The solution to a generalized Toda lattice and representation theory.(English)Zbl 0433.22008
MSC:
| 22E30 | Analysis on real and complex Lie groups |
| 22E60 | Lie algebras of Lie groups |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 70F10 | \(n\)-body problems |
| 70H99 | Hamiltonian and Lagrangian mechanics |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
split Lie algebra;split Cartan subalgebra;Hamiltonian system;symplectic structure;isospectral manifolds;Toda latticeCitations:
Zbl 0323.70012Cite
Full Text:DOI
References:
| [1] | M. Adler;M. Adler ·Zbl 0414.58020 |
| [2] | Auslander, L.; Kostant, B., Polarization and unitary representations of solvable Lie groups, Invent. Math., 14, 255-354 (1971) ·Zbl 0233.22005 |
| [3] | Bogoyavlensky, O. I., On perturbations of the periodic Toda lattice, Comm. Math. Phys., 51, 201-209 (1976) |
| [4] | Dixmier, J., Algèbres enveloppantes, (Cahiers scientifiques no. 27 (1974), Gauthier-Villars: Gauthier-Villars Paris) ·Zbl 0422.17003 |
| [5] | Dynkin, E. B., The structure of semi-simple Lie algebras, Amer. Math. Soc. Transl. Ser., 1, No. 17 (1950) ·Zbl 0052.26202 |
| [6] | Dynkin, E. B., Semi-simple subalgebras of semi-simple Lie algebras, Amer. Math. Soc. Transl. Ser. 2, 6, 111-244 (1957) ·Zbl 0077.03404 |
| [7] | Dynkin, E. B., The maximal subgroups of the classical groups, Amer. Math. Soc. Transl. Ser. 2, 6, 245-378 (1957) ·Zbl 0077.03403 |
| [8] | Flaschka, H., On the Toda lattice, II, Progr. Theor. Phys., 51, 703-716 (1974) ·Zbl 0942.37505 |
| [9] | Gelfand, I. M.; Kazhdan, D. A., Representations of the group \(Gl (n, K)\) where \(K\) is a local field, (Lie Groups and Their Representations (1975), Wiley: Wiley New York), 95-118 ·Zbl 0348.22011 |
| [10] | R. Hermann;R. Hermann ·Zbl 0381.35001 |
| [11] | Humphreys, J., Introduction to Lie Algebras and Representation Theory, (Graduate Texts in Mathematics No. 9 (1972), Springer-Verlag: Springer-Verlag New York/Berlin) ·Zbl 0254.17004 |
| [12] | Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., 81, 973-1032 (1959) ·Zbl 0099.25603 |
| [13] | Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math., 74, 329-387 (1961) ·Zbl 0134.03501 |
| [14] | Kostant, B., Lie group representations on polynomial rings, Amer. J. Math., 85, 327-404 (1963) ·Zbl 0124.26802 |
| [15] | Kostant, B., Quantization and unitary representations, (Lecture Notes in Mathematics No. 170 (1970), Springer-Verlag: Springer-Verlag New York/Berlin), 87-208 ·Zbl 0223.53028 |
| [16] | Kostant, B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Normale Supérieure, 6, 413-455 (1973) ·Zbl 0293.22019 |
| [17] | Kostant, B., On Whittaker vectors and representation theory, Invent. Math., 48, 101-184 (1978) ·Zbl 0405.22013 |
| [18] | Kostant, B.; Rallis, S., Orbits and representations associated with symmetric spaces, Amer. J. Math., 93, 753-809 (1971) ·Zbl 0224.22013 |
| [19] | Moser, J., Finitely many mass points on the line under the influence of an exponential potential—an integrable system, (Battelle Rencontres Summer Lectures. Battelle Rencontres Summer Lectures, Lecture Notes in Mathematics (1974), Springer-Verlag: Springer-Verlag New York/Berlin), 467-487 ·Zbl 0323.70012 |
| [20] | Olshanetzki, M. A.; Perelomov, A. M., Completely integrable Hamiltonian systems connected with semi-simple Lie algebras, Invent. Math., 37, 93-108 (1976) ·Zbl 0342.58017 |
| [21] | Pukanszky, L., On the theory of exponential groups, Trans. Amer. Math. Soc., 126, 487-507 (1967) ·Zbl 0207.33605 |
| [22] | Shafarevich, I. R., Basic Algebraic Geometry (1977), Springer-Verlag: Springer-Verlag New York/Berlin ·Zbl 0362.14001 |
| [23] | Suguira, M., Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan, 11, 374-434 (1959) ·Zbl 0204.04201 |
| [24] | Toda, M., Studies of a non-linear lattice, Phys. Rep. C, Phys. Lett., 8, 1-125 (1975) |
| [25] | van Moerbeke, P., The spectrum of Jacobi matrices, Invent. Math., 37, 45-81 (1976) ·Zbl 0361.15010 |
| [26] | Varadarajan, V. S., Lie Groups, Lie Algebras and Their Representations (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J ·Zbl 0371.22001 |
| [27] | Adler, M., On a trace functional for formal pseudodifferential operators and the symplectic structure for the Korteweg-de Vries type equations, Invent. Math., 50, 219-248 (1979) ·Zbl 0393.35058 |
| [28] | Olshanetzki, M. A.; Perelomov, M. A., Explicit solution of the classical generalized Toda models (1978), preprint |
| [29] | Leznov, A. N.; Saveliev, M. V., Representation of zero curvature for the system of nonlinear partial differential equations \(xa,zz̄ = exp(kx)a\) and its integrability (1979), Serpurchev, USSR ·Zbl 0415.35017 |
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