Hermitian Structures on Lie Groups with Two-dimensional Commutator Subgroups
Hamid Reza Salimi Moghaddam
TL;DR
The paper studies left-invariant Hermitian structures on Lie groups with a two-dimensional commutator subalgebra ($\dim\mathfrak{g}'=2$), introducing two structured, tractable types (Type I and Type II) and deriving explicit Hermitian and Kähler criteria. It develops a detailed algebraic framework using a decomposition with $\mathfrak{g}'=\mathrm{span}\{\mathsf{e}_1,\mathsf{e}_2\}$ and $\Gamma$, encoded by $\mathsf{a}_i,\mathsf{b}_i$ and skew-adjoint maps $f_i$, and provides precise conditions for Hermitian and Kähler structures, along with the corresponding Bismut connections. The results include necessary and sufficient Hermitian criteria for both types, explicit Kähler conditions ($f_1=f_2=0$ with $\mathsf{b}_2=-\mathsf{a}_1$ for Type I and $\mathsf{b}_1=\mathsf{a}_2$ for Type II), and exhaustive formulas for the Bismut connections governing torsion. The paper confirms these theoretical findings through concrete 4D and 6D examples (abelian and nilpotent) that realize weak and strong Kähler with torsion, illustrate nonexistence results, and show that different weak Kähler structures can share the same Bismut data, thereby enriching the landscape of structured Hermitian geometries on solvable Lie groups.
Abstract
This article studies left-invariant Hermitian structures on Lie groups with two-dimensional commutator subgroups. We provide an explicit classification for two specific types of such structures, which we designate as Type I and Type II. Furthermore, we classify the Kahler structures within these two types and compute their associated Bismut connections. Finally, we present examples of Kahler and strong (respectively, weak) Kahler with torsion structures.