Toda flows and isospectral manifolds
HTML articles powered by AMS MathViewer
- by Leonid Faybusovich
- Proc. Amer. Math. Soc. 115 (1992), 837-847
- DOI: https://doi.org/10.1090/S0002-9939-1992-1128727-2
- PDF | Request permission
Abstract:
We apply Bott’s method to the calculation of Betti numbers of isospectral manifolds. Necessary properties of Toda flows, including a description of the phase portrait, are given with complete proofs.References
- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
- Raoul Bott, An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. France 84 (1956), 251–281. MR 87035
- G. L. Luke (ed.), Representation theory of Lie groups, Cambridge University Press, Cambridge, 1979. MR 568880
- Michael W. Davis, Some aspherical manifolds, Duke Math. J. 55 (1987), no. 1, 105–139. MR 883666, DOI 10.1215/S0012-7094-87-05507-4
- L. E. Faĭbusovich, $\textrm {QR}$-type factorizations, the Yang-Baxter equation, and an eigenvalue problem of control theory, Linear Algebra Appl. 122/123/124 (1989), 943–971. MR 1020016, DOI 10.1016/0024-3795(89)90681-2
- L. E. Faĭbusovich, The $QR$-algorithm and generalized Toda flows, Ukrain. Mat. Zh. 41 (1989), no. 7, 944–952, 1007 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 7, 806–813 (1990). MR 1024291, DOI 10.1007/BF01060698 —, Generalized Toda flows, the Riccati equations on Grassmannian and the QR-algorithm, Funct. Anal. Appl. 21 (1987), 166-168.
- David Fried, The cohomology of an isospectral flow, Proc. Amer. Math. Soc. 98 (1986), no. 2, 363–368. MR 854048, DOI 10.1090/S0002-9939-1986-0854048-6
- I. M. Gel′fand and V. V. Serganova, Combinatorial geometries and the strata of a torus on homogeneous compact manifolds, Uspekhi Mat. Nauk 42 (1987), no. 2(254), 107–134, 287 (Russian). MR 898623
- Roe Goodman and Nolan R. Wallach, Classical and quantum mechanical systems of Toda-lattice type. II. Solutions of the classical flows, Comm. Math. Phys. 94 (1984), no. 2, 177–217. MR 761793
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561 R. Hermann, Toda lattices, cosymplectic manifolds, kinds, Part B, Interdisciplinary Math., vol. 18, Math. Sci. Press, Brookline, MA, 1978.
- Bertram Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), no. 3, 195–338. MR 550790, DOI 10.1016/0001-8708(79)90057-4
- J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197–220. MR 375869, DOI 10.1016/0001-8708(75)90151-6 M. Shayman and F. De Mari, Hessenberg varieties and generalized Eulerian numbers for semisimple Lie groups: The classical cases, preprint 1988.
- W. W. Symes, The $QR$ algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D 4 (1981/82), no. 2, 275–280. MR 653781, DOI 10.1016/0167-2789(82)90069-0
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 837-847
- MSC: Primary 58F09; Secondary 17B20, 35Q58, 58F07
- DOI: https://doi.org/10.1090/S0002-9939-1992-1128727-2
- MathSciNet review: 1128727