Metric Lie algebras and quadratic extensions

Abstract

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H2 Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.