On polynomial products in nilpotent and solvable Lie groups
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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
- Louis Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. MR 486307, DOI 10.1090/S0002-9904-1973-13134-9
- Louis Auslander, Simply transitive groups of affine motions, Amer. J. Math. 99 (1977), no. 4, 809–826. MR 447470, DOI 10.2307/2373867
- Benoist, Y. and Dekimpe, K. The Uniqueness of Polynomial Crystallographic Actions. Math. Ann., 2002, 322, pp. 563–571.
- Karel Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures, Forum Math. 12 (2000), no. 1, 77–96. MR 1736093, DOI 10.1515/form.1999.030
- Karel Dekimpe, Solvable Lie algebras, Lie groups and polynomial structures, Compositio Math. 121 (2000), no. 2, 183–204. MR 1757881, DOI 10.1023/A:1001738932743
- Karel Dekimpe and Paul Igodt, Polynomial alternatives for the group of affine motions, Math. Z. 234 (2000), no. 3, 457–485. MR 1774093, DOI 10.1007/PL00004807
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
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