Anti-Kählerian Geometry on Lie Groups

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Abstract

Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure (g, J) where J is abelian then the Lie algebra of G is unimodular and (G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple (G, g, J) is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor 𝜃 on its Lie algebra and prove that such structure is anti-Kähler if and only if 𝜃 is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).

Keywords

Anti-Hermitian geometry Norden metrics B-manifolds Anti-Kähler manifold Lie groups Abelian complex structure Bi-invariant complex structure 

Mathematics Subject Classification (2010)

22F30 22F50 53C50 32M10 53C55 53C15 53C56 

Notes

Acknowledgements

The authors wish to extend their sincerest appreciation and thanks to Isabel Dotti and Marcos Salvai for their corrections, comments and constructive criticisms.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Edison Alberto Fernández-Culma
    • 1
  • Yamile Godoy
    • 1
  1. 1.CIEM - FaMAFCONICET - Universidad Nacional de CórdobaCórdobaArgentina

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