Invertible linear maps on simple Lie algebras preserving commutativity
Invertible linear maps on simple Lie algebras preserving commutativity
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DOI：
10.1090/s0002-9939-2011-10834-7
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发表时间：
2011-11
影响因子：
6.7
通讯作者：
Dengyin Wang;Zhengxin Chen
中科院分区：
文献类型：
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作者：
Dengyin Wang;Zhengxin Chen
.设g是特征为零的代数闭域上秩为l的有限维单李代数。称g上的可逆线性映射保持双向交换性，如果对任意x，y ∈ g，[ x，y ] = 0惠[<$（x），<$（y）] = 0. g上所有这样的映射的群记为Pzp（g）。本文证明了：若l = 1，则Pzp（g）= GL（g），否则Pzp（g）= Aut（g）× F <$Ig，其中F <$Ig表示g上所有非零标量乘映射的群.
. Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic zero. An invertible linear map ϕ on g is called preserving commutativity in both directions if, for any x,y ∈ g , [ x,y ] = 0 ⇔ [ ϕ ( x ) ,ϕ ( y )] = 0. The group of all such maps on g is denoted by Pzp ( g ). It is shown in this paper that, if l = 1, then Pzp ( g ) = GL ( g ); otherwise, Pzp ( g ) = Aut ( g ) × F ∗ I g , where F ∗ I g denotes the group of all non-zero scalar multiplication maps on g .