Monatshefte für Mathematik

, Volume 180, Issue 1, pp 1–37 | Cite as

Characterization of regularity for a connected Abelian action

  • Didier Arnal
  • Bradley CurreyEmail author
  • Vignon Oussa


Let V be a finite dimensional real vector space, let \(\mathfrak g\) be the real span of a finite set of commuting endomorphisms of V, and \(G = \exp \mathfrak g\). We study the orbit structure in elements of a finite partition of V into explicit G-invariant connected sets. In particular, we prove that either there is an open conull G-invariant subset \(\Omega \) of V in which every G-orbit is regular, or there is a G-invariant, conull, \(\mathcal G_\delta \) subset of V in which every orbit is not regular. We present an explicit computable necessary and sufficient condition for almost everywhere regularity. Finally in the case of regularity we construct an explicit topological cross-section for the orbits in \(\Omega \).


Regular and not regular orbits Lie algebra roots Linear Lie group action 

Mathematics Subject Classification

57Sxx 22Exx 22E25 22E27 17B45 17B08 58E40 



We thank the referee for a number of helpful suggestions. The second author would like to thank the Université de Bourgogne for their hospitality during his stay.


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Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Inst. de Mathematiques de Bourgogne, UMR CNRS 5584Université de BourgogneDijonFrance
  2. 2.Department of Mathematics and Computer ScienceSt. Louis UniversitySt. LouisUSA
  3. 3.Department of MathematicsBridgewater State UniversityBridgewaterUSA

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