Geodesic
Bifurcation diagram and the discriminant of a spectral curve of integrable systems on Lie algebras
Sbornik. Mathematics, Tome 201 (2010) no. 9, pp. 1273-1305 Cet article a éte moissonné depuis la source Math-Net.Ru

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A bifurcation diagram is a stratified (in general, nonclosed) set and is one of the efficient tools of studying the topology of the Liouville foliation. In the framework of the present paper, the coincidence of the closure of a bifurcation diagram $\overline\Sigma$ of the moment map defined by functions obtained by the method of argument shift with the closure of the discriminant $\overline D_z$ of a spectral curve is proved for the Lie algebras $\operatorname{sl}(n+1)$, $\operatorname{sp}(2n)$, $\operatorname{so}(2n+1)$, and $\operatorname{g}_2$. Moreover, it is proved that these sets are distinct for the Lie algebra $\operatorname{so}(2n)$. Bibliography: 22 titles.
Keywords: method of argument shift, Lie algebra, bifurcation diagram, spectral curve. 
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A. Yu. Konyaev. Bifurcation diagram and the discriminant of a spectral curve of integrable systems on Lie algebras. Sbornik. Mathematics, Tome 201 (2010) no. 9, pp. 1273-1305. http://geodesic.mathdoc.fr/item/SM_2010_201_9_a1/

[1] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall, CRC, Boca Raton, FL, 2004 | MR | MR | Zbl | Zbl

[2] M. Audin, Spinning tops. A course on integrable systems, Cambridge Stud. Adv. Math., 51, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[3] V. V. Trofimov, A. T. Fomenko, “Non-invariant symplectic group structures and Hamiltonian flows on symmetric spaces”, Selecta Math. Soviet., 7:4 (1988), 356–414 | MR | Zbl

[4] A. S. Miščenko, A. T. Fomenko, “Euler equations on finite-dimensional Lie groups”, Math. USSR-Izv., 12:2 (1978), 371–389 | DOI | MR | Zbl | Zbl

[5] A. S. Mishchenko, A. T. Fomenko, “Integrability of Euler equations on semisimple Lie algebras”, Selecta Math. Soviet., 2:4 (1982), 207–291 | MR | Zbl | Zbl

[6] S. V. Manakov, “Note on the integration of Euler's equations of the dynamics of an $n$-dimensional rigid body”, Funct. Anal. Appl., 10:4 (1976), 328–329 | DOI | MR | Zbl

[7] A. T. Fomenko, Symplectic geometry, Adv. Stud. Contemp. Math., 5, Gordon and Breach, New York, 1988 | MR | MR | Zbl | Zbl

[8] Yu. A. Brailov, “Geometry of translations of invariants on semisimple Lie algebras”, Sb. Math., 194:11 (2003), 1585–1598 | DOI | MR | Zbl

[9] Yu. A. Brailov, A. T. Fomenko, “Lie groups and integrable Hamiltonian systems”, Recent advances in Lie theory (Vigo, Spain, 2000), Res. Exp. Math., 25, Heldermann, Lemgo, 2002, 45–76 | MR | Zbl

[10] N. Jacobson, Lie algebras, Intersci. Publ., New York–London, 1962 | MR | MR | Zbl | Zbl

[11] A. T. Fomenko, Differential geometry and topology, Contemp. Soviet Math., Consultants Bureau, New York–London, 1987 | MR | Zbl | Zbl

[12] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., 9, Springer-Verlag, New York–Berlin, 1978 | MR | Zbl

[13] J. F. Kurtzke, jr., “Centralizers of irregular elements in reductive algebraic groups”, Pacific J. Math., 104:1 (1983), 133–154 | MR | Zbl

[14] D. Panyushev, A. Premet, O. Yakimova, “On symmetric invariants of centralisers in reductive Lie algebras”, J. Algebra, 313:1 (2007), 343–391 | DOI | MR | Zbl

[15] A. V. Bolsinov, “A completeness criterion for a family of functions in involution constructed by the argument shift method”, Soviet Math. Dokl., 38:1 (1989), 161–165 | MR | Zbl

[16] A. V. Bolsinov, A. A. Oshemkov, “Bi-Hamiltonian structures and singularities of integrable systems”, Regul. Chaotic Dyn., 14:4–5 (2009), 431–454 | DOI | MR

[17] A. G. Elashvili, V. G. Kac, “Classification of good gradings of simple Lie algebras”, Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 85–104 | MR | Zbl

[18] A. V. Bolsinov, K. M. Zuev, “A formal Frobenius theorem and argument shift”, Math. Notes, 86:1–2 (2009), 10–18 | DOI | MR | Zbl

[19] B. Kostant, “Lie group representations on polynomial rings”, Amer. J. Math., 85:3 (1963), 327–404 | DOI | MR | Zbl

[20] A. A. Tarasov, “The maximality of certain commutative subalgebras in the Poisson algebra of a semisimple Lie algebra”, Russian Math. Surveys, 57:5 (2002), 1013–1014 | DOI | MR | Zbl

[21] D. I. Panyushev, “The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer”, Math. Proc. Cambridge Philos. Soc., 134:1 (2003), 41–59 | DOI | MR | Zbl

[22] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern geometry – methods and applications. Part I. The geometry of surfaces, transformation groups, and fields, Grad. Texts in Math., 93, Springer-Verlag, New York, 1992 | MR | MR | Zbl | Zbl