A complete classification of four-dimensional paraKähler Lie algebras

Giovanni Calvaruso

Complex Manifolds (2015)

  • Volume: 2, Issue: 1, page 733-748
  • ISSN: 2300-7443

Abstract

top
We consider paraKähler Lie algebras, that is, even-dimensional Lie algebras g equipped with a pair (J, g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental 2-form Ω(X, Y) = g(X, JY) is symplectic. A complete classification is obtained in dimension four.

How to cite

top

Giovanni Calvaruso. "A complete classification of four-dimensional paraKähler Lie algebras." Complex Manifolds 2.1 (2015): 733-748. <http://eudml.org/doc/275886>.

@article{GiovanniCalvaruso2015,
abstract = {We consider paraKähler Lie algebras, that is, even-dimensional Lie algebras g equipped with a pair (J, g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental 2-form Ω(X, Y) = g(X, JY) is symplectic. A complete classification is obtained in dimension four.},
author = {Giovanni Calvaruso},
journal = {Complex Manifolds},
keywords = {Lie algebras; paraKähler structures; pseudo-Riemannian homogeneous spaces; para-Kähler structure; para-Kähler Lie algebra; pseudo-Riemannian; homogeneous space; para-Kähler Ricci soliton},
language = {eng},
number = {1},
pages = {733-748},
title = {A complete classification of four-dimensional paraKähler Lie algebras},
url = {http://eudml.org/doc/275886},
volume = {2},
year = {2015},
}

TY - JOUR
AU - Giovanni Calvaruso
TI - A complete classification of four-dimensional paraKähler Lie algebras
JO - Complex Manifolds
PY - 2015
VL - 2
IS - 1
SP - 733
EP - 748
AB - We consider paraKähler Lie algebras, that is, even-dimensional Lie algebras g equipped with a pair (J, g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental 2-form Ω(X, Y) = g(X, JY) is symplectic. A complete classification is obtained in dimension four.
LA - eng
KW - Lie algebras; paraKähler structures; pseudo-Riemannian homogeneous spaces; para-Kähler structure; para-Kähler Lie algebra; pseudo-Riemannian; homogeneous space; para-Kähler Ricci soliton
UR - http://eudml.org/doc/275886
ER -

References

top
  1. [1] D.V. Alekseevsky, C. Medori, A. Tomassini, Homogeneous para-Kähler Einstein manifolds, Russian Math. Surveys, 64 (2009), 1–43. [Crossref][WoS] Zbl1179.53050
  2. [2] A. Andrada, M.L. Barberis, I.G. Dotti, G. Ovando, Product structures on four-dimensional solvable Lie algebras, Homology, Homotopy and Applications, 7 (2005), 9–37. Zbl1165.17303
  3. [3] P. Baird and L. Danielo, Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew.Math., 608 (2007), 65–91. [WoS] Zbl1128.53020
  4. [4] N. Blazić, S. Vukmirović, Four-dimensional Lie algebras with a para-hypercomplex structure, Rocky Mountain J. Math., 40 (2010), 1391–1439. Zbl1207.53071
  5. [5] M. Brozos-Vazquez, G. Calvaruso, E. Garcia-Rio and S. Gavino-Fernandez, Three-dimensional Lorentzian homogeneous Ricci solitons, Israel J. Math., 188 (2012), 385–403. Zbl1264.53052
  6. [6] G. Calvaruso, Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces, Diff. Geom. Appl., 29 (2011), 758–769. [Crossref] Zbl1228.53037
  7. [7] G. Calvaruso, Four-dimensional paraKähler Lie algebras: classification and geometry, Houston J. Math., to appear. 
  8. [8] G. Calvaruso and A. Fino, Complex and paracomplex structures on homogeneous pseudo-Riemannian four-manifolds, Int. J. Math., 24 (2013), 1250130, 28 pp. [Crossref][WoS] Zbl1266.53033
  9. [9] G. Calvaruso and A. Fino, Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces, Canad. J. Math., 64 (2012), 778–804. Zbl1252.53056
  10. [10] G. Calvaruso and A. Fino, Four-dimensional pseudo-Riemannian homogeneous Ricci soliton, Arxiv: 1111.6384. To appear in Int. J. Geom. Methods Mod. Phys. [WoS] 
  11. [11] H.-D. Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, 1–38, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010. 
  12. [12] V. Cruceanu, P. Fortuny and P.M. Gadea, A survey on paracomplex geometry, Rocky Mount. J. Math., 26 (1996), 83–115. Zbl0856.53049
  13. [13] B.Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197 (1974), 145–159. Zbl0261.53039
  14. [14] A.S. Dancer and M.Y. Wang, Some new examples on non-Ka¨ hler Ricci solitons, Math. Res. Lett., 16 (2009), no. 2, 349–363. [Crossref] 
  15. [15] A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259–280. Zbl0378.53018
  16. [16] J. Lauret, Ricci solitons solvmanifolds, J. Reine Angew. Math., 650 (2011), 1–21. 
  17. [17] G. Ovando, Invariant complex structures on solvable real Lie groups, Manuscripta Math., 103, (2000), 19–30. Zbl0972.32017
  18. [18] G. Ovando, Four-dimensional symplectic Lie algebras, Beiträge Algebra Geom., 47(2006), no. 2, 419–434. Zbl1155.53042
  19. [19] G. Ovando, Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory, 16 (2006), 371–391. Zbl1102.32011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.