Geodesic
Construction of invariants of the coadjoint representation of Lie groups using linear algebra methods
Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 1, pp. 3-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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We offer a method for constructing invariants of the coadjoint representation of Lie groups that reduces this problem to known problems of linear algebra. This method is based on passing to symplectic coordinates on the coadjoint representation orbits, which play the role of local coordinates on those orbits. The corresponding transition functions are their parametric equations. Eliminating the symplectic coordinates from the transition functions, we can obtain the complete set of invariants. The proposed method allows solving the problem of constructing invariants of the coadjoint representation for Lie groups with an arbitrary dimension and structure.
Mots-clés : invariant, coadjoint representation, Lie group
Keywords: Lie algebra, polarization, symplectic coordinate. 
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O. L. Kurnyavko; I. V. Shirokov. Construction of invariants of the coadjoint representation of Lie groups using linear algebra methods. Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/TMF_2016_188_1_a0/

[1] H. G. B. Casimir, Proc. R. Soc. Amsterdam, 34 (1931), 844–846 | Zbl

[2] A. S. Mischenko, A. T. Fomenko, Dokl. AN SSSR, 231:3 (1976), 536–538 | MR | Zbl

[3] A. V. Bolsinov, Izv. AN SSSR. Ser. matem., 55:1 (1991), 68–92 | DOI | MR | Zbl

[4] G. Racah, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., 8 (1950), 108–112 | MR | Zbl

[5] K. Shevalle, Teoriya grupp Li, IL, M., 1948, 1958

[6] I. M. Gelfand, Matem. sbornik, 26(68) (1950), 103–112 | MR | Zbl

[7] A. L. Perelomov, V. S. Popov, YaF, 3 (1966), 924–931 | MR

[8] A. L. Perelomov, V. S. Popov, Dokl. AN SSSR, 174 (1967), 287–290 | MR | Zbl

[9] R. Campoamor-Stursberg, J. Phys. A: Math. Gen., 35:30 (2002), 6293–6306, arXiv: math/0202006 | DOI | MR | Zbl

[10] V. M. Boyko, J. Patera, R. Popovych, J. Phys. A: Math. Gen., 39:20 (2006), 5749–5762, arXiv: math-ph/0602046 | DOI | MR | Zbl

[11] A. N. Leznov, I. A. Malkin, V. I. Manko, “Kanonicheskie preobrazovaniya i teoriya predstavlenii grupp Li”, Tr. FIAN, 96, Nauka, M., 1977, 24–71 | MR

[12] J. Guerrero, V. I. Manko, G. Marmo, A. Simoni, Geometrical aspects of Lie groups representations and their optical applications, arXiv: quant-ph/0112069

[13] Zh. Diksme, Universalnye obertyvayuschie algebry, Mir, M., 1978 | MR | MR | Zbl

[14] A. T. Fomenko, Simplekticheskaya geometriya. Metody i prilozheniya, Izd-vo Mosk. un-ta, M., 1988 | MR | MR | Zbl

[15] I. V. Shirokov, Izv. vuzov. Fizika, 6 (1997), 25–32 | DOI | MR

[16] I. V. Shirokov, TMF, 123:3 (2000), 407–423 | DOI | DOI | MR | Zbl