American Institute of Mathematical Sciences

Advanced
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

[1]

Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
[2]

Daniel Guan. Classification of compact homogeneous spaces with invariant symplectic structures. Electronic Research Announcements, 1997, 3: 52-54.

[3]

William D. Kalies, Konstantin Mischaikow, Robert C.A.M. Vandervorst. Lattice structures for attractors I. Journal of Computational Dynamics, 2014, 1 (2) : 307-338. doi: 10.3934/jcd.2014.1.307
[4]

Paulo Antunes, Joana M. Nunes da Costa. Hypersymplectic structures on Courant algebroids. Journal of Geometric Mechanics, 2015, 7 (3) : 255-280. doi: 10.3934/jgm.2015.7.255
[5]

Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021
[6]

Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553
[7]

Michel Cahen, Simone Gutt, John Rawnsley. On twistor almost complex structures. Journal of Geometric Mechanics, 2021, 13 (3) : 313-331. doi: 10.3934/jgm.2021006
[8]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[9]

Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120.

[10]

Matthias Liero, Alexander Mielke, Mark A. Peletier, D. R. Michiel Renger. On microscopic origins of generalized gradient structures. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 1-35. doi: 10.3934/dcdss.2017001
[11]

Partha Guha, Indranil Mukherjee. Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1677-1695. doi: 10.3934/dcdsb.2018287
[12]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
[13]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[14]

Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619
[15]

S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577
[16]

Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
[17]

Giovanni Rastelli, Manuele Santoprete. Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures. Journal of Geometric Mechanics, 2015, 7 (4) : 483-515. doi: 10.3934/jgm.2015.7.483
[18]

Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
[19]

Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233
[20]

Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (314)
  • HTML views (321)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences