Osaka Journal of Mathematics

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

Luis C. de Andrés, M. Laura Barberis, Isabel Dotti, and Marisa Fernández

Full-text: Open access


We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of [8] and [10], it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then $T^*G$ has no hermitian structure or, equivalently, $G$ has no left invariant generalized complex structure.

Article information

Osaka J. Math., Volume 44, Number 4 (2007), 765-793.

First available in Project Euclid: 7 January 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B30: Solvable, nilpotent (super)algebras 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 22E25: Nilpotent and solvable Lie groups
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 53D17: Poisson manifolds; Poisson groupoids and algebroids


de Andrés, Luis C.; Barberis, M. Laura; Dotti, Isabel; Fernández, Marisa. Hermitian structures on cotangent bundles of four dimensional solvable Lie groups. Osaka J. Math. 44 (2007), no. 4, 765--793.

Export citation


  • D. Alekseevsky, J. Grabowsky, G. Marmo, P.W. Michor: Poisson structures on double Lie groups, J. Geom. Phys. 26 (1998), 340--379.
  • A. Andrada, M.L. Barberis, I. Dotti, G. Ovando: Product structures on four dimensional solvable Lie algebras, Homology Homotopy Appl. 7 (2005), 9--37, arXiv:math.RA/0402234.
  • A. Andrada, M.L. Barberis, G. Ovando: Lie bialgebras of complex type and associated Poisson Lie groups, preprint, arXiv:math.DG/0610415.
  • M.L. Barberis, I. Dotti: Complex structures on affine motion groups, Q.J. Math. 55 (2004), 375--389.
  • O. Ben-Bassat: Mirror symmetry and generalized complex manifolds. I. The transform on vector bundles, spinors, and branes, J. Geom. Phys. 56 (2006), 533--558, arXiv:math.AG/0405303 v1.
  • O. Ben-Bassat, M. Boyarchenko: Submanifolds of generalized complex manifolds, J. Symplectic Geom. 2 (2004), 309--355, arXiv:math.DG/0309013.
  • G.R. Cavalcanti: New aspects of the $dd^c$-lemma, Ph.D. Thesis, University of Oxford, 2004.
  • G.R. Cavalcanti, M. Gualtieri: Generalized complex structures on nilmanifolds, J. Symplectic Geom. 2 (2004), 393--410.
  • J. Dozias: Sur les algèbres de Lie résolubles réelles de dimension inférieure ou égale à 5, Thèse de 3 cycle, novembre 1963, Faculté des Sciences de Paris.
  • M. Fernández, M. Gotay, A. Gray: Compact parallelizable four dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), 1209--1212.
  • M. Gualtieri: Generalized complex geometry, Ph.D. Thesis, University of Oxford, 2003, arXiv:math.DG/0401221.
  • N. Hitchin: Generalized Calabi-Yau manifolds, Q.J. Math. 54 (2003), 281--308.
  • S. Kobayashi, K. Nomizu: Foundations of Differential Geometry II, Interscience, 1969.
  • L. Magnin: Sur les algebres de Lie nilpotentes de dimension $\leq 7$, J. Geom. Phys. 3 (1986), 119--131.
  • A. Medina, P. Revoy: Groupes de Lie à structure symplectique invariante; in Symplectic Geometry, Grupoids and Integrable Systems, Séminaire Sud-Rhodanien de Géométrie (P. Dazord et A. Weinstein eds.), Mathematical Sciences Research Institute publications, New York-Berlin-Heidelberg, Springer Verlag, 1991, 247--266.
  • G.M. Mubarakzyanov: On solvable Lie algebras, Izv. Vysš. Učehn. Zaved. Matematika 32 (1963), 114--123.
  • G. Ovando: Invariant complex structures on solvable real Lie groups, Manuscripta Math. 103 (2000), 19--30.
  • G. Ovando: Complex, symplectic and Kähler structures on four dimensional Lie groups, Rev. Un. Mat. Argentina, 45 (2004), 55--67.
  • J. Patera, R.T. Sharp, P. Winternitz, H. Zassenhaus: Invariants of real low dimension Lie algebras, J. Mathematical Phys.\ 17 (1976), 986--994.
  • S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311--333.
  • H. Samelson: A class of complex analytic manifolds, Portugaliae Math. 12 (1953), 129--132.
  • T. Sasaki: Classification of left invariant complex structures on $GL(2,\mathbbR)$ and $U(2)$, Kumamoto J. Sci. (Math.) 14 (1980/81), 115--123.
  • J.E. Snow: Invariant complex structures on four dimensional solvable real Lie groups, Manuscripta Math. 66 (1990), 397--412.