Orbit nonproper dynamics on Lorentz manifolds
type: Item
instance of: scholarly article
title: Orbit nonproper dynamics on Lorentz manifolds
zbMATH Open document ID: 1017.53060
author: Scot Adams
published in: Illinois Journal of Mathematics
publication date: 2002-04-28T00:00:00Z
review text: This carefully written and highly readable paper is the last of a long series by author and \textit{G. Stuck}, originally motivated by a paper of \textit{N. Kowalsky} [Ann. Math. (2), 144, No. 3, 611-640 (1996; Zbl 0871.53048)]. The author characterizes in it the connected Lie groups \(G\) with simply connected nilradical which admit an orbit nonproper action by isometries of some connected Lorentz manifold. Here, orbit nonproper means that for some point \(x\), the evaluation map \(g\mapsto g(x)\) is nonproper. Such an action exists if and only if \(G\) satisfies one of the following conditions: the adjoint action \(G\to \text{GL} ({\mathfrak g})\) is nonproper, or \({\mathfrak g}\) has a direct summand \({\mathfrak so} (n,i)\), \(i\leq 2\leq n\), or \({\mathfrak g}\) has a nonzero abelian ideal \({\mathfrak a}\) admitting a positive definite or Minkowski quadratic form such that the adjoint action of \({\mathfrak g}\) on \({\mathfrak a}\) is conformal with respect to this form. There is another, equivalent formulation of the third condition which can be checked directly by looking at the adjoint representation of a Levi factor on the center of the nilradical. The proof of the theorem depends on a detailed analysis of the structure of \({\mathfrak so}(n,1)\) and of the representations of \({\mathfrak sl}_2\) and of noncompact simple or reductive groups.
reviewed by: Rainer Löwen
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 57S20
zbMATH DE Number: 1734190
zbMATH Keywords: orbit nonproper action
zbMATH Keywords: Lorentz manifold
zbMATH Keywords: Levi factor
zbMATH Keywords: nilradical
zbMATH Keywords: representations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1734190