Abelian Subalgebras And Abelian Ideals Of N-Lie Algebras

The paper mainly concerns the n Lie algebras with abelian subalgebras and abelianideals over an algebraically closed feld of characteristic zero. Denote α(L) the dimensionof the maximal dimensional abelian subalgebras and β(L) of the dimensional of theabelian ideals of a n Lie algebra L. The relationships between α(L) and β(L) of generaln Lie algebras are discussed. It is proved that β(L)≤m-n+1for any m dimensionaln Lie algebras. Moreover, m-dimensional n-Lie algebras for β(L)=m-2, dimL~1=1,2are completely classifed. And the metric structures of Low dimensional3Lie algebrasare studied.The main cantent are as follows. Section1introduce the basic concepts of n Liealgebras. Section2study structures of the maximal dimensional abelian subalgebrasand the maximal dimensional abelian ideals of low dimensional n Lie algebras, and classifcations of n Lie algebras. Section3study structures of the maximal dimensionalabelian subalgebras and the maximal dimensional abelian ideals of3Lie algebras, andclassifcations of m-dimensional3Lie algebras. At last discuss the metric structures oflow dimensional3Lie algebras.