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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).

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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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Authors and Affiliations

  1. CIEM - FaMAF, CONICET - Universidad Nacional de Córdoba, Ciudad Universitaria, 5000, Córdoba, Argentina

    Edison Alberto Fernández-Culma & Yamile Godoy

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  1. Edison Alberto Fernández-Culma
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  2. Yamile Godoy
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Correspondence to Edison Alberto Fernández-Culma.

Additional information

Partially supported by Conicet (PIP 112-2012-01-00300), Secyt Univ. Nac. Córdoba.

Partially supported by Conicet (PIP 112-2011-01-00670), Foncyt (PICT 2010 cat 1proyecto 1716), Secyt Univ. Nac. Córdoba.

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Fernández-Culma, E.A., Godoy, Y. Anti-Kählerian Geometry on Lie Groups. Math Phys Anal Geom 21, 8 (2018). https://doi.org/10.1007/s11040-018-9266-4

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  • DOI: https://doi.org/10.1007/s11040-018-9266-4

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