Invariant Generalized Complex Structures on Flag Manifolds

• Main Authors: Varea, Carlos A. B, Martin, Luiz A. B. San
• Format: Journal Article
• Language: English
• Published: 05-09-2019
• Subjects: Mathematics - Differential Geometry
• DOI: 10.48550/arxiv.1810.09532

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold
$\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact
real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to
study invariant generalized complex structures on $\mathbb{F}$. We describe the
invariant generalized almost complex structures on $\mathbb{F}$ and classify
which one is integrable. The problem reduces to the study of invariant
$4$-dimensional generalized almost complex structures restricted to each root
space, and for integrability we analyse the Nijenhuis operator for a triple of
roots such that its sum is zero. We also conducted a study about twisted
generalized complex structures. We define a new bracket `twisted' by a closed
$3$-form $\Omega $ and also define the Nijenhuis operator twisted by $\Omega $.
We classify the $\Omega $-integrable generalized complex structure.