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Invariant Generalized Complex Structures on Flag Manifolds

Research paper by Carlos A. B. Varea, Luiz A. B. San Martin

Indexed on: 22 Oct '18Published on: 22 Oct '18Published in: arXiv - Mathematics - Differential Geometry



Abstract

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.