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 | References | Similar Articles | Additional Information

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

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