矩阵代数上的2-局部Lie导子

2-Local Lie Derivations on Matrix Algebras

设Mn(C),Tn(C)分别是矩阵代数和上三角矩阵代数。本文证明若L:Mn(C)→Mn(C)是2-局部Lie导子,则存在T∈Mn(C)和映射τ:Mn(C)→CIn使得L(A)=TA-AT+τ(A), ∀A∈Mn(C) (*)其中τ(A+F)=τ(A), F=[A,B], ∀A, B∈Mn(C) 。利用该结论证明了Mn1(C)⊕Mn2(C)⊕---⊕Mnm(C)到自身的每个2-局部Lie导子具有形式(*)。证明了若L:Tn(C)→Tn(C)是2-局部Lie导子,且L(A+B)-L(A)-L(B)∈CIn, ∀A, B∈Tn(C),则L具有形式(*),并举例说明条件L(A+B)-L(A)-L(B)∈CIn不可去。本文还刻画了Tn1(C)⊕Tn2(C)⊕---⊕Tnm(C)到自身的2-局部Lie导子。

Let Mn(C) be a matrix algebra, and Tn(C) be an upper triangular matrix algebra. In this paper, we show that, if L:Mn(C)→Mn(C) is a 2-local Lie derivation, then there exist a matrix T∈Mn(C) and a map τ:Mn(C)→CIn such that L(A)=TA-AT+τ(A), ∀A∈Mn(C) (*) where τ(A+F)=τ(A), F=[A,B], ∀A, B∈Mn(C) . As its application, we show every 2-local Lie derivation from Mn1(C)⊕Mn2(C)⊕---⊕Mnm(C) into itself has the form (*). In addition, we show that, if L:Tn(C)→Tn(C) is a 2-local Lie derivation and satisfiesL(A+B)-L(A)-L(B)∈CIn, ∀A, B∈Tn(C), then L has the form (*). An example is given to show that the condition L(A+B)-L(A)-L(B)∈CIn is necessary. 2-local Lie derivations from Tn1(C)⊕Tn2(C)⊕---⊕Tnm(C) into itself are also characterized.