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.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11871014).
Disclosure statement
No potential conflict of interest was reported by the authors.