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Invariant structures on Lie groups
Departamento de Matemáticas Fundamentales, Universidad Nacional de Educación a Distancia, Paseo Senda del Rey 9, 28040 Madrid, Spain |
We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.
References:
| [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.
|
show all references
References:
| [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.
|
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