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).
ACKNOWLEDGMENT
We thank the referee for pointing imprecision in Corollary 4.4.
Notes
Communicated by M. Bresar.