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June  2020, 12(2): 141-148. doi: 10.3934/jgm.2020007

Invariant structures on Lie groups

Javier Pérez Álvarez

Departamento de Matemáticas Fundamentales, Universidad Nacional de Educación a Distancia, Paseo Senda del Rey 9, 28040 Madrid, Spain

Received  July 2019 Revised  December 2019 Published  June 2020 Early access  March 2020

We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.

Keywords: Symplectic structures, contact structures, Hamiltonian structures.
Mathematics Subject Classification: Primary: 22E15; Secondary: 53D05, 53D10.
Citation: Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007
References:
[1]

D. V. Alekseevskiǐ, Contact homogeneous spaces, Funktsional. Anal. i Prilozhen., 24 (1990), 74-75.  doi: 10.1007/BF01077337.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2), 68 (1958), 721-734.  doi: 10.2307/1970165.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

B. Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197 (1974), 145-159.  doi: 10.1090/S0002-9947-1974-0342642-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

J. W. Gray, Some global properties of contact structures, Ann. of Math. (2), 69 (1959), 421-450.  doi: 10.2307/1970192.  Google Scholar

[14]

M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 33 (1969), 707-734.   Google Scholar

[15]

J.-I. Hano, On Kaehlerian homogeneous spaces of unimodular Lie groups, Amer. J. Math., 79 (1957), 885-900.  doi: 10.2307/2372440.  Google Scholar

[16]

J. Helmstetter, Radical d'une algèbre symétrique à gauche, Ann. Inst. Fourier (Grenoble), 29 (1979), 17-35.  doi: 10.5802/aif.764.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Lutz, Quelques remarques historiques et prospectives sur la géométrie de contact, Rend. Sem. Fac. Sci. Univ. Caligari, 58 (1988), 361-393.   Google Scholar

[21]

A. Medina, Structure of symplectic Lie groups and momentum map, Tohoku Math. J. (2), 67 (2015), 419-431.  doi: 10.2748/tmj/1446818559.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

D. V. Alekseevskiǐ, Contact homogeneous spaces, Funktsional. Anal. i Prilozhen., 24 (1990), 74-75.  doi: 10.1007/BF01077337.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2), 68 (1958), 721-734.  doi: 10.2307/1970165.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

B. Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197 (1974), 145-159.  doi: 10.1090/S0002-9947-1974-0342642-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

J. W. Gray, Some global properties of contact structures, Ann. of Math. (2), 69 (1959), 421-450.  doi: 10.2307/1970192.  Google Scholar

[14]

M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 33 (1969), 707-734.   Google Scholar

[15]

J.-I. Hano, On Kaehlerian homogeneous spaces of unimodular Lie groups, Amer. J. Math., 79 (1957), 885-900.  doi: 10.2307/2372440.  Google Scholar

[16]

J. Helmstetter, Radical d'une algèbre symétrique à gauche, Ann. Inst. Fourier (Grenoble), 29 (1979), 17-35.  doi: 10.5802/aif.764.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Lutz, Quelques remarques historiques et prospectives sur la géométrie de contact, Rend. Sem. Fac. Sci. Univ. Caligari, 58 (1988), 361-393.   Google Scholar

[21]

A. Medina, Structure of symplectic Lie groups and momentum map, Tohoku Math. J. (2), 67 (2015), 419-431.  doi: 10.2748/tmj/1446818559.  Google Scholar

[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.   Google Scholar

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