SPLIT EXTENSIONS IN LIE ALGEBRA

We introduce the -extension and this extension induces the given -module structure on . In most of the cases treated here, will be abelian. Then the extension gives rise to -modules structure on , which is welldefined by , where , and is any element with . Now let be a -module. We call a -extension of if is an extension of by which induces the given -module structure on by the method above. we study the concepts of Lie algebra extension with abelian kernel and introduce forward and backward induced Lie algebra extensions. Further, we show any morphism can be factored uniquely. Then we can get solution set for diagram (2.3). we provides of congruent two lie algebra extensions and prove some important theorem.