上三角矩阵李代数的反交换映射

Anti-commuting Maps of Upper Triangular Matrix Lie Algebras

设F是特征不为2的域, Tn(F)是域F上所有n阶上三角矩阵全体构成的李代数,φ:Tn(F)→Tn(F)为线性映射.若对任意X,Y∈Tn(F),[φ(X),Y]=-[X,φ(Y)],称φ为Tn(F)上线性反交换映射.证明当n≥3时, Tn(F)上线性映射φ为反交换映射当且仅当φ为一中心反交换映射与一极端内导子的和.

Let F be a field with char(F)≠2,let Tn( F) be the Lie algebra consisting of all upper triangular matrices over F,and let φ: Tn( F)→ Tn( F) be a linear map. If [φ( X),Y]=-[X,φ( Y)] for all X,Y in Tn(F),φ is called an anti-commuting map of Tn(F). In this paper,if n ≥ 3,it proves that a linear map φ of Tn(F) is an anti-commuting map if and only if φ is a sum of a central anti-commuting map and an extremal inner derivation.