On the problem of classifying solvable Lie algebras having small codimensional derived algebras

Abstract

In this paper, we study the classification of finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. We present an effective method to classify $(n+1)$-dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given. In addition, the problem of classifying $(n+2)$-dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation.

Keywords:

• Lie algebra
• derived algebra
• wild problem

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A21
• 16G60
• 17B30
• 20G05

Acknowledgements

The authors would like to thank the referees for pointing-out references [23, 24, 31, 32] as well as helpful suggestions that help us to improve the exposition of the paper.