Expanding Automorphisms and Affine Structures on Nilpotent Lie Algebras with Few Generators

Abstract

We construct ℤ-gradings for all 3-generated 3-step nilpotent and 2-generated 4-step nilpotent Lie algebras. This shows that all 3-generated 4-step nilpotent and 2-generated 5-step nilpotent Lie algebras have a complete affine structure. We also obtain a lower bound on the number of generators of a c-step nilpotent Lie algebra with singular derivation algebra. Examples indicate this bound is sharp.

Key Words:

• Nilpotent Lie algebra
• Expanding automorphism
• Graded Lie algebra
• Affine structure

Acknowledgments

Notes

^#Research Assistant of the Fund for Scientific Research – Flanders (F.W.O.).