On the theta completions for maximal subalgebras

ABSTRACT

Let L be a finite dimensional Lie algebra. Then for a maximal subalgebra M of L, a 𝜃-completion for M is a subalgebra C of L such that CM and M_L⊆C and C∕M_L contains no non-zero ideal of L∕M_L, properly. And a 𝜃-completion C of M is said to be a strong 𝜃-completion, if C = L or there exists a subalgebra B of L such that C be maximal in B and B is not a 𝜃-completion for M. These are analogous to the concepts of 𝜃-completion and strong 𝜃-completion of a maximal subgroup of a finite group. Now, we consider the influence of these concepts on the structure of a finite dimensional Lie algebra.

KEYWORDS:

• 𝜃-Completion
• Lie algebra
• nilpotent
• solvable
• strong 𝜃-completion

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B05
• 17B20
• 17B30
• 17B50

Acknowledgment

The author would like to thank the referee and Professor A.R. Salemkar for their valuable suggestions.