International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 2, Pages 71-77

Noncomplete affine structures on Lie algebras of maximal class

E. Remm and Michel Goze

Faculté des Sciences et Techniques, 4, Rue des Frères Lumière, Mulhouse Cedex F. 68093, France

Received 29 January 2001

Copyright © 2002 E. Remm and Michel Goze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Every affine structure on Lie algebra 𝔤 defines a representation of 𝔤 in aff(n). If 𝔤 is a nilpotent Lie algebra provided with a complete affine structure then the corresponding representation is nilpotent. We describe noncomplete affine structures on the filiform Lie algebra Ln. As a consequence we give a nonnilpotent faithful linear representation of the 3-dimensional Heisenberg algebra.