izv. 2, 1271–1311 (1968), Zbl.222.17007Google Scholar
  17. Kamalin, S.A., Perelomov, A.M. [1985]: Construction of canonical coordinates on polarized coadjoint orbits of graded Lie groups. Commun. Math. Phys. 97, 553–568, Zbl.578.58016MathSciNetCrossRefMATHGoogle Scholar
  18. Kazhdan, D., Kostant, B., Sternberg, S. [1978]: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31, 491–508, Zbl.368.58008MathSciNetGoogle Scholar
  19. Kirillov, A.A. [1972]: Elements of the Theory of Representations. Nauka: Moscow, 336 pp. English transl.: Grundlehren der Mathematischen Wissenschaften 220. Springer-Verlag: New York-Berlin-Heidelberg 1976, 315 pp., Zbl.264.22011Google Scholar
  20. Kostant, B. [1979]: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338, Zbl.433.22008MathSciNetCrossRefMATHGoogle Scholar
  21. Krichever, I.M. [1978a]: Algebraic curves and nonlinear difference equations. Usp. Mat. Nauk 33, No. 4, 215–216. English transl.: Russ. Math. Surv. 33, No. 4, 255–256 (1978), Zbl.382.39003MathSciNetMATHGoogle Scholar
  22. Krichever, I.M. [1980]: Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Funkts. Anal. Prilozh. 14, No. 4, 45–54. English transl.: Funct. Anal. Appl. 14, 282–290 (1981), Zbl.462.35080MathSciNetMATHGoogle Scholar
  23. Manakov, S.V. [1974]: Complete integrability and stochastization of discrete dynamical systems. Zh. Exp. Teor. Fiz. 40, 269–274. English transl.: Sov. Phys. ZETP 40, 269–274 (1975)MathSciNetGoogle Scholar
  24. Marsden, J., Weinstein, A. [1974]: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130, Zbl.327.58005MathSciNetCrossRefMATHGoogle Scholar
  25. Moody, R.V. [1968]: A new class of Lie algebras. J. Algebra 10, 221–230, Zbl.191,30MathSciNetMATHGoogle Scholar
  26. Moser, J. [1975]: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220, Zbl.303.34019CrossRefMATHGoogle Scholar
  27. Moser, J. [1980]: Various aspect of integrable Hamiltonian systems. In: Dynamical Systems, C.I.M.E. Lect. Bressanone 1978. Prog. Math. 8, 233–289, Zbl.468.58011Google Scholar
  28. Olshanetsky, M.A. (= Olshanetskij, M.A.), Perelomov, A.M. [1976a]: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108, Zbl.342.58017MathSciNetCrossRefMATHGoogle Scholar
  29. Olshanetsky, M.A., Perelomov, A.M. [1976b]: Explicit solutions of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature. Lett. Nuovo Cim. 16, 333–339MathSciNetCrossRefGoogle Scholar
  30. Olshanetsky, M.A., Perelomov, A.M. [1976c]: Explicit solutions of some completely integrable systems. Lett. Nuovo Cim. 17, 97–101MathSciNetCrossRefGoogle Scholar
  31. Olshanetsky, M.A., Perelomov, A.M. [1979]: Explicit solutions of the classical generalized Toda models. Invent. Math. 54, 261–269, Zbl.419.58008MathSciNetCrossRefMATHGoogle Scholar
  32. Olshanetsky, M.A., Perelomov, A.M. [1980]: The Toda lattice as a reduced system. Teor. Mat. Fiz. 45, No. 1, 3–18. English transl.: Theor. Math. Phys. 45, 843–854 (1981)Google Scholar
  33. Olshanetsky, M.A., Perelomov, A.M. [1981]: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71C, No. 5, 313–400MathSciNetCrossRefGoogle Scholar
  34. Olshanetsky, M.A., Perelomov, A.M. [1983]: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, 313–404MathSciNetCrossRefGoogle Scholar
  35. Olshanetsky, M.A., Rogov, V.B. [1978]: Bound states in completely integrable systems with two types of particles. Ann. Inst. Henri Poincare, Sect. A 29,169–177, Zbl.416.58014Google Scholar
  36. Perelomov, A.M. [1976]: Completely integrable classical systems connected with semisimple Lie algebras. III. Lett. Math. Phys. 1, 531–534MathSciNetCrossRefGoogle Scholar
  37. Perelomov, A.M. [1978]: The simple relation between certain dynamical systems. Commun. Math. Phys. 63, 9–11, Zbl.435.70013MathSciNetCrossRefMATHGoogle Scholar
  38. Perelomov, A.M. [1983a]: Systems with constraints. Preprint ITEP No. Ill, Moscow, 26 pp.Google Scholar
  39. Perelomov, A.M. [1983b]: Motion of a rigid body about a fixed point. Preprint ITEP No. 147, Moscow, 42 pp.Google Scholar
  40. Perelomov, A.M. [1990]: Integrable Systems of Classical Mechanics and Lie Algebras. Nauka: Moscow, 240 pp., English transi.: Birkhäuser: Basel - Boston - Berlin, 307 pp., Zbl.699.70003, Zbl.717.70003CrossRefGoogle Scholar
  41. Reyman, A.G. [1980]: Integrable Hamiltonian systems connected with graded Lie algebras. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 95, 3–54. English transi.: J. Sov. Math. 19, No. 5, 1507–1545 (1982), Zbl.488.70013MATHGoogle Scholar
  42. Reyman, A.G., Semenov-Tian-Shansky, M.A. [1979]: Reduction of Hamiltonian system, affine Lie algebras and Lax equations. I. Invent. Math. 54, 81–100, Zbl.403.58004MathSciNetGoogle Scholar
  43. Reyman, A.G., Semenov-Tian-Shansky, M.A. [1981]: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. II. Invent. Math. 63,423–432, Zbl.442.58016MathSciNetCrossRefMATHGoogle Scholar
  44. Reyman, A.G., Semenov-Tian-Shansky, M.A., Frenkel, LB. [1979]: Graded Lie algebras and completely integrable systems. Dokl. Akad. Nauk SSSR 247, No. 4, 802–805. English transi: Sov. Math., Dokl. 20, 811–814 (1979), Zbl.437.58008Google Scholar
  45. Sawada, K., Kotera, T. [1975]: Integrability and a solution for the one-dimensional n-particle system with inversely quadratic pair potential. J. Phys. Soc. Japan 39,1614–1618MathSciNetCrossRefGoogle Scholar
  46. Sutherland, B. [1971]: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019–2021CrossRefGoogle Scholar
  47. Toda, M. [1967]: Wave propagation in anharmonic lattices. J. Phys. Soc. Japan 23, 501–506CrossRefGoogle Scholar
  48. Weinstein, A. [1983]: The local structure of Poisson manifolds. J. Differ. Geom. 18, 523–557, Zbl.524.58011MATHGoogle Scholar
  49. Wojciechowski, S. [1977]: Involutive set of integrals for completely integrable many body problems with pair interactions. Lett. Nuovo Cim. 18, 103–107CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • M. A. Olshanetsky
  • A. M. Perelomov

There are no affiliations available

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