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TMF, 2000, Volume 123, Number 3, Pages 407–423 (Mi tmf612)  

This article is cited in 20 scientific papers (total in 20 papers)

Darboux coordinates on $K$-orbits and the spectra of Casimir operators on Lie groups

I. V. Shirokov

Omsk State University

Abstract: We propose an algorithm for obtaining the spectra of Casimir ce Lie groups. We prove that the existence of the normal polarization associated with a linear functional on the Lie algebra is necessary and sufficient for the transition to local canonical Darboux coordinates $(p,q)$ on the coadjoint representation orbit that is linear in the “momenta”. We show that the $\lambda$-representations of Lie algebras are used, in particular, in integrating differential equationsthe quantization of the Poisson bracket on the coalgebra in canonical coordinates.


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English version:
Theoretical and Mathematical Physics, 2000, 123:3, 754–767

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Received: 14.07.1999

Citation: I. V. Shirokov, “Darboux coordinates on $K$-orbits and the spectra of Casimir operators on Lie groups”, TMF, 123:3 (2000), 407–423; Theoret. and Math. Phys., 123:3 (2000), 754–767

Citation in format AMSBIB
\by I.~V.~Shirokov
\paper Darboux coordinates on $K$-orbits and the spectra of Casimir operators on Lie groups
\jour TMF
\yr 2000
\vol 123
\issue 3
\pages 407--423
\jour Theoret. and Math. Phys.
\yr 2000
\vol 123
\issue 3
\pages 754--767

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    This publication is cited in the following articles:
    1. I. V. Shirokov, “Identities and Invariant Operators on Homogeneous Spaces”, Theoret. and Math. Phys., 126:3 (2001), 326–338  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. S. P. Baranovskii, V. V. Mikheyev, I. V. Shirokov, “Quantum Hamiltonian Systems on K-Orbits: Semiclassical Spectrum of the Asymmetric Top”, Theoret. and Math. Phys., 129:1 (2001), 1311–1319  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Klishevich, VV, “Exact solution of Dirac and Klein-Gordon-Fock equations in a curved space admitting a second Dirac operator”, Classical and Quantum Gravity, 18:17 (2001), 3735  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    4. Bagrov, VG, “New solutions of relativistic wave equations in magnetic fields and longitudinal fields”, Journal of Mathematical Physics, 43:5 (2002), 2284  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    5. S. P. Baranovskii, I. V. Shirokov, “Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces”, Theoret. and Math. Phys., 135:1 (2003), 510–519  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. A. Magazev, I. V. Shirokov, “Integration of Geodesic Flows on Homogeneous Spaces: The Case of a Wild Lie Group”, Theoret. and Math. Phys., 136:3 (2003), 1212–1224  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. A. Magazev, I. V. Shirokov, “Hamiltonian systems in variations and the integration of the Jacobi equation on homogeneous spaces”, Russian Math. (Iz. VUZ), 50:8 (2006), 38–49  mathnet  mathscinet  elib
    8. S. P. Baranovskii, I. V. Shirokov, “Deformations of vector fields and canonical coordinates on coadjoint orbits”, Siberian Math. J., 50:4 (2009), 580–586  mathnet  crossref  mathscinet  isi  elib
    9. M. M. Goncharovskiy, I. V. Shirokov, “An integrable class of differential equations with nonlocal nonlinearity on Lie groups”, Theoret. and Math. Phys., 161:3 (2009), 1604–1615  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. Breev, AI, “POLARIZATION OF A SPINOR FIELD VACUUM ON MANIFOLDS OF THE Lie GROUPS”, Russian Physics Journal, 52:8 (2009), 823  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    11. Breev A.I., “Vacuum Polarization of a Scalar Field on the Nonunimodular Lie Groups”, Russian Physics Journal, 53:4 (2010), 421–430  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    12. A. I. Breev, I. V. Shirokov, A. A. Magazev, “Vacuum polarization of a scalar field on Lie groups and homogeneous spaces”, Theoret. and Math. Phys., 167:1 (2011), 468–483  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    13. A. A. Magazev, “Integrating Klein–Gordon–Fock equations in an external electromagnetic field on Lie groups”, Theoret. and Math. Phys., 173:3 (2012), 1654–1667  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    14. V. V. Mikheev, “Vysokotemperaturnoe razlozhenie matritsy plotnosti i ego prilozheniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 369–378  mathnet  crossref
    15. Breev A.I. Goncharovskii M.M. Shirokov I.V., “Klein-Gordon Equation with a Special Type of Nonlocal Nonlinearity in Commutative Homogeneous Spaces with Invariant Metric”, Russ. Phys. J., 56:7 (2013), 731–739  crossref  mathscinet  isi  scopus  scopus  scopus
    16. A. I. Breev, “Scalar field vacuum polarization on homogeneous spaces with an invariant metric”, Theoret. and Math. Phys., 178:1 (2014), 59–75  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. Breev A.I., “Schrodinger Equation With Convolution Nonlinearity on Lie Groups and Commutative Homogeneous Spaces”, Russ. Phys. J., 57:8 (2014), 1050–1058  crossref  zmath  isi  scopus  scopus  scopus
    18. Breev A.I. Kozlov A.V., “Vacuum Averages of the Energy-Momentum Tensor of a Scalar Field in Homogeneous Spaces With a Conformal Metric”, Russ. Phys. J., 58:9 (2016), 1248–1257  crossref  zmath  isi  scopus  scopus  scopus
    19. O. L. Kurnyavko, I. V. Shirokov, “Construction of invariants of the coadjoint representation of Lie groups using linear algebra methods”, Theoret. and Math. Phys., 188:1 (2016), 965–979  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. Mikheev V., “Method of Orbits of Co-Associated Representation in Thermodynamics of the Lie Non-Compact Groups”, Geometric Science of Information, Gsi 2017, Lecture Notes in Computer Science, 10589, eds. Nielsen F., Barbaresco F., Springer International Publishing Ag, 2017, 425–431  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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