Derivations 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. In this article, we give an explicit description of any derivation of N(R). We prove that under some conditions for R, any derivation of N(R) can be expressed as a sum of inner, central, diagonal, extremal derivations of N(R).

Key Words:

• Commutative ring
• Chevalley algebra
• Derivation of Lie algebra
• Nilpotent Lie algebra

2000 Mathematics Subject Classification:

• 17B30

ACKNOWLEDGMENT

This work is supported by the National Nature Science Foundation of China (Grant No. 11071040). We thank the referee for his/her advice to improve the article.

Notes

Communicated by K. Misra.