We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.
Citation: |
[1] | D. V. Alekseevskiǐ, Contact homogeneous spaces, Funktsional. Anal. i Prilozhen., 24 (1990), 74-75. doi: 10.1007/BF01077337. |
[2] | J. M. Ancochea-Bermúdez and M. Goze, Classification des algèbres de Lie filiformes de dimension $8$, Arch. Math., 50 (1988), 511-525. doi: 10.1007/BF01193621. |
[3] | O. Baues and V. Cortés, Symplectic Lie groups: Symplectic reduction, Lagrangian extensions, and existence of Lagrangian normal subgroups, Astérisque, 379 (2016), ⅵ+90 pp. |
[4] | W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2), 68 (1958), 721-734. doi: 10.2307/1970165. |
[5] | M. Bordemann, A. Medina and A. Ouadfel, Le group affine comme variété symplectique, Tohoku Math. J. (2), 45 (1993), 423-436. doi: 10.2748/tmj/1178225893. |
[6] | L. Boza, F. J. Echarte and J. Núñez, Classification of complex filiform Lie algebras of dimension 10, Algebras Groups Geom., 11 (1994), 253-276. |
[7] | C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124. doi: 10.1090/S0002-9947-1948-0024908-8. |
[8] | B. Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197 (1974), 145-159. doi: 10.1090/S0002-9947-1974-0342642-7. |
[9] | J.-M. Dardié and A. Medina, Double extension symplectique d'un groupe de Lie symplectique, Adv. Math., 117 (1996), 208-227. doi: 10.1006/aima.1996.0009. |
[10] | A. Diatta, Left invariant contact structures on Lie groups, Differential Geom. Appl., 26 (2008), 544-552. doi: 10.1016/j.difgeo.2008.04.001. |
[11] | J. R. Gómez and F. J. Echarte, Classification of complex filiform nilpotent Lie algebras of dimension $9$, Rend. Sem. Fac. Sci. Univ. Cagliari, 61 (1991), 21-29. |
[12] | J. R. Gómez, A. Jiménez-Merchán and Y. Khakimdjanov, Symplectic structures on the filiform Lie algebras, J. Pure Appl. Algebra, 156 (2001), 15-31. doi: 10.1016/S0022-4049(99)90120-2. |
[13] | J. W. Gray, Some global properties of contact structures, Ann. of Math. (2), 69 (1959), 421-450. doi: 10.2307/1970192. |
[14] | M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 33 (1969), 707-734. |
[15] | J.-I. Hano, On Kaehlerian homogeneous spaces of unimodular Lie groups, Amer. J. Math., 79 (1957), 885-900. doi: 10.2307/2372440. |
[16] | J. Helmstetter, Radical d'une algèbre symétrique à gauche, Ann. Inst. Fourier (Grenoble), 29 (1979), 17-35. doi: 10.5802/aif.764. |
[17] | Y. Khakimdjanov, M. Goze and A. Medina, Symplectic or contact structures on Lie Groups, Differential Geom. Appl., 21 (2004), 41-54. doi: 10.1016/j.difgeo.2003.12.006. |
[18] | M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989. |
[19] | A. Lichnerowicz and A. Medina, On Lie groups with left-invariant symplectic or Kählerian structures, Lett. Math. Phys., 16 (1988), 225-235. doi: 10.1007/BF00398959. |
[20] | R. Lutz, Quelques remarques historiques et prospectives sur la géométrie de contact, Rend. Sem. Fac. Sci. Univ. Caligari, 58 (1988), 361-393. |
[21] | A. Medina, Structure of symplectic Lie groups and momentum map, Tohoku Math. J. (2), 67 (2015), 419-431. doi: 10.2748/tmj/1446818559. |
[22] | M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application à l'étude de la variété des algèbres de Lie nilpotentes, Bull. Soc. Math. France, 98 (1970), 81-116. |