Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Abstract

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ϕ on 𝔤 is said to preserve solvability in both directions if ϕ, as well as ϕ⁻¹, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ϕ on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.

Key Words:

• Automorphisms of Lie algebras
• Simple Lie algebra
• Solvability

2010 AMS Subject Classification:

• 17B20
• 17B30
• 17B40

ACKNOWLEDGMENT

This study is supported by “the Fundamental Research Funds for the Central Universities (2012LWA08)” and “the National Natural Science Foundation of China (No. 11171343).”

Notes

Communicated by M. Bresar.