Semigroups in the simply connected covering of \(SL(2)\)
type: Item
instance of: scholarly article
title: Semigroups in the simply connected covering of \(SL(2)\)
zbMATH Open document ID: 0776.22001
DOI: 10.1007/BF02573580
author: Dirk Mittenhuber
published in: Semigroup Forum
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/135244
review text: This article contributes to the investigation of subsemigroups of the simply connected covering group \(G=\text{Sl}(2,\mathbb{R})\tilde{ }\) of \(\text{Sl}(2,\mathbb{R})\) which are Lie semigroups, i.e., closed and topologically generated by one-parameter semigroups. In the Lie algebra \({\mathfrak {sl}}(2)\) there exists a one-parameter family \(W_ r\), \(r>0\) of Lorentzian cones which are invariant under the adjoint action of SO(2). In [Semigroup Forum 43, No. 1, 33-43 (1991; Zbl 0736.22005)] the reviewer proved that \(\exp W_ r\) generates the whole group \(G\) if and only if \(r>1\). Later \textit{J. Hilgert} showed [in Math. Z. 209, 463-466 (1992)] that these results can be derived from rather general arguments which are derived from those ideas which led Gödel to his non-causal cosmological model. In this article the author uses the Pontrjagin Maximum Principle to exhibit smooth causal loops for \(r>1\) (the non-causal case) and he also derives an explicit description of the semigroups \(S_ r:=\langle\exp W_ r\rangle\) for \(r\leq 1\). His method also leads to a more explicit description of the exponential function of \(G\).
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 164975
zbMATH Keywords: reachable sets
zbMATH Keywords: controllability
zbMATH Keywords: left-invariant Lorentzian metric
zbMATH Keywords: geodesics
zbMATH Keywords: simply connected covering group
zbMATH Keywords: \(\text{Sl}(2,\mathbb{R})\)
zbMATH Keywords: Lie semigroups
zbMATH Keywords: one-parameter semigroups
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lorentzian cones
zbMATH Keywords: adjoint action
zbMATH Keywords: Pontrjagin Maximum Principle
zbMATH Keywords: smooth causal loops
zbMATH Keywords: exponential function
reviewed by: Karl-Hermann Neeb
MaRDI profile type: Publication
cites work: Old and new on Sl(2)
cites work: Controllability on real reductive Lie groups
cites work: Semigroups in the universal covering group of \(SL(2)\)
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description: scientific article; zbMATH DE number 164975