Anti-K\"ahlerian geometry on Lie groups

Abstract

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-K\"ahler structure on $G$. In this article we study anti-K\"{a}hler 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-K\"ahler 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-K\"ahler 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-K\"ahler 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-K\"ahler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-K\"ahler structures).