The category of generalized Lie groups

Abstract:

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