Anti-Kählerian geometry on Lie groups

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).