矩阵代数上的2-局部Lie导子

2-Local Lie Derivations on Matrix Algebras

摘要: 设M_n(C)，T_n(C)分别是矩阵代数和上三角矩阵代数。本文证明若L:M_n(C)→M_n(C)是2-局部Lie导子，则存在T∈M_n(C)和映射τ:M_n(C)→CI_n使得L(A)=TA-AT+τ(A), ∀A∈M_n(C) (*)其中τ(A+F)=τ(A),  F=[A,B], ∀A, B∈M_n(C) 。利用该结论证明了M_n1(C)⊕M_n2(C)⊕---⊕M_nm(C)到自身的每个2-局部Lie导子具有形式(*)。证明了若L:Tn(C)→T_n(C)是2-局部Lie导子，且L(A+B)-L(A)-L(B)∈CI_n, ∀A, B∈T_n(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T_n1(C)⊕T_n2(C)⊕---⊕T_nm(C)到自身的2-局部Lie导子。

Abstract: Let M_n(C) be a matrix algebra, and T_n(C) be an upper triangular matrix algebra. In this paper, we show that, if L:M_n(C)→M_n(C) is a 2-local Lie derivation, then there exist a matrix T∈M_n(C) and a map τ:M_n(C)→CIn such that L(A)=TA-AT+τ(A), ∀A∈M_n(C) (*) where τ(A+F)=τ(A),  F=[A,B], ∀A, B∈M_n(C) . As its application, we show every 2-local Lie derivation from M_n1(C)⊕M_n2(C)⊕---⊕M_nm(C) into itself has the form (*). In addition, we show that, if L:T_n(C)→T_n(C) is a 2-local Lie derivation and satisfies L(A+B)-L(A)-L(B)∈CIn, ∀A, B∈T_n(C), then L has the form (*). An example is given to show that the condition L(A+B)-L(A)-L(B)∈CI_n is necessary. 2-local Lie derivations from T_n1(C)⊕T_n2(C)⊕---⊕T_nm(C) into itself are also characterized.