Invariant generalized complex structures on flag manifolds

摘要

Let G be a complex semi-simple Lie group and form its maximal flag manifold F = G/P = U/T where P is a minimal parabolic subgroup, U a compact real form and T = U boolean AND P a maximal torus of U. The aim of this paper is to study invariant generalized complex structures on F. We describe the invariant generalized almost complex structures on 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 analyze 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. (C) 2020 Elsevier B.V. All rights reserved.