Fields of tangent sets and Hofmann cones
type: Item
instance of: scholarly article
title: Fields of tangent sets and Hofmann cones
zbMATH Open document ID: 0619.22007
DOI: 10.1007/BF02573087
author: Jimmie D. Lawson
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134853
review text: The Bony-Brezis theorem states that a closed subset F of a differentiable manifold M is invariant under the flow associated with a locally Lipschitzian vector field A if and only if for every \(p\in F\) the tangent vector A(p) belongs to the subtangent space of F at p. In the present paper the author generalizes this result to Lipschitz fields of subsets of tangent vectors on M; applying this generalization in the Lie semigroup setting he obtains a number of remarkable results concerning Lie semialgebras (the author proposes the term 'Hofmann wedge' instead of semialgebra). We only cite the following: (Theorem 6.2) Let W be a generating semialgebra in a finite-dimensional real Lie algebra, and suppose that p is a \(C^ 1\)-point of W such that the characteristic function does not vanish at p. If (p) is an eigenvalue of multiplicity one for ad p then the tangent hyperplane \(T_ p\) of W at p is a subalgebra of L. (Theorem 10.1) If every point in a generating finite-dimensional real Lie semialgebra W is a \(C^ 1\)-point (or if every point of W is an \(E^ 1\)- point) then either W is an invariant cone or L is almost abelian. [The latter result generalizes earlier results by \textit{A. V. Levichev}].
Mathematics Subject Classification ID: 22A20
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 4004500
zbMATH Keywords: Bony-Brezis theorem
zbMATH Keywords: Lipschitzian vector field
zbMATH Keywords: tangent vectors
zbMATH Keywords: Lie semigroup
zbMATH Keywords: Lie semialgebras
zbMATH Keywords: Hofmann wedge
reviewed by: Wolfgang M. Ruppert
MaRDI profile type: Publication
cites work: On a characterization of flow‐invariant sets
cites work: Semigroups in Lie Groups, Semialgebras in Lie Algebras
cites work: Lorentzian cones in real Lie algebras
cites work: Old and new on Sl(2)
cites work: The local theory of semigroups in nilpotent Lie groups
cites work: Q4110164
cites work: Q4060241
OpenAlex ID: W1975494055
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description: scientific article; zbMATH DE number 4004500