Biderivations and strong commutativity-preserving maps on parabolic subalgebras of simple Lie algebras

Abstract

ABSTRACT A linear map ψ on a Lie algebra over a field F with char is called to be commuting (resp., skew-commuting) if (resp., ) for all , and to be strong commutativity-preserving if for all . Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, firstly, we improve existing results about skew-symmetric biderivations on P by determining related linear commuting maps. Secondly, we classify the linear skew-commuting maps and the related symmetric biderivations on P, and so the biderivations of P are characterized. Finally, we classify the invertible linear strong commutativity-preserving maps of P.