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

ABSTRACT

A linear map ψ on a Lie algebra $\mathfrak{g}$ over a field F with char$\left(F\right)\ne 2$ is called to be commuting (resp., skew-commuting) if $\left[\psi \right(x),y]=[x,\psi (y\left)\right]$ (resp., $\left[\psi \right(x),y]=-[x,\psi (y\left)\right]$) for all $x,y\in \mathfrak{g}$, and to be strong commutativity-preserving if $\left[\psi \right(x),\psi (y\left)\right]=[x,y]$ for all $x,y\in \mathfrak{g}$. 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.

Keywords:

• Simple Lie algebras
• parabolic subalgebras
• symmetric biderivations
• skew-symmetric biderivations
• strong commutativity-preserving maps

2010 Mathematics Subject Classifications:

• 17B05
• 17B20
• 17B40

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.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11871014).