Generalized derivations on parabolic subalgebras of general linear Lie algebras

Abstract

Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, ▪) over a field ▪, where n ≥ 3, ▪ contains at least n different elements, and char(▪) ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.