Extensions of Lie algebras and Hamiltonian systems

V. V. Trofimov

Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 561-578.

Résumé

An extension $\Omega(G)$ is constructed for a Lie algebra $G$, and an algorithm is proposed which converts functions in involution on $G^*$ into functions in involution on $\Omega(G)^*$. Operators of “rigid body” type are constructed for $\Omega(G)$ in the case of a semisimple Lie algebra $G$; complete integrability is proved for the Euler equations on $\Omega(G)^*$ with these operators. Bibliography: 21 titles.

Bibliographie

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[5] Trofimov V. V., Fomenko A. T., “Metodika postroeniya gamiltonovykh potokov na simmetricheskikh prostranstvakh i integriruemost nekotorykh gidrodinamicheskikh sistem”, Dokl. AN SSSR, 254:6 (1980), 1349–1353 | MR

[6] Mischenko A. S., Fomenko A. T., “Integriruemost uravnenii Eilera na poluprostykh algebrakh Li”, Trudy seminara po vektornomu i tenzornomu analizu, XIX, 1979, 3–94

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[9] Trofimov V. V., “Uravneniya Eilera na konechnomernykh razreshimykh gruppakh Li”, Izv. AN SSSR. Ser. matem., 44:5 (1980), 1191–1199 | MR | Zbl

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[14] Fomenko A. T., “Gruppovye simplekticheskie struktury na odnorodnykh prostranstvakh”, Dokl. AN SSSR, 253:5 (1980), 1062–1067 | MR | Zbl

[15] Trofimov V. V., “Konechnomernye predstavleniya algebr Li i vpolne integriruemye sistemy”, Matem. sbornik, 111:4 (1980), 610–621 | MR | Zbl

[16] Arkhangelskii A. A., “Vpolne integriruemye gamiltonovye sistemy na gruppe treugolnykh matrits”, Matem. sbornik, 108:1 (1979), 134–142 | MR

[17] Vergne M., “La structure de Poisson sur l'algèbre sumétrique d'une algèbre de Lie nilpotente”, Bull. Soc. Math. France, 100 (1972), 301–335 | MR | Zbl

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[19] Trofimov V. V., Fomenko A. T., “Gruppovye neinvariantnye simplekticheskie struktury i gamiltonovy potoki na simmetricheskikh prostranstvakh”, Trudy seminara po vektornomu i tenzornomu analizu, no. 21, 1983, 23–83 | MR | Zbl

[20] Trofimov V. V., Fomenko A. T., “Dinamicheskie sistemy na orbitakh lineinykh predstavlenii grupp Li i polnaya integriruemost nekotorykh gidrodinamicheskikh sistem”, Funkts. analiz, 17:1 (1983), 31–39 | MR | Zbl

[21] Trofimov V. V., “Vpolne integriruemye geodezicheskie potoki levoinvariantnykh metrik na gruppakh Li, svyazannye s kommutativnymi graduirovannymi algebrami s dvoistvennostyu Puankare”, Dokl. AN SSSR, 263:4 (1982), 812–816 | MR | Zbl