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.

摘要

李代数上的线性映射ψ$\mathfrak{G}$在带有 char的字段F上$\left(F\right)\ne 2$被称为通勤（resp.，skew-commuting）如果$\left[\psi \right(X),\mathrm{是的}]=[X,\psi (\mathrm{是的}\left)\right]$（分别，$\left[\psi \right(X),\mathrm{是的}]=-[X,\psi (\mathrm{是的}\left)\right]$） 对所有人$X,\mathrm{是的}\in \mathfrak{G}$，并且是强交换性保持的，如果$\left[\psi \right(X),\psi (\mathrm{是的}\left)\right]=[X,\mathrm{是的}]$对所有人$X,\mathrm{是的}\in \mathfrak{G}$. 令L是特征 0的代数闭域F上的有限维简单李代数， P是L的抛物线子代数。在本文中，首先，我们通过确定相关的线性通勤图来改进关于P上的斜对称双推导的现有结果。其次，我们对P上的线性倾斜交换映射和相关的对称双推导进行了分类，从而表征了P的双推导。最后，我们对P的可逆线性强交换性保持映射进行分类。