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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
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.

References [Enhancements On Off] (What's this?)

  • 1. Auslander, L.
    An exposition of the structure of solvmanifolds. Part I: Algebraic Theory.
    Bull. Amer. Math. Soc., 1973, 79 2, pp. 227-261. MR 58:6066a
  • 2. Auslander, L.
    Simply Transitive Groups of Affine Motions.
    Amer. J. Math., 1977, 99 (4), pp. 809-826. MR 56:5782
  • 3. Benoist, Y. and Dekimpe, K.
    The Uniqueness of Polynomial Crystallographic Actions. Math. Ann., 2002, 322, pp. 563-571.
  • 4. Dekimpe, K.
    Semi-simple splittings for solvable Lie groups and polynomial structures.
    Forum Math., 2000, 12, pp. 77-96. MR 2001h:22009
  • 5. Dekimpe, K.
    Solvable Lie algebras, Lie groups and polynomial structures.
    Compositio Mathematica, 2000, 121, pp. 183-204. MR 2001e:17014
  • 6. Dekimpe, K. and Igodt, P.
    Polynomial Alternatives for the Group of Affine Motions.
    Math. Zeit., 2000, 234, pp. 457-485. MR 2001k:57042
  • 7. Malcev, A. I.
    On a class of homogeneous spaces.
    Translations A.M.S., 1951, 39, pp. 1-33. MR 12:589e

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Additional Information

Karel Dekimpe
Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium

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