Primitive Actions and Maximal Subgroups of Lie Groups

The classification of the primitive transitive and effective actions of Lie groups on manifolds is a problem dating back to Lie. The classification of the infinite dimensional infinitesimal actions was originally done by Cartan [3] and was made rigorous by some joint work of Guillemin, Quillen and Sternberg [8], whose proof was further simplified by Guillemin [7] recently by using some results of Veisfieler [19]. The classification of the primitive actions of a given finite dimensional Lie group is equivalent to that of the Lie subgroups of that group, which satisfy a certain maximality condition (see Prop. 1.5). This correspondence although more or less known seems never to have been stated in the literature (under the assumption that the leaves of a foliation are connected) so in § 1 we state it. The rest of § 1 is devoted to showing that the isotropy subalgebras of primitive actions are an intrinsically well-defined class, namely, they are the Lie algebras which correspond to maximal Lie subgroups and contain no proper ideals. We call these subalgebras primitive and hasten to add that this terminology does not agree with the use of "primitive" in [7], [8], [11], [12], [13] and [16]. In these articles a "primitive subalgebra" is a maximal Lie subalgebra which contains no proper ideals. In light of Theorem 1.10 we do feel that this is a more reasonable terminology. Also we show that every subalgebra which is "primitive" in the old sense is primitive in the new sense. The main result of this paper is that there exist primitive subalgebras which are not maximal subalgebras, i.e., there exist maximal Lie subgroups whose Lie algebras are not maximal subalgebras. In § 3 we classify the primitive, maximal rank, reductive subalgebras of the (complex) classical algebras giving many examples of primitive subalgebras which are, in fact, not maximal. § 2 and § 4 combined show that non-maximal primitive algebras exist only when the containing algebra is simple and the primitive subalgebra is reductive. The proofs of this involves essentially classifying the primitive subalgebras. In doing so we duplicate results of Morozov [15] on the classification of the maximal primitive subalgebras of non-simple algebras and results of Karpelevich [10] and Ochiai [16] on the