On j-algebras and homogeneous Kähler manifolds
Let G be a connected Lie group with Lie algebra \(g\) and let K be a connected subgroup with Lie algebra \(k\) such that \(k\) contains no non- zero ideal of \(g\). Let j be an endomorphism of \(g\) which preserves \(k\) and induces an endomorphism of \(g/k\) which commutes with the adjoint action of K, has square -1 and satisfies \([jx,jy]=[x,y] + j[jx,y] + j[x,jy] modulo k\) for all x,y. Let \(\omega\) be a linear form on \(g\) which satisfies \(\omega ([k,g])=0,\) \(\omega ([jx,jy])=\omega ([x,y])\) for all x,y and \(\omega ([jx,x])>0\) if \(x\not\in k\). Then (\(g,k,j,\omega)\) is called an effective j-algebra and induces a G-invariant Kähler structure on the homogeneous space G/K. In this paper it is proved that if (\(g,k,j,\omega)\) is an effective j-algebra then G/K is biholomorphic to \(M_ 1\times M_ 2\) where \(M_ 1\) is a bounded homogeneous domain and \(M_ 2\) is a compact simply-connected homogeneous complex manifold [cf. \textit{S. G. Gindikin}, \textit{I. I. Pyatetskij-Shapiro} and \textit{E. B. Vinberg}, Trans. Mosc. Math. Soc. 12, 404-437 (1963)]. Using similar arguments it is shown that the same is true whenever G/K is a homogeneous Kähler manifold which contains no locally flat homogeneous Kähler submanifold. Conversely it is shown that if \(M_ 1\) is a homogeneous bounded domain and \(M_ 2\) is a compact simply connected homogeneous complex manifold such that a connected Lie group G acts transitively, effectively and holomorphically on \(M_ 1\times M_ 2\) and \(M_ 1\times M_ 2\) has a G-invariant Kähler metric then there is an effective j-algebra (\(g,k,j,\omega)\) such that the associated homogeneous manifold is biholomorphic to \(M_ 1\times M_ 2\).
scientific article; zbMATH DE number 3937548