The category of generalized Lie groups

Abstract

We consider the category Γ \Gamma of generalized Lie groups. A generalized Lie group is a topological group G G such that the set L G = H o m ( R , G ) LG = Hom({\mathbf {R}},G) of continuous homomorphisms from the reals R {\mathbf {R}} into G G has certain Lie algebra and locally convex topological vector space structures. The full subcategory Γ r {\Gamma ^r} of r r -bounded ( r r positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in Γ \Gamma . Various properties of generalized Lie groups G G and their locally convex topological Lie algebras L G = H o m ( R , G ) LG = Hom({\mathbf {R}},G) are investigated.