一般线性李代数和有限维单李代数的抛物子代数上非线性强交换映射

Nonlinear Strongly Commuting Maps on Parabolic Subalgebras of General Linear Lie Algebras and Finite-dimensional Simple Lie Algebras

设F为域且char F≠2,L为域F上李代数.L上的一个映射φ:L→L称为非线性强交换映射,如果对任意的x,y∈L,有[φ(x),y]=[x,φ(y)].当P为一般线性李代数gl(n,F)(n≥2)的抛物子代数时,证明了P上映射φ为非线性强交换映射当且仅当φ是P上数乘映射与中心映射之和;又当P是有限维单李代数L的抛物子代数时,证明了P上映射φ是非线性强交换映射当且仅当φ是P上数乘映射.

Let F be a field whose characteristic is not equal to 2,L a Lie algebra over F.A map φ on L is called a nonlinear strongly commuting map,if for any x,y ∈ L,[φ（ x）,y] = [x,φ（ y） ].If P is a parabolic subalgebra of a general linear Lie algebra gl（ n,F）（ n ≥2）,it is shown that any nonlinear strongly commuting map on P is a sum of a scalar multiplication map and a central quasiderivation.And if P is a parabolic subalgebra of a finite dimensional simple Lie algebra L over F,it is shown that any nonlinear strongly commuting map is a scalar multiplication map.