On rational symplectic parametrization of the coadjoint orbit of $\mathrm {GL}(N)$. Diagonalizable case
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M. V. Babich and S. E. Derkachov
Translated by: the authors - St. Petersburg Math. J. 22 (2011), 347-357
- DOI: https://doi.org/10.1090/S1061-0022-2011-01145-8
- Published electronically: March 17, 2011
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Abstract:
A method for constructing birational Darboux coordinates on a coadjoint orbit of the general linear group is presented. This method is based on the Gauss decomposition of a matrix in the product of an upper-triangular and a lower-triangular matrix. The method works uniformly for the orbits formed by the diagonalizable matrices of any size and for arbitrary dimensions of the eigenspaces.References
- Nigel Hitchin, Geometrical aspects of Schlesinger’s equation, J. Geom. Phys. 23 (1997), no. 3-4, 287–300. MR 1484592, DOI 10.1016/S0393-0440(97)80005-8
- I. M. Gel′fand, Lektsii po lineĭnoĭ algebre, Izdat. “Nauka”, Moscow, 1971 (Russian). Fourth edition, augmented. MR 0352111
- F. R. Gantmaher, Teoriya matrits, Second supplemented edition, Izdat. “Nauka”, Moscow, 1966 (Russian). With an appendix by V. B. Lidskiĭ. MR 0202725
- V. I. Arnol′d, Matematicheskie metody klassicheskoĭ mekhaniki, 3rd ed., “Nauka”, Moscow, 1989 (Russian). MR 1037020
- A. T. Fomenko, Simplekticheskaya geometriya, Moskov. Gos. Univ., Moscow, 1988 (Russian). Metody i prilozheniya. [Methods and applications]. MR 964470
- Kazuo Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.) 5 (1979), no. 1, 1–79 (French). MR 614694, DOI 10.4099/math1924.5.1
- M. V. Babich, Coordinates on the phase spaces of the system of Schlesinger equations and the system of Garnier-Painlevé 6 equations, Dokl. Akad. Nauk 412 (2007), no. 4, 439–443 (Russian); English transl., Dokl. Math. 75 (2007), no. 1, 71–75. MR 2451331, DOI 10.1134/S1064562407010206
- Mikhail Vasilievich Babich, Rational symplectic coordinates on the space of Fuchs equations $m\times m$-case, Lett. Math. Phys. 86 (2008), no. 1, 63–77. MR 2460728, DOI 10.1007/s11005-008-0274-3
- Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4
- S. P. Novikov (ed.), Dynamical systems. VII, Encyclopaedia of Mathematical Sciences, vol. 16, Springer-Verlag, Berlin, 1994. Integrable systems, nonholonomic dynamical systems; A translation of Current problems in mathematics. Fundamental directions, Vol. 16 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987 [ MR0922069 (88g:58004)]; Translation by A. G. Reyman [A. G. Reĭman] and M. A. Semenov-Tian-Shansky [M. A. Semenov-Tyan-Shanskiĭ]; Translation edited by V. I. Arnol′d and S. P. Novikov. MR 1256257, DOI 10.1007/978-3-642-57884-7
- Sergey É. Derkachov and Alexander N. Manashov, $\scr R$-matrix and Baxter $\scr Q$-operators for the noncompact $\textrm {SL}(N,{\Bbb C})$ invariant spin chain, SIGMA Symmetry Integrability Geom. Methods Appl. 2 (2006), Paper 084, 20. MR 2264900, DOI 10.3842/SIGMA.2006.084
- I. M. Gel′fand and M. A. Naĭmark, Unitarnye predstavleniya klassičeskih grupp, Izdat. Nauk SSSR, Moscow-Leningrad, 1950 (Russian). Trudy Mat. Inst. Steklov. no. 36,. MR 0046370
- A. Alekseev, L. Faddeev, and S. Shatashvili, Quantization of symplectic orbits of compact Lie groups by means of the functional integral, J. Geom. Phys. 5 (1988), no. 3, 391–406. MR 1048508, DOI 10.1016/0393-0440(88)90031-9
- A. Gerasimov, S. Kharchev, and D. Lebedev, Representation theory and quantum inverse scattering method: the open Toda chain and the hyperbolic Sutherland model, Int. Math. Res. Not. 17 (2004), 823–854. MR 2040074, DOI 10.1155/S1073792804132595
- Boris Dubrovin and Marta Mazzocco, Canonical structure and symmetries of the Schlesinger equations, Comm. Math. Phys. 271 (2007), no. 2, 289–373. MR 2287909, DOI 10.1007/s00220-006-0165-3
- Igor Moiseevich Krichever, An analogue of the d’Alembert formula for the equations of a principal chiral field and the sine-Gordon equation, Dokl. Akad. Nauk SSSR 253 (1980), no. 2, 288–292 (Russian). MR 581396
- A. P. Veselov and S. P. Novikov, Poisson brackets and complex tori, Trudy Mat. Inst. Steklov. 165 (1984), 49–61 (Russian). Algebraic geometry and its applications. MR 752932
Bibliographic Information
- M. V. Babich
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia
- Email: mbabich@pdmi.ras.ru, misha.babich@gmail.com
- S. E. Derkachov
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia
- Email: derkach@pdmi.ras.ru
- Received by editor(s): February 15, 2010
- Published electronically: March 17, 2011
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 347-357
- MSC (2010): Primary 53D05
- DOI: https://doi.org/10.1090/S1061-0022-2011-01145-8
- MathSciNet review: 2729938
Dedicated: Dedicated to L. D. Faddeev on the occasion of his 75th birthday