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).2010 Mathematics Subject Classification.

Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds

“…Vice versa, let the latter equalities be satisfied. Then, applying (6) and (11) to them, we deduce (14) and (15). Therefore, g is Killing.…”

“…A number of authors have studied Lie groups as manifolds equipped with various tensor structures and metrics that are compatible with the structures (including in the lowest-dimensional cases) -for example, [3] and [4] for almost contact metric manifolds, [13] and [10] for almost contact B-metric manifolds, [1] and [6] for almost complex manifolds with Hermitian metric, [8] and [24] for almost complex manifolds with Norden metric, [2] and [5] for hypercomplex hyper-Hermitian manifolds, [7] and [12] for almost hypercomplex Hermitian-Norden manifolds, [9] and [22] for Riemannian almost product manifolds, [16] and [25] for almost paracontact metric manifolds.…”

“…In a series of papers, e.g. [1,3,5,6,7,8,9,10,11,19,20], the authors consider Lie groups as manifolds equipped with different additional tensor structures and metrics compatible with them. Furthermore, in our previous work [13], we construct and characterize a family of 3-dimensional Lie algebras corresponding to Lie groups considered as almost paracontact almost paracomplex Riemannian manifolds.…”

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

“…When the manifold is a Lie group G, the metric and the complex structure are considered left-invariant, where they are both determined by their restrictions to the Lie algebra g of G. In this situation, (g, g e , J e ) is called an anti-Kähler Lie algebra. Anti-Kähler geometry on Lie groups have been studied by Edison Alberto Fernández-Culma and Yamile Godoy [6].…”

Kähler-Norden structures on Hom-Lie group and Hom-Lie algebras

“…When the manifold is a Lie group G, the metric and the complex structure are considered left-invariant, where they are both determined by their restrictions to the Lie algebra g of G. In this situation, (g, g e , J e ) is called a anti-Kähler Lie algebra. Anti-Kähler geometry on Lie groups have been studied by Edison Alberto Fernández-Culma and Yamile Godoy [6].…”