Dipolarization in semisimple Lie algebras and homogeneous parakähler manifolds

Abstract: A dipolarization in a Lie algebra ${\scriptstyle{\frak g}}$ is two polarizations ${\scriptstyle{\frak g}^\pm}$ in ${\scriptstyle{\frak g}}$ at a common linear form on ${\scriptstyle{\frak g}}$ satisfying ${\scriptstyle{\frak g}={\frak g}^++{\frak g}^-}$. We study dipolarizations in semisimple Lie algebras, especially, the relation between dipolarizations and gradations. As an application, we give a relation between semisimple homogeneous parak$\ddot a$hler manifolds and hyperbolic semisimple orbits. For ${\scriptstyle{\frak g}}$ real semisimple, we determine the characteristic elements, from which dipolarizations can be constructed.

Keywords: Semisimple Lie algebra, graded Lie algebra, parabolic subalgebra, dipolarization, parak$\ddot a$hler manifold.

Classification (MSC91): 17B20, 22E46, 53C12, 15, 30

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