QUADRATIC LIE ALGEBRAS AND COMMUTATIVE ASSOCIATIVE ALGEBRAS

Abstract

This paper is concerned with the tensor product ⊗ R of a quadratic Lie algebra and a commutative associative algebra R, the dimension of the space of invariant symmetric bilinear forms on a quadratic Lie algebra and a way to classify the quadratic Lie algebras. When is quadratic, some conditions are obtained such that ⊗ R is also quadratic and an optimistic lower bound for the dimensions of space invariant symmetric bilinear forms on a quadratic Lie algebra is given by means of their centers and Levi factors. At last, we give a way to classify the irreducible quadratic Lie algebras and realize a class of them by the tensor products of quadratic Lie algebras and commutative associative algebras.

ACKNOWLEDGMENT

The first author would like to express his deep gratitude to Professor S.P. Wang for his guidance and help. He also thank Dr. F.H. Zhu for fruitful discussions. We also thank a referee for his or her valuable comments. This work is supported by (19971044) the National Science Foundation of China, (97005511) the Doctoral Programme Foundation of Institution of Higher Education and the Foundation of Jiangsu Educational Committee.