Homogeneous Siegel Domains

Abstract

In this chapter, we will introduce the notions of homogeneous Kähler manifolds, homogeneous bounded domains and homogeneous Siegel domains. In particular, we will give a sufficient condition for the Lie algebra of the isometric transformation group acting transitively on a homogeneous Kähler manifold. We will introduce the notions of K Lie algebras, J Lie algebras, effective proper J Lie algebras and normal J Lie Algebras, respectively. We will prove that the Lie algebra of the holomorphic automorphism group acting transitively on a homogeneous bounded domain is an effective proper J Lie algebra. On the other hand, we will give a J basis of a normal J Lie algebra, and prove that the classification of effective proper J Lie algebras up to a J isomorphism is equivalent to the classification of normal matrix sets up to a unitarily equivalent.