Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Abstract

Let L be a finite-dimensional complex simple Lie algebra, L _ℤ be the ℤ-span of a Chevalley basis of L, and L _R  = R ⊗_ℤ L _ℤ be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ϕ of 𝒩(R) is called commuting if [ϕ(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).

Key Words:

• Commuting derivation
• Commuting automorphism
• Commutative ring
• Chevalley algebra
• Nilpotent Lie algebra

2010 Mathematics Subject Classification:

• 17B30
• 17B40

ACKNOWLEDGMENT

We thank the referee for pointing imprecision in Corollary 4.4.

Notes

Communicated by M. Bresar.