On Sophus Lie's fundamental theorem
type: Item
instance of: scholarly article
title: On Sophus Lie's fundamental theorem
zbMATH Open document ID: 0597.22005
DOI: 10.1016/0022-1236(86)90028-5
author: Joachim Hilgert
author: Karl H. Hofmann
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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)].
reviewed by: Jimmie D. Lawson
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 22E05
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 3962000
zbMATH Keywords: local Lie groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: local semigroup
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90028-5
OpenAlex ID: W2018968538
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description: scientific article; zbMATH DE number 3962000