On the Lengths of Certain Chains of Subalgebras in Lie Algebras

Abstract

In this paper we study the lengths of certain chains of subalgebras of a Lie algebra L: namely, a chief series, a maximal chain of minimal length, a chain of maximal length in which each subalgebra is modular in L, and a chain of maximal length in which each subalgebra is a quasi-ideal of L. In particular we show that, over a field F of characteristic zero, a Lie algebra L with radical R has a maximal chain of subalgebras and a chain of subalgebras all of which are modular in L of the same length if and only if L = R, or and L/R is a direct sum of isomorphic three-dimensional simple Lie algebras.

Key Words:

• Chief series
• Lie algebras
• Maximal chain
• Modular subalgebra
• Quasi-ideal

2010 Mathematics Subject Classification:

• 17B05
• 17B20
• 17B30
• 17B50

Notes

Communicated by A. Elduque.