Geometries of Orthogonal Groups and Their Contractions: a Unified Classical Deformation Viewpoint

The general aim of this paper is to describe a particular case of a classical scheme which involves a whole class of spaces, and geometries associated to a family of Lie groups. At all different levels of this scheme, either the spaces, the Lie groups or their Lie algebras are related among themselves by contractions, yet their properties can be dealt with in a completely unified way. The family we will consider here comprises the classical real geometries of spaces with a projective metric (Cayley-Klein or CK geometries) , together with their Lie groups and Lie algebras. These algebras will be called here orthogonal CK algebras, as they include all simple orthogonal real Lie algebras so(p, q), as well as many Lie algebras of great physical relevance, (Poincaré, Galilei, Euclidean, etc) which are obtained by different contractions from so(p, q). All two-point homogeneous symmetric spaces of real type (the Riemannian spaces of constant curvature) appear related to CK algebras, but pseudoRiemannian and degenerate Riemannian spaces (which strictu senso are not two-point homogeneous) are included as well; indeed this class of spaces is more natural than the class of two-point homogeneous spaces as usually considered in the literature. The orthogonal CK scheme is one of the several (though in reduced number) possible schemes of a similar kind. Each include some simple real Lie algebras, as well as some non-simple contracted algebras, which are however ‘near’ to the simple ones so that most properties of simple algebras (geometries, groups), when suitably reformulated, still survive for them; this makes ‘quasi-simple’ an apt name