On Diverse Forms of Homogeneity of Lie Algebras

In the paper, several different ways to introduce the notion of homogeneity in the case of finite-dimensional Lie algebras are considered. Among these notions, we have homogeneity, almost homogeneity, weak homogeneity, and projective homogeneity. Constructions and examples of Lie algebras of diverse forms of homogeneity are presented. It is shown that the notions of weak homogeneity and of weak projective homogeneity are the most nontrivial and interesting for a detailed investigation. Some structural properties are proved for weakly homogeneous and weakly projectively homogeneous Lie algebras.