Special algebraic properties of Kähler algebras (Q1174664)

A Kähler algebra corresponds to the Lie algebra of infinitesimal automorphisms of a homogeneous Kähler manifold; it can be defined as a 4-uple (\({\mathfrak g,k,;}j,\rho\)) with suitable properties, where \(\mathfrak g\) is the Lie algebra of infinitesimal automorphisms, \(\mathfrak k\) the isotropy subalgebra of a point, \(j\) and \(\rho\) representing the complex structure and the Kähler form; if \(\rho\) is the differential of a 1-form \(\omega\) on \(\mathfrak g\), the Kähler algebra is called a \(j\)-algebra. The main results of this paper are: 1) if \(\mathfrak g\) is the Lie algebra of infinitesimal automorphisms of a homogeneous Kähler manifold, then \(\text{ad}{\mathfrak g}\) is algebraic; 2) \(\mathfrak g\) has an orthogonal decomposition \({\mathfrak a}+{\mathfrak h}\), where \(\mathfrak a\) is a \(j\)-invariant abelian ideal, \({\mathfrak k}\subset{\mathfrak h}\), \(\mathfrak h\) is an invariant subalgebra, (\({\mathfrak h,k},j\)) is a \(j\)-algebra and there is a linear form \(\omega\) on \(\mathfrak h\) such that \(\rho\mid {\mathfrak h}=d\omega\). Applications are various characterizations of homogeneous Kähler manifolds without locally flat homogeneous submanifolds.