Structures Of Low Dimensional Nilpotent Lie Algebras

It is well know that nilpotent Lie algebras and solvable Lie algebras are two impor-tant classes in Lie algebras. Since structures of them provide scientific basis for studying Lie algebras, they are an imoprtant researching subjet all the while. Up to now, lots of problems on such two classes of Lie algebras are still open. The paper mainly studies 5 dimensional nilpotent Lie algebras. It provides the concrete expression of the cen-troid of ever class of 5 dimensional nilpotent Lie algebras and the structures of invariant symmetric bilinear forms on them. And it is proved that there exists only one class of quadratic nilpotent 5 dimensional Lie algebra.The paper consists of four sections. The back ground and development of nilpotent Lie algebras are introduced in Section 1. The section 2 gives some definitions and results used in the paper. The section 3 studies the structure of centroid of 5-dimensional nilpotent Lie algebras. The section 4 discusses the structure of invariant symmetrical bilinear forms on 5-dimensional nilpotent Lie algebras.