半单李代数单理想直和分解式的性质

Property of Decomposition of Semisimple Lie Algebra

本文作为推论给出了半单李代数L上的一个元素x的半单部分,幂零部分与对应的单理想值和分解式中x的每一个分量的半单部分,幂零部分之间的关系.然后再给出了半单李代数L的Killing型与单理想值和分解式中的每一个单理想的Killing型之间的关系.

In this paper,as a corollary it is shown that the semisimlpe and nilpotent parts of x ∈ L are the sums of thesemisimple and nilpotent parts in the various Li （i =1,…,t） of the components of x,where L =L1 （@）L2（@）…（@） Lt be the decomposition of a semisimple Lie algebra L into its simple ideals.It is also shown that the Killing form of L is sum of the Killing forms of various Li （i =1,…,t）.