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On polynomial products in nilpotent and solvable Lie groups


Author: Karel Dekimpe
Journal: Proc. Amer. Math. Soc. 131 (2003), 973-978
MSC (1991): Primary 22E15
DOI: https://doi.org/10.1090/S0002-9939-02-06572-3
Published electronically: July 17, 2002
MathSciNet review: 1937436
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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

DOI: https://doi.org/10.1090/S0002-9939-02-06572-3
Keywords: Nilpotent and solvable Lie groups
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
Article copyright: © Copyright 2002 American Mathematical Society

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