Dipolarizations in compact Lie algebras and homogeneous parakähler manifolds
A parakähler manifold \(M\) is a symplectic manifold which admits a pair of transversal Lagrangian foliations. If a Lie group \(G\) acts transitively on \(M\) by symplectomorphisms which preserve each of the foliations, then \(M\) is called homogeneous parakähler manifold. The authors characterize homogeneous parakähler manifolds by the notion of dipolarization of compact Lie algebras. They prove that a compact semisimple Lie algebra has no nontrivial dipolarizations. They also prove that a \(G\)-invariant parakähler structure may exist on \(G/H\), where \(G\) is a connected compact Lie group and \(H\) is a closed subgroup of \(G\), if and only if \(G/H\) is an even-dimensional torus.
scientific article; zbMATH DE number 1130803