On Sophus Lie's fundamental theorem
In this paper the fundamental correspondence of Sophus Lie between local Lie groups and Lie algebras is extended to a local semigroup setting. It is known that the set of vectors L(S) tangent to the identity in a local semigroup S form a Lie wedge, i.e., a closed convex set closed under positive scalar multiplication which is invariant (as a set) under automorphisms of the form exp(ad x) for \(x\in H\), the largest subspace of L(S). The main result of this paper provides the much more difficult converse: namely that to each Lie wedge W in a Lie algebra L there exists a local semigroup S with \(L(S)=W.\) \textit{G. I. Ol'shanskij} [Funct. Anal. Appl. 15, 275-285 (1982; Zbl 0503.22011)] had outlined a proof in the case the subgroup corresponding to the subalgebra H is closed, and \textit{K. H. Hofmann} and the reviewer had done the special cases that W is a cone or a ''split'' wedge [Indagationes Math. 45, 453-466 (1983; Zbl 0525.22004) and ibid. 46, 255- 265 (1984; Zbl 0564.22006)].
scientific article; zbMATH DE number 3962000
scientific article