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.
Citation:
M. V. Babich, S. E. Derkachov, “On rational symplectic parametrization of the coadjoint orbit of GL(N)GL(N). Diagonalizable case”, Algebra i Analiz, 22:3 (2010), 16–31; St. Petersburg Math. J., 22:3 (2011), 347–357
\Bibitem{BabDer10}
\by M.~V.~Babich, S.~E.~Derkachov
\paper On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case
\jour Algebra i Analiz
\yr 2010
\vol 22
\issue 3
\pages 16--31
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\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2729938}
\zmath{https://zbmath.org/?q=an:1222.53080}
\transl
\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 3
\pages 347--357
\crossref{https://doi.org/10.1090/S1061-0022-2011-01145-8}
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Linking options:
https://www.mathnet.ru/eng/aa1184
https://www.mathnet.ru/eng/aa/v22/i3/p16
This publication is cited in the following 11 articles:
Ilia Gaiur, Marta Mazzocco, Vladimir Rubtsov, “Isomonodromic Deformations: Confluence, Reduction and Quantisation”, Commun. Math. Phys., 400:2 (2023), 1385
N. Belousov, S. Derkachov, “Regular Representation of the Group GL(N, ℝ): Factorization, Casimir Operators and Toda Chain”, J Math Sci, 264:3 (2022), 215
N. M. Belousov, S. E. Derkachev, “Regulyarnoe predstavlenie gruppy GL(N,R): faktorizatsiya, operatory Kazimira i tsepochka Tody”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 27, Zap. nauchn. sem. POMI, 494, POMI, SPb., 2020, 23–47
Babich V M., “On Canonical Parametrization of Phase Spaces of Isomonodromic Deformation Equations”, Geometric Methods in Physics Xxxvii, Trends in Mathematics, eds. Kielanowski P., Odzijewicz A., Previato E., Birkhauser Verlag Ag, 2020, 3–12
M. V. Babich, “On extensions of canonical symplectic structure from coadjoint orbit of complex general linear group”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 26, Zap. nauchn. sem. POMI, 487, POMI, SPb., 2019, 28–39
J. Math. Sci. (N. Y.), 242:5 (2019), 587–594
J. Math. Sci. (N. Y.), 238:6 (2019), 763–768
M. V. Babich, “Birational Darboux Coordinates on (Co)Adjoint Orbits of GL(N,C)”, Funct. Anal. Appl., 50:1 (2016), 17–30
J. Math. Sci. (N. Y.), 209:6 (2015), 830–844
M. V. Babich, “Young tableaux and stratification of space of complex square matrices”, J. Math. Sci. (N. Y.), 213:5 (2016), 651–661
Babich M.V., “On birational Darboux coordinates of isomonodromic deformation equations phase space”, Painleve Equations and Related Topics (St. Petersburg, June 17–23, 2011), DeGruyter Proceedings in Mathematics, eds. Bruno A., Batkhin A., Walter de Gruyter & Co, 2012, 91–94