内余分裂李代数的唯一性

Inner co-split Lie algebras and its uniqueness

复数域上的有限维李代数L是弱余分裂的当且仅当[L,L]=L,而当L半单一定是余分裂的.但复数域上余分裂李代数是否一定半单仍然是开问题.证明了复数域C上的有限维李代数是内余分裂李代数当且仅当它是半单的,而且它的内余分裂结构是唯一的.

A finite-dimensional Lie algebra L over C is weak co-split if and only it [L,L]=L,anct eacn semi-simple L is co-split. However,it is still open that if a co-split Lie algebra over C must be semi- simple. In the present short note,we prove that a finite-dimensional Lie algebra over complex number field C is an inner co-split Lie algebra if and only if it is semi-simple. Moreover, its inner co-split structure must be unique.