Hermitian and Kähler Structures of Two Types on Lie Groups with Two-Dimensional Derived Algebras

Abstract

We study left-invariant Hermitian structures on Lie groups whose derived algebra is two-dimensional. We consider two natural types of almost Hermitian structures, according to the position of the derived algebra with respect to the almost complex structure. For each type, we obtain explicit necessary and sufficient conditions for integrability. In particular, for the second type, integrability is equivalent to the abelianity of the complex structure. We then characterize the Kähler structures of both types and show that, in each case, the Kähler condition imposes strong restrictions on the Lie bracket. Moreover, we compute the corresponding Bismut connections and torsion forms explicitly. As applications, we construct examples of Kähler, WKT, and SKT structures in dimensions four, six, and in all even dimensions greater than six. In particular, we obtain a family of abelian SKT structures on ${\mathfrak{r}}_2\oplus {\mathfrak{h}}_3\oplus {\mathbb{R}}^{2n-5}$ for $n\ge 3$.