Anti-Kählerian geometry on Lie groups

EDISON ALBERTO FERNÁNDEZ-CULMA; YAMILE GODOY

MATHEMATICAL PHYSICS, ANALYSIS AND GEOMETRY

SPRINGER

Lugar: Berlin; Año: 2018 vol. 21 p. 1 - 24

1385-0172

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"{a}hler 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 $heta$ on its Lie algebra and prove that such structure is anti-K"{a}hler if and only if $heta$ 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"{a}hler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-K"{a}hler structures).