Dipolarizations in semisimple Lie algebras and homogeneous parakähler manifolds. (English) Zbl 0938.17008
Let \(g\) be a real or complex Lie algebra. The concept of dipolarization was introduced by the third author. It is a pair \((g^+,g^-)\) of subalgebras such that \(g^+\oplus g^- =g\), \([g^+,g^+]\subset V\), \([g^-,g^-]\subset V\), \([X,g]\subset V\) iff \(X\in g^+\cap g^-\), where \(\operatorname{codim} V=1\). By definition, \(Z\in g\) is a characteristic element if \(V=\{x\in g\mid b(Z,x)=0\}\) where \(b(\cdot,\cdot)\) is the Killing form. In complex semisimple algebras each non-zero element is a characteristic element of some dipolarization if and only if it is a semisimple one. Each dipolarization in a complex semisimple algebra comes from a gradation \(g=\sum_k g_k\).
As an application, the following theorem is proved. The coset space \(G/H\) of a semisimple group is a parakähler manifold if and only if \(H\) is an open subset of a Levi subgroup of a parabolic subgroup of \(G\). The authors also give the relation between semisimple parakähler homogeneous manifolds and hyperbolic semisimple orbits.
As an application, the following theorem is proved. The coset space \(G/H\) of a semisimple group is a parakähler manifold if and only if \(H\) is an open subset of a Levi subgroup of a parabolic subgroup of \(G\). The authors also give the relation between semisimple parakähler homogeneous manifolds and hyperbolic semisimple orbits.
Reviewer: P.Grushko (Irkutsk)
MSC:
17B20 | Simple, semisimple, reductive (super)algebras |
53C12 | Foliations (differential geometric aspects) |