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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On polynomial products in nilpotent and solvable Lie groups
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by Karel Dekimpe PDF
Proc. Amer. Math. Soc. 131 (2003), 973-978 Request permission

Abstract:

We are dealing with Lie groups $G$ which are diffeomorphic to ${\mathbb R}^n$, for some $n$. After identifying $G$ with ${\mathbb R}^n$, the multiplication on $G$ can be seen as a map $\mu :{\mathbb R}^n\times {\mathbb R}^n\rightarrow {\mathbb R}^n: (\mathbf {x},\mathbf {y})\mapsto \mu (\mathbf {x},\mathbf {y})$. We show that if $\mu$ is a polynomial map in one of the two (sets of) variables $\mathbf {x}$ or $\mathbf {y}$, then $G$ is solvable. Moreover, if one knows that $\mu$ is polynomial in one of the variables, the group $G$ is nilpotent if and only if $\mu$ is polynomial in both its variables.
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Additional Information
  • Karel Dekimpe
  • Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium
  • Email: Karel.Dekimpe@kulak.ac.be
  • Received by editor(s): March 9, 2001
  • Received by editor(s) in revised form: October 23, 2001
  • Published electronically: July 17, 2002
  • Additional Notes: This research was conducted while the author was a Postdoctoral Fellow of the Fund for Scientific Research – Flanders (F.W.O.)
  • Communicated by: Ronald A. Fintushel
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 973-978
  • MSC (1991): Primary 22E15
  • DOI: https://doi.org/10.1090/S0002-9939-02-06572-3
  • MathSciNet review: 1937436