A rank-three condition for invariant (1,2)-symplectic almost Hermitian structures on flag manifolds
The authors consider invariant \((1,2)\)-symplectic almost Hermitian structures on the maximal flag manifold associated to a complex semi-simple Lie group \(G\). An invariant almost complex structure \(J\) is called \((1,2)\)-admissible if there exists an invariant Riemannian metric \(\Lambda\) such that \((J, \Lambda)\) is \((1,2)\)-symplectic. It is shown that a \((1,2)\)-admissible almost complex structure is necessarily cone-free. The property of being cone-free is a condition on the rank three subsystems of the root system of \(G\) if this is not \(G_{2}\). It extends the property of being cone-free for tournaments related to the \( Sl(n,\mathbb{C})\) case [see \textit{N. Cohen, C. J. C. Negreiros} and \textit{L. A. B. San Martin}, Bull. Lond. Math. Soc. 34, No. 6, 641-649 (2002)]. If the semi-simple Lie group does not contain components of type \(B_{l}\), \(l \geq 3, G_{2}, F_{4}\), then this property is also sufficient.
scientific article; zbMATH DE number 1890487