Biderivations of the Parabolic Subalgebras of Simple Lie Algebras

Abstract

Let 𝔤 be a simple Lie algebra of rank l over an algebraically closed field of characteristic zero, 𝔭 an arbitrary parabolic subalgebra of 𝔤. A bilinear map ϕ: 𝔭 × 𝔭 → 𝔭 is called a biderivation of 𝔭 if it is a derivation with respect to both components, meaning that

for all x, y, z ∈ 𝔭. It is shown in this article that a bilinear map ϕ: 𝔭 × 𝔭 → 𝔭 is a biderivation if and only if it is a sum of an inner and an extremal biderivation.

Key Words:

• Biderivations
• Parabolic subalgebras
• Simple Lie algebras

2000 Mathematics Subject Classification:

• 17B20
• 17B30
• 17B40

ACKNOWLEDGMENT

The authors thank the referee for his (or her) careful examination and helpful suggestions.

Supported by “National Natural Science Foundation of China” (No. 11171343).

Notes

Communicated by M. Bresar.