Anti-Kahlerian Geometry on Lie Groups

作者：Alberto Fernandez Culma Edison; Godoy Yamile

来源：Mathematical Physics, Analysis and Geometry, 2018, 21(1): 8.

DOI：10.1007/s11040-018-9266-4

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

Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kahler structure on G. In this article we study anti-Kahler 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-Kahler 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-Kahler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor theta on its Lie algebra and prove that such structure is anti-Kahler if and only if theta 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-Kahler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kahler structures).

• 出版日期2018-3-6