Lie triple derivations on the parabolic subalgebras of simple Lie algebras

Abstract

A linear map ϕ on a Lie algebra L is called a Lie triple derivation if ϕ([x, [y, z]]) = [ϕ(x), [y, z]] + [x, [ϕ(y), z]] + [x, [ y, ϕ(z)]] for all x, y, z ∈ L. Let be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field F of characteristic zero, and an arbitrary parabolic subalgebra of . It is shown in this note that a linear map ϕ on is a Lie triple derivation if and only if it is an inner derivation, which generalizes the main result of Tolpygo [A.K. Tolpygo, On chomology of parabolic Lie algebras, Math. Notes 12 (1973), pp. 585–587] saying that every derivation of is inner derivation.

Keywords:

• simple Lie algebras
• parabolic subalgebras
• Lie triple derivations

AMS Subject Classifications:

• 17B20
• 17B30
• 17B40

Acknowledgements

We thank the referee for his careful examination and helpful suggestions. His suggestions make our proofs more transparent.