交换环上的严格上三角矩阵代数上的Lie导子

Lie Derivations of Strictly Upper Triangular Matrices over Commutative Rings

设R是任意含单位元的交换环,N(R)为R上(n+1)×(n+1)严格上三角矩阵构成的代数．本文证明了当n≥3且2是R的单位时,N(R)上任意Lie导子D可以唯一的表示为D=D_d+D_b+D_c+D_x,其中D_d,D_b,D_c,D_x分别是N(R)上的对角,极端,中心和内Lie导子,在n=2的情况,我们也证明了N(R)上任意Lie导子D可以表示为对角,极端,内Lie导子的和。

Let R be an arbitrary commutative ring with＇identity. Denote by N（R） the algebra over R consisting of all strictly upper triangular （n ＋ 1） × （n ＋ 1） matrices over R. We prove that any Lie derivation D of N（R） can be uniquely expressed as D = Dd ＋ Db ＋ Dc ＋ Dx, where Dd, Db, Dc, Dx are diagonal, extremal, central and inner Lie derivations, respectively, of N（R） when n ≥ 3 and R contains 2 as a unit. In the case n = 2, we also prove that any Lie derivation D of N（R） can be expressed as a sum of diagonal, extremal and inner Lie derivations.