The Generating Index Of N-lie Algebras

The structural study of n-Lie algebras is an imprtant subject in n-Lie theory re-searching. Through several dozens years eforts, the basic results on n-Lie algebras areobtained, but there are lots of open problems. In this paper, structures of n-Lie algebrasare studied by means of generating index and eigenvalue number, and the classificationof the complex n-Lie algebras with the generating index n is given. A notion of eigen-value number of the complex n-Lie algebras is introduced, which is used to interpret thecomplex n-Lie algebras with the generating index n+1. In particular, classifications ofthe complex3-Lie algebras with the generating index4are discussed. Moreover, it isproved that the generating index of a metric3-Lie algebras which is neither simple nornilpotent is bigger than4.In this paper, all n-Lie algebras are finite-dimensional and over the complex field C.In section1, some basic concepts and operations of n-Lie algebras are given.In section2, the structure of m-dimensional n-Lie algebras with I(A)=n is studied.In section3, the eigenvalue number of n-Lie algebras with I(A)=n+1is studied.In section4, the structure of the3-Lie algebras with the generating index4isstudied, and classification results of eigenvalue number2and3are given.In section5, some properties of the generating index of metric3-Lie algebras arediscussed.