IJPAM: Volume 30, No. 1 (2006)

TRANSLATIONS IN SIMPLY TRANSITIVE AFFINE
ACTIONS OF FREE 2-STEP NILPOTENT LIE GROUPS

Tine De Cat$^1$, Karel Dekimpe$^2$
Department of Mathematics
Subfaculty of Sciences
Campus Kortrijk
Katholieke Universiteit Leuven
Sabbelaan 53, BE-8500 Kortrijk, BELGIUM
$^2$e-mail: [email protected]
$^2$url: http://www.kulak.ac.be/ dekimpe/


Abstract.In this paper we study simply transitive affine actions of free 2-step nilpotent Lie groups. We show that for the free 2-step nilpotent Lie group on 3 generators there is always a non-trivial subgroup, which will act as a group of pure translations. We prove that for any odd number of generators $\geq 5$, one can find a simply transitive action without translations. Finally, we also show that for an even number of generators, there does not exist a similar kind of action.

Received: May 31, 2006

AMS Subject Classification: 17B30, 22E25

Key Words and Phrases: free nilpotent Lie algebra, left symmetric structure, simply transitive affine action

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 30
Issue: 1