Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras

Abstract

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of finite-dimensional Lie algebras, starting from the non-zero brackets in its law. In order to implement this method, we use the symbolic computation package MAPLE 12. Moreover, we also give a brief computational study considering both the computing time and the memory used in the two main routines of the implementation. Finally, we determine the maximal dimension of abelian subalgebras and ideals for non-decomposable solvable non-nilpotent Lie algebras of dimension 6 over both the fields ℝ and ℂ, showing the differences between these fields.

Keywords:

• abelian Lie subalgebra
• abelian ideal
• α invariant
• β invariant
• algorithm

2000 AMS Subject Classifications :

• 17B30
• 17B05
• 68W40
• 68Q25

Acknowledgements

This work has been partially supported by MTM2010-19336 and FEDER. Additionally, the authors want to thank the referees for their helpful and useful comments and suggestions, which have allowed us to improve the quality of this paper.